#  Holography of Higher Codimension Submanifolds: Riemannian and Conformal

Samuel Blitz♭ & Josef Šilhan♯ ♭ Department of Mathematics and Statistics
Masaryk University
Building 08, Kotlářská 2
Brno, CZ 61137<EMAIL_ADDRESS>♯ Department of Mathematics and Statistics
Masaryk University
Building 08, Kotlářská 2
Brno, CZ 61137<EMAIL_ADDRESS>
###### Abstract.

We provide a natural generalization to submanifolds of the holographic method
used to extract higher-order local invariants of both Riemannian and conformal
embeddings, some of which depend on a choice of parallelization of the normal
bundle. Qualitatively new behavior is observed in the higher-codimension case,
giving rise to new invariants that obstruct the order-by-order construction of
unit defining maps. In the conformal setting, a novel invariant (that vanishes
in codimension 1) is realized as the leading transverse-order term appearing
in a holographically-constructed Willmore invariant. Using these same tools,
we also investigate the formal solutions to extension problems off of an
embedded submanifold.

Keywords: Riemannian geometry, conformal geometry, submanifold embeddings,
holography

###### 2010 Mathematics Subject Classification:

53C18, 53A55, 53C21, 58J32.

###### Contents

  1. 1 Acknowledgements
  2. 2 Introduction
    1. 2.1 Notations and Conventions
  3. 3 Orthonormal frames for normal bundles of Riemannian submanifold embeddings
    1. 3.1 Classical Submanifold Geometry
    2. 3.2 Rotation Minimizing Frames
    3. 3.3 Gauge-Fixed Frames
  4. 4 Defining Maps for Parallelized Riemannian Submanifold Embeddings
    1. 4.1 First Order
    2. 4.2 Second Order
    3. 4.3 Higher Orders
  5. 5 Conformal Geometry
  6. 6 Orthonormal Frames for Normal Bundles of Conformal Submanifold Embeddings
  7. 7 Defining Densities for Parallelized Conformal Submanifold Embeddings
    1. 7.1 First Order
    2. 7.2 Second Order
    3. 7.3 Third Order
  8. 8 A Holographic Submanifold Willmore Invariant
  9. 9 The Extension Problem
    1. 9.1 The Symmetric Extension Problem
    2. 9.2 The Restricted Extension Problem
    3. 9.3 The Conformal Extension Problem

##  1\. Acknowledgements

SB and JŠ are supported by the Czech Science Foundation (GACR) grant
GA22-00091S. SB is also supported by the Operational Programme Research
Development and Education Project No. CZ.02.01.01/00/22-010/0007541. SB and JŠ
would like to thank A. Rod Gover and Andrew Waldron for helpful comments and
insights.

##  2\. Introduction

The local geometry of embedded submanifolds $\Lambda^{d-k}$ in Riemannian
$d$-manifolds $(M^{d},g)$ has been studied extensively for over a century,
with special focus being given to curves (codimension $d-1$ submanifolds) and
hypersurfaces (codimension $1$ submanifolds). These cases stand out as
describing particularly useful systems: curves can describe the trajectory of
a particle in space and hypersurfaces can provide boundary data for elliptic
PDEs. On the other hand, conformal manifolds—smooth manifolds $M^{d}$ endowed
with an equivalence class of Riemannian metrics $\bm{c}:=\\{\Omega^{2}g\\}$
for every $\Omega\in C^{\infty}_{+}M$—have also been studied since the early
20th century. Early investigations include work by Cartan [7], Thomas [31],
and Fialkow [9], among others. In particular, conformal invariants—those
quantities which are equivariant under the rescaling group action—are central
to any description of conformal geometry. In the last thirty years, a method
of Thomas [3] has been rediscovered that allows for the construction of such
invariants in the same way that the Ricci calculus allows for the construction
of invariants of a Riemannian manifold: this is called the tractor calculus,
which will be discussed in more detail in Section 5. A confluence of
submanifold geometry and conformal geometry has gained popularity in the last
few decades [19, 36, 16, 12, 27, 29, 28], having been driven in part by
physics applications [25, 23]. Specifically, conformal hypersurface geometry
has taken center stage, with a focus on an extrinsic characterization of the
embedding $\Sigma^{d-1}\hookrightarrow(M^{d},\bm{c})$ itself [17, 18]. Recent
work conducted by Curry, Gover, and Snell [8] also investigated conformal
submanifold geometry using tractor methods, however their approach differed
significantly from the methods applied here.

One popular method to study any submanifold embedding structure, whether
Riemannian, conformal, or otherwise, is called holography. Given a submanifold
embedding into some geometric structure
$\Lambda\hookrightarrow(M,\mathcal{S})$, one can study the invariants of the
embedding by studying solutions to geometrically-natural PDEs relating data on
$\Lambda$ to data on $M$. In the study of Riemannian hypersurface embeddings
$\Sigma\hookrightarrow(M,g)$, this method uses the existence of a unique unit
defining function for $\Sigma$, i.e. a function $s\in C^{\infty}M$ such that
$\Sigma=\\{p\in M|s(p)=0\\}$ and that $|ds|^{2}_{g}=1$. It is this second
condition—the eikonal equation—that distinguishes this function from any other
defining function, which need only have that $|ds|^{2}_{g}>0$. As $s$ is
uniquely determined by the embedding, its jets can be used to construct
invariants of the embedding structure.

In the case of separable conformal hypersurface embeddings
$\Sigma\hookrightarrow(M,\bm{c})$, a similar method bears fruit. In that case,
one seeks a double equivalence class $\sigma=[g;s]=[\Omega^{2}g;\Omega s]$ of
metrics $g\in\bm{c}$ and functions $s\in C^{\infty}M$ such that for any
representative $[g;s]$, we have that on $M/\Sigma$, the metric
$g^{o}:=g/s^{2}$ has constant scalar curvature. (More detail is provided on
these double equivalence classes in Section 5.) Finding such a double
equivalence class solves the so-called “singular Yamabe problem,” which can be
shown [24] to be equivalent to a conformal eikonal equation, given by

$|ds|^{2}_{g}-\tfrac{2s}{d}(\Delta^{g}s+\tfrac{1}{2(d-1)}Sc^{g}s)=1\,.$

While a one-sided global solution to this problem always exists [1, 2, 26], it
turns out that one cannot (in general) solve this to arbitrarily high order
[1, 11]. Nonetheless, one may still find a unique unit defining density up to
order $d+1$ in $\sigma$ that is entirely determined by the embedding and the
conformal structure. As it is unique, its jets below this order will therefore
characterize the conformal embedding. A study of these jets was carried out in
[4].

The goal of this article is to investigate the holographic method of unit
defining functions as it applies to higher-codimension submanifold embeddings
in both the Riemannian and conformal settings. In general, we find that this
method does not yield true submanifold invariants, but rather invariants that
depend on a choice of parallelization of the normal bundle on $\Lambda$. In
the following subsection, we introduce notations and conventions that will be
used throughout this article. In Section 3, we first discuss when and how a
canonical frame for the normal bundle of a Riemannian submanifold embedding
can be constructed. In Section 4, we proceed with the holographic program in
the Riemannian setting. In Section 5, we give a brief review of conformal
geometry, before considering the frame construction in a conformal submanifold
embedding in Section 6. In Section 7 we investigate the use of the holographic
method to conformal submanifold embeddings. While we find that we cannot
proceed to particularly high order, we are still able to construct an
interesting parallelization-dependent conformal submanifold invariant in the
special case of a conformally-embedded surface. This is detailed in Section 8.
Finally, in Section 9 we examine the submanifold extension problems, which are
a natural problems to study given a holographically-constructed defining map.

###  2.1. Notations and Conventions

In this article, $M$ denotes a $d$-dimensional smooth manifold endowed with a
metric tensor $g$ or conformal class of metrics $\bm{c}$, both of which will
be positive definite. Furthermore, $\Lambda$ denotes a $d-k$-dimensional
smooth submanifold of $M$, and $0<k<d$. Throughout the entirety of this
article, we work only locally and so will assume that trivializations are
always available for any section of any bundle. In the Riemannian setting
$(M,g)$, we denote the Levi-Civita connection by $\nabla$. Our convention for
the curvature tensor is given by

$R(x,y)z=(\nabla_{x}\nabla_{y}-\nabla_{y}\nabla_{x})z-\nabla_{[x,y]}z\,.$

Here, $x,y,z$ are vector fields on $M$ and $[x,y]$ is the Lie bracket of $x$
and $y$. In general, for any bundle $\mathcal{V}M$ over $M$, we will denote
sections of this bundle by $\Gamma(\mathcal{V}M)$. That is,
$x,y,z\in\Gamma(TM)$. When necessary, we will refer to arbitrary tensor
products of a bundle (and its dual) using capital Greek letter, so that, for
example, $T^{\Phi}M$ is some unspecified tensor product of $TM$ and $T^{*}M$.

For ease of calculation, we will use Penrose’s abstract index notation,
labelling such indices with letters from the beginning of the Latin alphabet
$a$, $b$, $c$, etc., so that, for example, the Riemann curvature tensor may be
expressed as

$R_{ab}{}^{c}{}_{d}x^{d}=[\nabla_{a},\nabla_{b}]x^{c}\,.$

Using the metric and its inverse, indices can be raised and lowered, as usual;
the Einstein summation convention is used to indicate traces. (Sometimes, when
the notation is unambiguous, we will use dot product notation to represent
contraction; for example, when $\omega$ is some 1-form, we will sometimes
write $\nabla^{a}\omega_{a}=:\nabla\cdot\omega$.) In this language, the we
denote by $Ric_{ab}:=R_{ca}{}^{c}{}_{b}$ and its corresponding scalar
$Sc:=Ric_{a}^{a}$. When $d>2$, a particular trace-correction of the Ricci
tensor, known as the Schouten tensor, arises frequently in conformal
considerations:

$P_{ab}:=\tfrac{1}{d-2}\left(Ric_{ab}-\tfrac{1}{2(d-1)}Sc\,g_{ab}\right)\,.$

The trace of this tensor is denoted by $J:=P_{a}^{a}$. Written in terms of the
Riemann tensor and the Schouten tensor, we may express the Weyl tensor as

$W_{abcd}=R_{abcd}-g_{ac}P_{bd}+g_{bc}P_{ad}+g_{ad}P_{bc}-g_{bd}P_{ac}\,.$

In dimensions $d>2$, the Cotton tensor is given by the covariant curl of the
Schouten tensor,

$C_{abc}=\nabla_{a}P_{bc}-\nabla_{b}P_{ac}\,.$

When $d>3$, this can be reexpressed in terms of the divergence of the Weyl
tensor:

$C_{abc}=\tfrac{1}{d-3}\nabla^{d}W_{dcab}\,.$

In the context of submanifold embeddings, we will use Greek indices to
represent tensors on normal frame bundles. Objects that are along the
submanifold will be denoted by overbars $\bar{\bullet}$, so that, for example,
$\bar{\nabla}$ represents the induced Levi-Civita connection on
$(\Lambda,\bar{g})$, where $\bar{g}$ is the induced metric from
$\Lambda\hookrightarrow(M,g)$. In general for any index notation, we will use
round brackets $(\cdots)$ to indicate unit-normalized symmetrization and
square brackets $[\cdots]$ to indicate unit-normalized antisymmetrization.
Furthermore, we will use a subscript $\circ$ to indicate the tracefree part of
a particular symmetric index configuration. However, because Latin and Greek
indices represent different types of objects, we may sometimes write $(\alpha
ab\beta)$ to specifically mean the symmetrization over $\alpha$ and $\beta$,
and similarly for antisymmetrization. To be explicit, whichever type of index
is closest to the opening and closing brackets is the type of index that is
(anti)symmetrized over. On the other hand, when excluding an index from
(anti)symmetrization is necessary, we will use vertical bars, so that
$(\alpha|\beta|\gamma)$ refers to the symmetrization over $\alpha$ and
$\gamma$.

##  3\. Orthonormal frames for normal bundles of Riemannian submanifold
embeddings

We begin with the case of smooth Riemannian submanifold embeddings
$\iota:\Lambda^{d-k}\hookrightarrow(M,g)$ with $d\geqslant 3$. For notational
convenience, we will often view $\Lambda$ as a subset of $M$. We now review
classical submanifold geometry.

###  3.1. Classical Submanifold Geometry

At a point $p\in\Lambda$, every tangent space of the bulk space along
$\Lambda$ can be decomposed according to

(3.1) $\displaystyle T_{p}M=T_{p}\Lambda\oplus N_{p}\Lambda\,,$

where $N_{p}$ is the normal space at $p\in\Lambda$ and
$N\Lambda:=\sqcup_{p\in\Lambda}N_{p}\Lambda$ is the normal bundle over
$\Lambda$.

As the pullback map is well-defined, we can construct the intrinsic metric,
also sometimes called the first fundamental form, via the pullback:
$\bar{g}:=\iota^{*}g$. Note that this metric is compatible with the
decomposition above, and given an orthornormal frame $\\{n_{\alpha}\\}$ for
$N\Lambda$, we may express this invariant in index notation according to
$\bar{g}_{ab}=g_{ab}|_{\Lambda}-n_{a\alpha}n_{b\alpha}$. Closely related to
the first fundamental form is the projection operator given by
$\bar{g}^{a}_{b}:=\delta^{a}_{b}-n^{a}_{\alpha}n_{b\alpha}$.

With a metric in hand, one may construct the Levi-Civita connection on
$\Lambda$, denoted $\bar{\nabla}$. A classical result of submanifold geometry
is that this connection is directly obtainable from the Levi-Civita connection
$M$ and the projection operator:

$\bar{\nabla}_{\bar{v}}\bar{u}^{a}=\bar{g}^{a}_{b}\nabla_{\iota_{*}\bar{v}}(\iota_{*}\bar{u})^{b}\,,$

where $\bar{v},\bar{u}\in\Gamma(T\Lambda)$ and $\iota_{*}\bar{v}$,
$\iota_{*}\bar{u}$ are their respective pushforwards. Note that we will often
write $\nabla^{\top}$ for the pullback of the bulk Levi-Civita connection—this
way, we may use abstract index notation and we do not need to carry around
arbitrary vector fields. In a similar way, we use $\top$ both as an operator
and as a superscript to denote the projection to the submanifold, e.g. we will
write $P^{\top}$ to represent the projection of the Schouten tensor of $(M,g)$
to $\Lambda$.

The normal component of the right-hand side of the above display also yields a
classical invariant: the second fundamental form. Indeed, relative to an
orthonormal frame on $N\Lambda$, we define

$\bar{u}^{a}\bar{v}^{b}{\rm
II}_{ab\alpha}:=-n_{a\alpha}\nabla_{\iota_{*}\bar{v}}(\iota_{*}\bar{u})^{a}\,.$

A short calculation shows that this can be expressed in a simpler way
according to

${\rm
II}_{ab\alpha}=\bar{g}_{bc}\nabla^{\top}_{a}n^{c}_{\alpha}\in\Gamma(\odot^{2}T^{*}\Lambda)\,.$

Accordingly, one may express the Levi-Civita connection of $M$ restricted to
$\Lambda$ via the following identity:

$\nabla_{\iota_{*}\bar{v}}(\iota_{*}\bar{u})=\bar{\nabla}_{\bar{v}}\bar{u}-{\rm
II}_{\alpha}(\bar{u},\bar{v})n_{\alpha}\,.$

The first and second fundamental forms are sufficient to completely describe
an embedded hypersurface in $\mathbb{E}^{d}$ up to Euclidean motions. This is
because, for $k=1$, the normal bundle is one-dimensional and hence (up to
orientation) each normal space has a canonical basis given by $n\in
N_{p}\Lambda$ where $g(n,n)=1$. (Note that as we work locally for the entirety
of this article, we will neglect any issues that arise from orientations.)
This rigidity effectively removes all geometric structure from the normal
bundle. Unfortunately, this rigidity is lost entirely when we consider higher-
codimension submanifolds. Indeed, while orthonormality can certainly be
retained, there is (a priori) no uniquely preferred basis for $N\Lambda$, even
locally. In what follows, we will find that there is sometimes a preferred
basis up to constant $SO(k)$ rotations—but in general, it is not that clear
that any preferred basis can be constructed. We will now provide a
construction for the final piece of data required to describe embedded
submanifolds up to Euclidean motions and also provide notation and language
that will be necessary for later considerations.

Just as one can induce a Levi-Civita connection on the tangent bundle to
$\Lambda$, one may also induce on the normal bundle a connection

$D:\Gamma(N\Lambda)\rightarrow\Gamma(T^{*}\Lambda\otimes N\Lambda)$

given by

$D_{\bar{v}}n=\perp(\nabla_{\iota_{*}\bar{v}}\xi)\,,$

where $\perp$ is the orthogonal projection to $N\Lambda$,
$\bar{v}\in\Gamma(T\Lambda)$ is any vector field along $\Lambda$ and
$\xi\in\Gamma(N\Lambda)$ is any normal vector field. Note that this connection
is compatible with the metric on the normal bundle given by
$\eta=g|_{\Lambda}-\iota^{*}g$. Then, for any frame $\\{n_{\alpha}\\}$ for
$N\Lambda$, the connection coefficients are given by the relationship

$D_{a}n^{b}_{\beta}=\omega_{\gamma a\beta}n^{b}_{\gamma}\,.$

Given any orthonormal frame $\\{n_{\alpha}\\}_{\alpha=1}^{k}$ for the normal
bundle $N\Lambda$, $\omega$ is given precisely by the so-called normal
fundamental forms [30] (sometimes called the Hájíček one-form in the physics
literature [21, 22]):

$\beta_{a\alpha\beta}=n^{b}_{\alpha}\nabla^{\top}_{a}n_{b\beta}\,.$

Note that in general, $\beta$ depends on the choice of orthonormal frame.

Associated to the normal bundle is the orthonormal frame bundle $F(N\Lambda)$,
which is a principal $O(k)$-bundle. So in particular, any two orthonormal
frame fields are related to one another by the action of a section $m\in
C^{\infty}(\Lambda,O(k))$. Furthermore, the connection coefficients (and thus
the normal fundamental forms) transform according to

(3.2) $\displaystyle\beta\mapsto m\beta m^{-1}+md(m^{-1})\,,$

where adjacency is understood to imply matrix multiplication. Note that for
constant rotations, i.e. when $m$ is a constant matrix, $\beta$ transforms
tensorially.

Because $\beta$ is not covariant under the group action, it is not a true
invariant of the normal bundle (in that it does not transform linearly under
the group action). Rather, its curvature 2-form

(3.3) $\displaystyle\mathcal{R}=d\beta+[\beta,\beta]$

does transform linearly under the group action and hence is a true invariant
of the normal bundle. In terms of the connection itself, the curvature 2-form,
hereafter known as the normal curvature, obeys the standard relation:

$\mathcal{R}(\bar{u},\bar{v})\xi=D_{\bar{u}}(D_{\bar{v}}\xi)-D_{\bar{v}}(D_{\bar{u}}\xi)-D_{[\bar{u},\bar{v}]}\xi\,,$

for any $\bar{u},\bar{v}\in\Gamma(T\Lambda)$ and $\xi\in\Gamma(N\Lambda)$. The
normal curvature is the final ingredient necessary to completely specify an
embedded submanifold in $\mathbb{E}^{d}$ up to Euclidean motions.

To summarize, the three classical invariants of submanifold embeddings are
$\bar{g}$, ${\rm II}$, and $\mathcal{R}$. It is these tensors that appear in
classical submanifold identities as described by Spivak [30], and given below
for later use. First is the Gauss equation, relating the tangential component
of the Riemann curvature tensor to that of the submanifold $\Lambda$:

(3.4) $\displaystyle R_{abcd}^{\top}=\bar{R}_{abcd}+{\rm II}_{ad\alpha}{\rm
II}_{bc\alpha}-{\rm II}_{bd\alpha}{\rm II}_{ac\alpha}\,,$

where $R^{\top}:=\top(R)$, the projection of the Riemann curvature tensor to
the submanifold. Similar notation will be used throughout. Next is the
Codazzi–Mainardi equation, relating another component of the Riemann curvature
tensor to extrinsic invariants:

(3.5) $\displaystyle
R_{abcn_{\alpha}}^{\top}\stackrel{{\scriptstyle\Lambda}}{{=}}2\bar{\nabla}_{[a}{\rm
II}_{b]c\alpha}+2{\rm II}_{c[b\beta}\beta_{a]\alpha\beta}\,.$

Note that here, the right-hand side does not look explicitly
tensorial—however, a short calculation shows that it is indeed independent of
the choice of orthonormal basis $\\{n_{\alpha}\\}$. Finally, we have the Ricci
equation (which is trivial for hypersurfaces), relating another projection of
the Riemann curvature to the normal curvature and the second fundamental form:

$R_{ab\alpha\beta}^{\top}=\mathcal{R}_{ab\alpha\beta}+{\rm
II}^{c}_{a\beta}{\rm II}_{cb\alpha}-{\rm II}^{c}_{b\beta}{\rm
II}_{ca\alpha}\,.$

As noted above, all of these equations are covariant under the action of a
section $m\in C^{\infty}(\Lambda,O(k))$. It would be desireable, in principle,
to promote the normal fundamental forms to submanifold invariants so that
covariance would be more explicitly seen. However, as $\beta$ does not
transform covariantly under non-constant sections $m$, one might wish to
restrict the structure group just to those constant sections by employing a
geometric constraint on the normal frame bundle—a procedure known in the
physics literature as “gauge-fixing.” Alas, in general, there is no preferred
frame for the normal bundle fixed by a differential constraint. This is known
as the Gribov ambiguity [20]. Furthermore, even global minimizing conditions
of energy functionals depending on the normal connection are not necessarily
unique [33]. To our knowledge, the problem of finding a geometrically
meaningful canonical frame for the normal bundle (up to constant rotations) is
still an open question when the embedding is sufficiently generic. However, in
certain situations, canonical frames can be identified, at least up to global
gauge transformations.

###  3.2. Rotation Minimizing Frames

In the simplest case, where the normal curvature vanishes, it is easy to show
that a canonical frame (up to constant rotations) for the normal bundle can be
constructed. We capture this result in the following proposition.

######  Proposition 3.1.

Let $\Lambda^{d-k}\hookrightarrow(M,g)$ be a smooth submanifold embedding with
$1\leqslant k\leqslant d-1$. Then the normal curvature $\mathcal{R}$ vanishes
if and only if there exists in a neighborhood around every point $p\in\Lambda$
an orthonormal frame for the normal bundle $N\Lambda$ satisfying $\beta=0$.
Furthermore, this orthonormal frame is unique (up to constants in
$C^{\infty}(\Lambda,O(k))$).

###### Proof.

It follows trivially from Equation (3.3) that if there exists an orthonormal
frame such that $\beta=0$, then $\mathcal{R}=0$. Now suppose that
$\mathcal{R}=0$. We wish to show that there exists a frame such that
$\beta=0$. This follows from a standard result: namely, if the bundle
$N\Lambda\rightarrow\Lambda$ has a connection $D$ with vanishing curvature,
then there always exists locally a frame $\\{n_{\alpha}\\}$ consisting of
parallel sections, namely $Dn_{\alpha}=0$. But then by the Gram-Schmidt
procedure, it follows that (along $\Lambda$) there exists an orthonormal frame
$\\{n_{\alpha}\\}$ consisting of parallel sections, and so we have that for
that choice of frame, $\beta=0$.

Now suppose $\mathcal{R}=0$ and that two frames $\\{n_{\alpha}\\}$ and
$\\{n^{\prime}_{\alpha}\\}$ have vanishing normal fundamental forms. Because
they are both orthonormal frames, there exists a (possibly non-constant) $m\in
C^{\infty}(\Lambda,O(k))$ such that
$n^{\prime}_{\alpha}=m_{\alpha\beta}n_{\beta}$. It then follows from Equation
(3.2) that

$0=m_{\alpha\gamma}\nabla_{a}m_{\beta\gamma}\,.$

But clearly this identity only holds if $\nabla_{a}m_{\beta\gamma}=0$. Thus we
have that $m$ is a constant section of $O(k)\rightarrow\Lambda$. ∎

It is thus useful to think of this canonical orthonormal frame as, in some
sense, “maximally parallel.” From a geometrical perspective, the associated
$\beta$ for a given orthonormal frame $\\{n_{\alpha}\\}$ can be thought of as
a generalization of the torsion of a spacecurve, insofar as torsion describes
how one normal vector rotates into another normal vector while moving along a
spacecurve. In fact, for $\Lambda^{1}\hookrightarrow\mathbb{R}^{3}$, it is
straightforward to show that in the Frenet frame $\\{T,N,B\\}$ with torsion
$\tau$, we have that

(3.6) $\displaystyle\beta_{aBN}=\tau\,.$

It is because of this analogy that those orthonormal frames for the normal
bundle that fix $\beta=0$ are sometimes called rotation minimizing [34]. We
provide a formal definition below:

######  Definition 3.2.

Let $\Lambda^{d-k}\hookrightarrow(M,g)$ be a Riemannian submanifold embedding
and let $\\{n_{\alpha}\\}$ be an orthonormal frame for $N\Lambda$. Then, we
say that $\\{n_{\alpha}\\}$ is a rotation minimizing frame when, for each
$\alpha\in\\{1,\ldots,k\\}$ and $\bar{v}\in\Gamma(T\Lambda)$, we have that

$\nabla_{\iota_{*}\bar{v}}n_{\alpha}\in\Gamma(T\Lambda)\,.$

By construction, rotation minimizing frames are precisely the canonical frames
described in Proposition 3.1, as captured in the following lemma:

######  Lemma 3.3.

Let $\Lambda^{d-k}\hookrightarrow(M,g)$ have a rotation minimizing frame
$\\{n_{\alpha}\\}$. Then, modulo the action of constant sections of
$O(k)\rightarrow\Lambda$, this frame is the canonical frame described in
Proposition 3.1.

###### Proof.

Let $\\{n_{\alpha}\\}$ be a rotation minimizing frame for $N\Lambda$. From the
definition, we have that for any $\bar{v}\in\Gamma(T\Lambda)$,
$\nabla_{\iota_{*}\bar{v}}n_{\alpha}\in\Gamma(T\Lambda)$. But by the
orthogonal decomposition $TM|_{\Lambda}=T\Lambda\oplus N\Lambda$, it follows
that $n_{a\beta}\nabla_{\iota_{*}\bar{v}}n^{a}_{\alpha}=0$. But this implies
that $\bar{v}^{a}\beta_{a\beta\alpha}=0$ for all
$\alpha,\beta\in\\{1,\ldots,k\\}$ and any choice of $\bar{v}$. This is
precisely the frame described in Proposition 3.1. ∎

We thus have the following result for spacecurves:

######  Corollary 3.4.

Let $\Lambda^{1}\hookrightarrow(M^{d},g)$ be a curve embedded in a Riemannian
manifold. Then, there exists a unique (up to constant rotations) rotation
minimizing frame for $N\Lambda$.

###### Proof.

Because $\Lambda$ is 1-dimensional and the normal curvature is a 2-form, we
have that $\mathcal{R}$ necessarily vanishes. Hence from Proposition 3.1 and
Lemma 3.3, there exists a unique rotation minimizing frame. ∎

###  3.3. Gauge-Fixed Frames

When the normal curvature is non-vanishing, rotation minimizing frames do not
exist. However, that does not necessarily preclude the existence of a
preferred frame determined by some condition, differential or otherwise.

We begin by considering the simplest non-trivial case, where $k=2$. In that
setting, we can directly parametrize sections of $SO(2)\rightarrow\Lambda$ by
a single function $\theta\in C^{\infty}S^{1}$ so that if $m\in
C^{\infty}(\Lambda,SO(2))$, then we have that

$m_{\alpha\beta}=\cos\theta\delta_{\alpha\beta}-\sin\theta\epsilon_{\alpha\beta}\,.$

Given a orthonormal frame $\\{n_{\alpha}\\}$, the normal fundamental form is
specified entirely by a single 1-form, $\beta_{a}$. Under the group action, we
find that this function transforms according to

$\beta_{a}\mapsto\beta_{a}+\bar{\nabla}_{a}\theta\,.$

Because this gauge transformation is sufficiently simple, we may be able to
find a unique solution to some differential gauge-fixing condition. Consider
the Coulomb gauge $\bar{\nabla}\cdot\beta=0$. Fixing this gauge is equivalent
to finding $\theta$ such that

$\bar{\Delta}\theta=-\bar{\nabla}^{a}\beta_{a}\,.$

In general, any global solution to this equation will require a global frame
and hence requires that $F(N\Lambda)$ is trivial. In that case, if $\Lambda$
is compact and $\int_{\Lambda}\bar{\nabla}\cdot\beta=0$, then there exists
$\theta$ that solves the Poisson equation globally and allows the construction
of a preferred frame (up to a constant) $\\{m_{\alpha\beta}n^{a}_{\beta}\\}$.
Trivially, then, if $\Lambda$ is closed such a preferred frame exists.

More generally, for $k\geqslant 2$, Uhlenbeck showed [32] that when the normal
curvature is sufficiently small in a trivializing neighborhood
$U\subset\Lambda$ around a point $p$, there exists a gauge transformation
$m\in C^{\infty}(U,O(k))$ such that the basis
$\\{m_{\alpha\beta}n^{a}_{\beta}\\}$ has normal fundamental forms that satisfy
$\bar{\nabla}\cdot\beta_{\alpha\beta}=0$ and $u\cdot\beta=0$ on $\partial U$,
where $u$ is the normal to the boundary $\partial U$.

While the existence of a “clean” gauge choice can sometimes be taken for
granted, uniqueness is much harder to demonstrate—and indeed, it is not always
present (see the discussion about the Gribov ambiguity above). Regardless, we
can still study the extrinsic geometry of submanifold embeddings equipped with
an orthonormal basis for the normal bundle. When a global orthonormal frame
exists for $N\Lambda$, it is called a parallelization of $N\Lambda$. However,
when the normal bundle is non-trivial, no such parallelization exists.
Nonetheless, as the results of this article are entirely local, we may without
loss of generality assume that the normal bundle is trivial going forward. To
that end, for the remainder of this article, we assume that an orthonormal
frame is specified for $N\Lambda$. We call such an embedding equipped with an
orthonormal frame for the normal bundle a parallelized (Riemannian)
submanifold embedding, specified by the pair
$(\Lambda\hookrightarrow(M,g),\\{n_{\alpha}\\})$.

##  4\. Defining Maps for Parallelized Riemannian Submanifold Embeddings

Given a Riemannian submanifold embedding
$\iota:\Lambda^{d-k}\hookrightarrow(M^{d},g)$, we are interested in studying
the embedding by examining the jets of a geometrically-adapted defining map
$s_{\alpha}:M\rightarrow\mathbb{R}^{k}$, which is a $k$-tuple of functions
such that for any $p\in\Lambda$, we have that $s_{\alpha}(p)=0$ and
$ds_{\alpha}(p)\neq 0$ for every $\alpha\in\\{1,\ldots,k\\}$. That such
families exist is well-known, at least locally: for every $p\in\Lambda$, there
exists a neighborhood $U\subset M$ such that $U\cap\Lambda$ is the zero locus
of some defining map $s_{\alpha}:U\rightarrow\mathbb{R}^{k}$. However, there
are many such defining maps, and in order for the jets of such a defining map
to contain information about the embedding $\iota$, it must be determined
canonically by the geometry. To realize this canonical defining map, we
consider the Gram matrix $G_{\alpha\beta}:=g^{-1}(ds_{\alpha},ds_{\beta})$ of
a given defining map $s_{\alpha}$. In the case of hypersurfaces, we
canonically choose $G=\operatorname{Id}$, which is equivalent to the condition
that $|ds|^{2}=1$. Similarly, in the case of a higher-codimension submanifold
we would like to achieve $G=\operatorname{Id}$. Because we will work locally,
it is sufficient for our purposes to find a defining map $s_{\alpha}$ such
that $G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{\infty})$, meaning
we wish to find a defining map such that for any $m\in\mathbb{Z}_{\geqslant
1}$, we have that $G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{m})$.
We call such a defining map a unit defining map. Going forward, we will abuse
notation by also denoting $g^{-1}(ds_{\alpha},\cdot)$ by $n_{\alpha}$. Note
that, when clarification is necessary, we will decorate $n_{\alpha}$ with
abstract indices (such as $n^{a}_{\alpha}$ and $n_{a\alpha}$) to indicate the
whether we are referring to the vector or its metric dual.

###  4.1. First Order

Given the parallelization $\\{n_{\alpha}\\}$ of $N\Lambda$ and any defining
map $\hat{s}_{\alpha}$ for $\Lambda$, there exists a linear combination of
components of $\hat{s}_{\alpha}$ denoted by $s_{\alpha}$ such that that
$g^{-1}(ds_{\alpha},\cdot)\stackrel{{\scriptstyle\Lambda}}{{=}}n_{\alpha}$.
This linear combination provides a unique $s_{\alpha}$ modulo terms that are
at least second order in a defining function. The behavior of $ds_{\alpha}$
away from $\Lambda$, however, is harder to control. Our goal is to obtain a
canonical family of unit defining functions adapted to our parallelization
satisfying $g^{-1}(ds_{\alpha},ds_{\beta})=\delta_{\alpha\beta}$. However,
this will not be possible in general. Instead, we will choose $s_{\alpha}$ to,
in some sense, get as close to the identity Gram matrix condition as possible.
We begin with an iterative procedure. Because the parallelization of
$N\Lambda$ is defined to be orthonormal, we can certainly obtain
$G_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}\delta_{\alpha\beta}$, so
we assume this going forward. This constraint defines a unique family of
defining maps that are aligned to the parallelization of $N\Lambda$ and whose
elements have a difference given by
$s^{\prime}_{\alpha}-s_{\alpha}=\mathcal{O}(s^{2})$. For any of these defining
maps, it thus follows that

(4.1) $\displaystyle
G_{\alpha\beta}=\delta_{\alpha\beta}+F^{(1)}_{\alpha\beta\gamma}s_{\gamma}\,,$

where $F^{(1)}_{\alpha\beta\gamma}$ represents an array of smooth functions on
$M$. Our goal will be to pare down this family of defining maps by controlling
$F^{(1)}$.

However, note that Equation (4.1) does not uniquely specify
$F^{(1)}_{\alpha\beta\gamma}$ as an array of functions on $C^{\infty}M$. To
see this, consider the mapping

$F^{(1)}_{\alpha\beta\gamma}\mapsto
F^{(1)}_{\alpha\beta\gamma}+F^{(1,1)}_{\alpha\beta\gamma\delta}s_{\delta}\,,$

where $F^{(1,1)}$ is any array of functions so that
$F^{(1,1)}_{\alpha\beta\gamma\delta}=F^{(1,1)}_{\alpha\beta[\gamma\delta]}$.
This mapping clearly preserves Equation (4.1), and so we see that $F^{(1)}$ is
only uniquely determined along $\Lambda$. Nonetheless, we may choose $F^{(1)}$
in a collar neighborhood of $\Lambda$ in a non-canonical way. Going forward,
we will assume such a choice has been made; later, we will show that this
choice actually yields a unique result.

Because of the symmetry of $G_{\alpha\beta}$, we have that
$F^{(1)}_{\alpha\beta\gamma}=F^{(1)}_{(\alpha\beta)\gamma}$, and thus as a
representation of $S_{k}$, we can see that the tensor structure of $F^{(1)}$
in the normal bundle takes the form

$\raisebox{3.0pt}{\scalebox{0.4}{\ydiagram{2,1}}}\;\oplus\;\scalebox{0.4}{\ydiagram{3}}\,.$

For what follows, we will abuse notation and write, for example, that
$F^{(1)}\in
C^{\infty}(M,\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,1}}}\;\oplus\;\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{3}}})$.

Now observe that under the mapping

$s_{\alpha}\mapsto
s_{\alpha}+A^{(1)}_{\alpha\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}\,,$

where $A^{(1)}$ is a list of smooth functions on $M$, we have that the zeroth
order part of the Gram matrix condition given in Equation (4.1) is preserved
while changing $F^{(1)}$. Indeed, by inspection we may consider only those
$A^{(1)}$ satisfying
$A^{(1)}_{\alpha\gamma_{1}\gamma_{2}}=A^{(1)}_{\alpha(\gamma_{1}\gamma_{2})}$,
and so $A^{(1)}\in
C^{\infty}(M,\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,1}}}\;\oplus\;\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{3}}})$.
Evidently, it may be possible to choose $A^{(1)}$ so that it cancels
$F^{(1)}$.

Now, we can evaluate the new Gram matrix with improved defining map:

(4.2)
$\displaystyle\begin{split}G_{\alpha\beta}=\delta_{\alpha\beta}&+(F^{(1)}_{\alpha\beta\gamma}+4A^{(1)}_{(\alpha\beta)\gamma})s_{\gamma}\\\
&+\left(4A^{(1)}_{\alpha(\rho\gamma_{1})}A^{(1)}_{\beta(\rho\gamma_{2})}+2n_{a(\alpha}\nabla^{a}A^{(1)}_{\beta)(\gamma_{1}\gamma_{2})}+4A^{(1)}_{\alpha(\rho\gamma_{1})}F^{(1)}_{\beta\rho\gamma_{2}}\right)s_{\gamma_{1}}s_{\gamma_{2}}\\\
&+\left(4A^{(1)}_{\alpha(\rho\gamma_{1})}\nabla_{n_{\rho}}A^{(1)}_{\beta\gamma_{2}\gamma_{3}}+4A^{(1)}_{\alpha(\omega_{1}\gamma_{2})}A^{(1)}_{\beta(\omega_{2}\gamma_{2})}F^{(1)}_{\omega_{1}\omega_{2}\gamma_{3}}\right)s_{\gamma_{1}}s_{\gamma_{2}}s_{\gamma_{3}}\\\
&+\left(\nabla^{a}A^{(1)}_{\alpha(\gamma_{1}\gamma_{2})}\right)\left(\nabla_{a}A^{(1)}_{\beta(\gamma_{3}\gamma_{4})}\right)s_{\gamma_{1}}s_{\gamma_{2}}s_{\gamma_{3}}s_{\gamma_{4}}\,,\end{split}$

where symmetry in $\alpha$ and $\beta$ is sometimes implied. Observe that in
order to fix $A^{(1)}$, we must express it in terms of $F^{(1)}$. To that end,
we fix an ansatz:
$A^{(1)}_{\alpha\beta\gamma}=aF^{(1)}_{\alpha\beta\gamma}+bF^{(1)}_{\gamma\alpha\beta}+cF^{(1)}_{\beta\gamma\alpha}$
and then solve explicitly; these are the only three terms possible as
$F^{(1)}_{\alpha\beta\gamma}=F^{(1)}_{(\alpha\beta)\gamma}$. We find that

$A^{(1)}_{\alpha\beta\gamma}=-\tfrac{1}{4}(F^{(1)}_{\alpha\beta\gamma}+F^{(1)}_{\gamma\alpha\beta}-F^{(1)}_{\beta\gamma\alpha})$

sets the first order term in Equation 4.2 to zero identically, and thus we
have found $s_{\alpha}$ such that

$G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{2})\,,$

fixing $s_{\alpha}$ to second order in a way dependent on the original choice
of extension of $F^{(1)}$.

fix $F^{(1)}$ terms appearing in higher order terms and check the +4 term in
first order term

With this result, we find a useful identity for the normal fundamental form
$\beta$ associated with the parallelization of $N\Lambda$:

$\displaystyle
n^{b}_{\alpha}\nabla_{a}n_{b\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle
n^{b}_{[\alpha}\nabla_{a}n_{b\beta]}+n^{b}_{(\alpha}\nabla_{a}n_{b\beta)}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle
n^{b}_{[\alpha}(\nabla_{a}^{\top}+n_{a\gamma}n^{c}_{\gamma}\nabla_{c})n_{b\beta]}+\tfrac{1}{2}\nabla_{a}n^{b}_{(\alpha}n_{b\beta)}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle\beta_{a\alpha\beta}+n_{a\gamma}n^{c}_{\gamma}n^{b}_{[\alpha}\nabla_{c}n_{b\beta]}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle\beta_{a\alpha\beta}+n_{a\gamma}n^{c}_{\gamma}n^{b}_{[\alpha}\nabla_{b}n_{c\beta]}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle\beta_{a\alpha\beta}-n_{a\gamma}n^{b}_{[\alpha}n^{c}_{\beta]}\nabla_{b}n_{c\gamma}$
(4.3) $\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle\beta_{a\alpha\beta}\,.$

In the above computation, we used that when $n_{\alpha}=ds_{\alpha}$, we
evidently have that $\nabla_{[a}n_{b]\beta}=0$. Observe from this calculation
that
$[n_{\alpha},n_{\beta}]^{a}\stackrel{{\scriptstyle\Lambda}}{{=}}2\beta^{a}_{\alpha\beta}$.
Further, it thus follows that

(4.4)
$\displaystyle\nabla_{a}n_{b\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}{\rm
II}_{ab\beta}+n_{b\alpha}\beta_{a\alpha\beta}+n_{a\alpha}\beta_{b\alpha\beta}\,.$

###  4.2. Second Order

Subsection 4.1 established that for a given parallelized Riemannian
submanifold embedding, there exists a unique family of defining maps related
by $s^{\prime}_{\alpha}-s_{\alpha}=\mathcal{O}(s^{3})$ such that

(4.5) $\displaystyle
G_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}\,.$

As in that section, our goal is to pare down this family by controlling
$F^{(2)}$. From Equation (4.2), $F^{(2)}$ is uniquely determined in a collar
neighborhood of $\Lambda$ given the initial choice of $F^{(1)}$. Our goal will
be to adjust $s_{\alpha}$ so that we may remove as much of $F^{(2)}$ as
possible.

Using the same reasoning as in Subsection 4.1, we have that $F^{(2)}\in
C^{\infty}(M,\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,2}}}\;\oplus\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{3,1}}}\;\oplus\;\raisebox{1.5pt}{\scalebox{0.2}{\ydiagram{4}}})$.
So, we attempt to fix the family of defining functions to cancel $F^{(2)}$ by
applying the map

$s_{\alpha}\mapsto
s_{\alpha}+A^{(2)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}}s_{\gamma_{1}}s_{\gamma_{2}}s_{\gamma_{3}}\,,$

which clearly preserves the structure of the Gram matrix Condition (4.5) while
possibly changing $F^{(2)}$. However, note that $A^{(2)}\in
C^{\infty}(M,\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{3,1}}}\;\oplus\;\raisebox{1.5pt}{\scalebox{0.2}{\ydiagram{4}}})$.
Because $A^{(2)}$ and $F^{(2)}$ can live in different representations of
$S_{k}$, it is not always possible to construct $A^{(2)}$ such that
$F^{(2)}\mapsto 0$. Nonetheless, one can always find $A^{(2)}$ so that
$F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\mapsto
F^{(2)}_{(\alpha\beta)(\gamma_{1}\gamma_{2})}\in
C^{\infty}(M,\raisebox{3.5pt}{\scalebox{0.2}{\ydiagram{2,2}}})$.

From the above construction, for a given parallelization of $N\Lambda$, there
always exists at least one defining map (in fact, there exists an infinite
family of such maps) such that

(4.6) $\displaystyle
G_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}\,,$

where $F^{(2)}\in
C^{\infty}(M,\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,2}}})$ is entirely
determined by the parallelization of $N\Lambda$ and the choice of $F^{(1)}$.
We call defining maps that are in this infinite family associated defining
maps. Along $\Lambda$, $F^{(2)}|_{\Lambda}$ is a first obstruction to
producing a holonomic basis for the extension of the normal bundle. In fact,
$F^{(2)}|_{\Lambda}$ can be computed in terms of tensors specified by the
parallelization. We now carry out this computation, and going forward we will
work with the family of associated defining maps to the parallelization.

Now, observe that

$P_{\scalebox{0.2}{\ydiagram{2,2}}}\,n^{a}_{\gamma_{1}}n^{b}_{\gamma_{2}}\nabla_{a}\nabla_{b}G_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}2F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\,,$

where $P_{\scalebox{0.2}{\ydiagram{2,2}}}$ is the projector onto the  2,2
component of a rank-4 tensor in $S^{k}$. In calculations that will follow, it
is useful to have explicit formulas for relevant projection operators. To that
end, we find that

$\displaystyle
P_{\scalebox{0.2}{\ydiagram{2,2}}}\,F_{\alpha\beta\gamma\delta}=\tfrac{1}{12}($
$\displaystyle
F_{\alpha\beta\gamma\delta}-F_{\gamma\beta\alpha\delta}-F_{\alpha\delta\gamma\beta}+F_{\gamma\delta\alpha\beta}+F_{\beta\alpha\gamma\delta}-F_{\gamma\alpha\beta\delta}-F_{\beta\delta\gamma\alpha}+F_{\gamma\delta\beta\alpha}+F_{\alpha\beta\delta\gamma}$
$\displaystyle-
F_{\delta\beta\alpha\gamma}-F_{\alpha\gamma\delta\beta}+F_{\delta\gamma\alpha\beta}+F_{\beta\alpha\delta\gamma}-F_{\delta\alpha\beta\gamma}-F_{\beta\gamma\delta\alpha}+F_{\delta\gamma\beta\alpha})\,,$

and

$\displaystyle
P_{\scalebox{0.2}{\ydiagram{3,1}}}\,F_{\alpha\beta\gamma\delta}=\tfrac{1}{8}($
$\displaystyle
F_{\alpha\beta\gamma\delta}-F_{\delta\beta\gamma\alpha}+F_{\beta\alpha\gamma\delta}-F_{\delta\alpha\gamma\beta}+F_{\gamma\beta\alpha\delta}-F_{\delta\beta\alpha\gamma}+F_{\alpha\gamma\beta\delta}-F_{\delta\gamma\beta\alpha}+F_{\beta\gamma\alpha\delta}$
$\displaystyle-
F_{\delta\gamma\alpha\beta}+F_{\gamma\alpha\beta\delta}-F_{\delta\alpha\beta\gamma})\,.$

We can now compute directly

$\displaystyle
P_{\scalebox{0.2}{\ydiagram{2,2}}}\,n^{a}_{\gamma_{1}}n^{b}_{\gamma_{2}}\nabla_{a}\nabla_{b}n^{c}_{\alpha}n_{c\beta}=$
$\displaystyle
P_{\scalebox{0.2}{\ydiagram{2,2}}}\,n^{(a}_{\gamma_{1}}n^{b)}_{\gamma_{2}}\left[2(\nabla
n_{\alpha})^{c}_{(a}(\nabla
n_{\beta})_{b)c}+2n^{c}_{(\alpha|}\nabla_{a}\nabla_{b}n_{c|\beta)}\right]$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-2\beta_{c\alpha(\gamma_{1}}\beta^{c}_{\gamma_{2})\beta}-\tfrac{2}{3}R_{n_{\gamma_{1}}n_{(\alpha}n_{\beta)}n_{\gamma_{2}}}\,.$

Putting this together, we have that

(4.7) $\displaystyle
F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\stackrel{{\scriptstyle\Lambda}}{{=}}-\beta_{c\alpha(\gamma_{1}}\beta^{c}_{\gamma_{2})\beta}-\tfrac{1}{3}R_{n_{\gamma_{1}}n_{(\alpha}n_{\beta)}n_{\gamma_{2}}}\,.$

It is important to note that $F^{(2)}$ is (as yet) dependent on the choice of
extension of $F^{(1)}|_{\Lambda}$. However, what this computation verifies is
that this choice of extension yields a unique obstruction $F^{(2)}|_{\Lambda}$
that depends only on the parallelization of $N\Lambda$ and not on some choice
of how the parallelization or $F^{(1)}|_{\Lambda}$ extends off $\Lambda$.
Thus, the obstruction in Equation (4.7) is a true invariant of the
parallelized Riemannian submanifold embedding. As such, it may be of
independent interest to study those structures for which this obstruction
vanishes. While there is indeed an obstruction, we may proceed regardless.

###  4.3. Higher Orders

We now seek to generalize the result of the previous subsection to all orders.
As noted, at second order $F^{(2)}|_{\Lambda}$ is unique, despite the choice
made of extension of $F^{(1)}$ off $\Lambda$. We would like to show that this
phenomenon continues. This result is captured by the following theorem:

######  Theorem 4.1.

Let $\Lambda^{d-k}\hookrightarrow(M^{d},g)$ be parallelized by
$\\{n_{\alpha}\\}$. Then, in a collar neighborhood of $\Lambda$, there exists
formally a unique defining map determined by $\\{n_{\alpha}\\}$ such that

(4.8) $\displaystyle
G_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}+F^{(3)}_{\alpha\beta\gamma_{1}\gamma_{2}\gamma_{3}}s_{\gamma_{1}}s_{\gamma_{2}}s_{\gamma_{3}}+\cdots,$

where each $F^{(i)}$ is canonically determined to all orders and lives in a
generalized window tensor representation of $S^{k}$.

###### Proof.

We first show that one may find a defining map that satisfies Equation (4.8)
via induction. To do so, we begin by fixing an arbitrary extension of
$F^{(1)}|_{\Lambda}$ as described in Section 4.1. Given such an extension, the
base case is handled by the previous subsection, so that there exists a
defining map so that $G=\delta+\mathcal{O}(s^{2})$. Now, suppose that there
exists a defining map such that

$G_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}+\cdots+F^{(m)}_{\alpha\beta\gamma_{1}\cdots\gamma_{m}}s_{\gamma_{1}}\cdots
s_{\gamma_{m}}+F^{(m+1)}_{\alpha\beta\gamma_{1}\cdots\gamma_{m+1}}s_{\gamma_{1}}\cdots
s_{\gamma_{m+1}}\,,$

where for each $2\leqslant i\leqslant m$, $F^{(i)}$ occupies a generalized
window tensor representation. Now let

$s_{\alpha}\mapsto\tilde{s}_{\alpha}:=s_{\alpha}+A^{(m+1)}_{\alpha\gamma_{1}\cdots\gamma_{m+2}}s_{\gamma_{1}}\cdots
s_{\gamma_{m+2}}\,.$

Then,

$\tilde{G}_{\alpha\beta}=G_{\alpha\beta}+2(m+1)A^{(m+1)}_{(\alpha\beta)\gamma_{1}\cdots\gamma_{m+1}}s_{\gamma_{1}}\cdots
s_{\gamma_{m+1}}\,.$

From the representation theory of the normal bundle, we find that $A^{(m+1)}$
may thus be chosen in a collar neighborhood of $\Lambda$ so that $F^{(m+1)}$
occupies the generalized window tensor representation. This completes the
induction, and thus given an arbitrary extension of $F^{(1)}|_{\Lambda}$ off
$\Lambda$, we have uniquely determined a defining map $s_{\alpha}$ that
satisfies Equation (4.8).

Next, we must show that this defining map is independent of the choice of
$F^{(1)}$. Let $F^{(1)}$ and $F^{(1)^{\prime}}$ be distinct extensions of
$F^{(1)}|_{\Lambda}$. Then, we may produce unique defining maps $s$ and
$s^{\prime}$ respectively associated to each of these choices as described
above. Now because the differentials of the defining maps necessarily agree on
the submanifold, we must have that

$s^{\prime}_{\alpha}=s_{\alpha}+M_{\alpha\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}$

for some $M_{\alpha\gamma_{1}\gamma_{2}}\in
C^{\infty}M\times(\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,1}}}\;\oplus\;\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{3}}})$.
We may now compare the Gram matrices:

$G^{\prime}_{\alpha\beta}=G_{\alpha\beta}+2M_{(\alpha\beta)\gamma_{1}}s_{\gamma_{1}}+\mathcal{O}(s^{2})\,.$

However, observe that both $G^{\prime}_{\alpha\beta}$ and $G_{\alpha\beta}$
must be the identity to second order, and hence
$M_{(\alpha\beta)\gamma_{1}}\stackrel{{\scriptstyle\Lambda}}{{=}}0$. This
implies that $M\stackrel{{\scriptstyle\Lambda}}{{=}}0$, as
$M_{[\alpha\beta]\gamma_{1}}\stackrel{{\scriptstyle\Lambda}}{{=}}0$. To see
this, note that
$M_{(\alpha\beta)\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}0$, so
$M_{\alpha\beta\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}M_{[\alpha\beta]\gamma}$.
So,

$M_{\alpha\beta\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}-M_{\beta\alpha\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}-M_{\beta\gamma\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}M_{\gamma\beta\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}M_{\gamma\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}-M_{\alpha\gamma\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}-M_{\alpha\beta\gamma}\,.$

So this means that

$s^{\prime}_{\alpha}=s_{\alpha}+M_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}}s_{\gamma_{1}}s_{\gamma_{2}}s_{\gamma_{3}}\,.$

By the same considerations, we have that

$G^{\prime}_{\alpha\beta}-G_{\alpha\beta}=4M_{(\alpha\beta)\gamma_{1}\gamma_{2}}s_{\gamma_{1}}s_{\gamma_{2}}\,.$

However, at second order the left hand side necessarily occupies a window
tensor representation—but $M$ is symmetric in three indices, and thus cannot
contribute to this tensor representation. Therefore, it must vanish along
$\Lambda$, and hence $s^{\prime}-s=\mathcal{O}(s^{4})$. By a similar argument,
we may apply induction to find that $s^{\prime}_{\alpha}=s_{\alpha}$ to
arbitrarily high order. Thus, the defining map is independent of the choice of
the extension of $F^{(1)}|_{\Lambda}$. ∎

As the defining map constructed above is unique and fixed for a given
parallelization, we call any such defining maps canonical defining maps. Note
the canonical defining map for a given parallelization is also an associated
defining map.

When the submanifold embedding is particularly geometrically simple, we may do
better. To see this, we first need a technical lemma:

######  Lemma 4.2.

Suppose $\Lambda^{d-k}\hookrightarrow(M,g)$ has vanishing normal curvature,
that $(M,g)$ is flat, and that there exists a defining map satisfying
$\beta=0$ and $G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{m})$ for
some $m\geqslant 2$. Then

$n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}\nabla_{a_{1}}\cdots\nabla_{a_{j}}n^{b}_{\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}0\,,$

for any $1\leqslant j\leqslant m-1$.

###### Proof.

We prove by induction on $j$. For that, we first examine the case where $j=1$.
We must show that
$n^{a}_{\gamma}\nabla_{a}n^{b}_{\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}0$.
However, because $m\geqslant 2$, Equation (4.4) holds, and thus this condition
is equivalent to the condition that $\beta=0$, which is satisfied by
hypothesis. Note that this entirely covers the case where $m=2$, so going
forward we assume that $m\geqslant 3$.

Now fix some $1\leqslant j\leqslant m-2$, and suppose that the equation in the
lemma holds for all $1\leqslant\ell\leqslant j$. We wish to show that this
implies that the equation holds for $j+1$. We compute:

$\displaystyle n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j+1}}_{\gamma_{j+1}}\nabla_{a_{1}}\cdots\nabla_{a_{j+1}}n^{b}_{\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}(\nabla_{a_{1}}n^{a_{j+1}}_{\gamma_{j+1}})\nabla_{a_{2}}\cdots\nabla_{a_{j+1}}n^{b}_{\alpha}$
$\displaystyle+n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}\nabla_{a_{1}}n^{a_{j+1}}_{\gamma_{j+1}}\nabla_{a_{2}}\cdots\nabla_{a_{j+1}}n^{b}_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle\cdots$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle
n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}\nabla_{a_{1}}\cdots\nabla_{a_{j}}n^{a_{j+1}}_{\gamma_{j+1}}\nabla_{a_{j+1}}n^{b}_{\alpha}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle
n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}\nabla_{a_{1}}\cdots\nabla_{a_{j}}n^{a_{j+1}}_{\gamma_{j+1}}\nabla^{b}n_{a_{j+1}\alpha}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}(\nabla_{a_{1}}\cdots\nabla_{a_{j}}n^{a_{j+1}}_{\alpha}\nabla^{b}n_{a_{j+1}\gamma_{j+1}}-\nabla_{a_{1}}\cdots\nabla_{a_{j}}\nabla^{b}G_{\alpha\gamma_{j+1}})$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}\nabla_{a_{1}}\cdots\nabla_{a_{j}}n^{a_{j+1}}_{\alpha}\nabla^{b}n_{a_{j+1}\gamma_{j+1}}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle\cdots$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}n^{a_{j+1}}_{\alpha}\nabla_{a_{1}}\cdots\nabla_{a_{j}}\nabla^{b}n_{a_{j+1}\gamma_{j+1}}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j}}_{\gamma_{j}}n^{a_{j+1}}_{\alpha}\nabla_{a_{1}}\cdots\nabla_{a_{j-1}}\nabla_{a_{j+1}}\nabla^{b}n_{a_{j}\gamma_{j+1}}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle\cdots$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-n^{a_{1}}_{\gamma_{1}}\cdots
n^{a_{j-1}}_{\gamma_{j-1}}n^{a_{j+1}}_{\alpha}\nabla_{a_{1}}\cdots\nabla_{a_{j-1}}\nabla_{a_{j+1}}n^{a_{j}}_{\gamma_{j}}\nabla^{b}n_{a_{j}\gamma_{j+1}}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle 0\,.$

The inductive hypothesis is used repeatedly in the above computation when
moving normal vectors inside of covariant derivatives. Furthermore, note that
we can always rearrange covariant derivatives because $(M,g)$ is flat.
However, as a result of this, we know that the expression must be symmetric in
$\\{\gamma_{1},\ldots,\gamma_{j+1}\\}$. In the sixth equality, we used the
fact that $\nabla^{j+1}G\stackrel{{\scriptstyle\Lambda}}{{=}}0$ because
$j+1\leqslant m$. In the penultimate equality, we have
$n^{a}_{\gamma_{j}}\nabla^{b}n_{a\gamma_{j+1}}$, which by symmetry of the
Greek indices, is equivalent to
$\tfrac{1}{2}\nabla^{b}G_{\gamma_{j}\gamma_{j+1}}$. But by hypothesis the Gram
matrix is the identity up to order $s^{m}$. Counting derivatives, we see that
only $j+1$ derivatives are taken of the Gram matrix, which is strictly less
than $m$. Hence, along $\Lambda$ the expression vanishes. ∎

We are now equipped to produce a unique unit defining map.

######  Corollary 4.3.

Let $\Lambda^{d-k}\hookrightarrow(M^{d},g)$ have vanishing normal curvature
and let $(M,g)$ be flat. Then, there exists a unique unit defining map (up to
constant rotations).

###### Proof.

Observe from Lemma 3.3 that we may choose a parallelization such that
$\beta=0$, as the submanifold embedding has vanishing normal curvature. This
parallelization is unique up to constant rotations. Given this
parallelization, we may use Theorem 4.1 to determine the canonical defining
map for this parallelization. Then, it it suffices to show that each
obstruction $F^{(i)}$ vanishes for this parallelization. Evidently,
$F^{(2)}|_{\Lambda}$ vanishes because $\beta=0$ and $(M,g)$ is flat. We would
like to show that each subsequent obstruction vanishes along $\Lambda$. To do
so, we will induct on $i$.

Suppose that $i\geqslant 2$ and that
$G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{i+1})$. Then, from
Theorem 4.1, $F^{(i+1)}$ occupies a generalized window tensor representation
of $S^{k}$, and thus it can be computed by projecting appropriately the
expression
$\tilde{n}^{a_{1}}_{(\gamma_{1}}\cdots\tilde{n}^{a_{i+1}}_{\gamma_{i+1})}\nabla_{a_{1}}\cdots\nabla_{a_{i+1}}\tilde{G}_{\alpha\beta}$.
By Lemma 4.2 and the flatness condition on $(M,g)$, the only term that may
appear in $F^{(i+1)}$ is of the form

$\tilde{n}^{a_{1}}_{(\gamma_{1}}\cdots\tilde{n}^{a_{i+1}}_{\gamma_{i+1})}n^{b}_{\alpha}\nabla_{a_{1}}\cdots\nabla_{a_{i+1}}\nabla_{b}s_{\beta}\,.$

But as this expression is symmetric in Latin indices, it is necessarily
symmetric on all the Greek indices except $\beta$. Thus, the generalized
window projection of this expression necessarily vanishes, as this expression
is symmetric in $i+2$ Greek indices. Thus, $F^{(i+1)}|_{\Lambda}=0$, and hence
$G_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(s^{i+2})$, completing the
induction. ∎

Notably, the unique unit defining map is that which produces a parallelization
with vanishing $\beta$ and is as described in Theorem 4.1.

##  5\. Conformal Geometry

We now take a detour to explain the conformal calculus that will be used in
the following sections. Much of what is summarized here can be found in more
detail in any of [3, 5, 6, 13, 15].

Consider a conformal manifold $(M^{d},\bm{c})$ with $d\geqslant 3$. We may
view this conformal manifold as a smooth ray subbundle
$\mathcal{Q}\subset\odot^{2}T^{*}M$ coordinatized by $(p,t^{2}g)$ for $p\in
M$, $t>0$, and $g\in\bm{c}$ a metric on $M$. Indeed, this ray subbundle is a
principal $\mathbb{R}_{+}$ bundle and hence, for each $w\in\mathbb{R}$,
induces a natural associated line bundle, called a density bundle of weight
$w$, associated to $\mathcal{Q}$ via the irreducible representation
$\mathbb{R}_{+}\ni t\mapsto
t^{-\tfrac{w}{2}}\in\operatorname{End}(\mathbb{R})$. Such a density bundle is
denoted $\mathcal{E}M[w]$. Sections of this bundle can then be viewed as
functions $\varphi$ on $\mathcal{Q}$ with the property that

$\varphi(p,\Omega^{2}g)=\Omega^{w}\varphi(p,g)\,.$

We call such a function a (conformal) density. For practical purposes, it is
often more convenient to view a density as a double equivalence class of
functions, $\varphi=[g;f]=[\Omega^{2}g;\Omega^{w}f]$, where $f$ is some
function in $C^{\infty}M$. Note that we can tensor a density bundle of weight
$w$ with any bundle $\mathcal{V}M$ over $M$ to obtain a weighted bundle
denoted by $\mathcal{V}M[w]$.

Of particular importance is the so-called conformal metric $\bm{g}$ which is
the tautological section of $\odot^{2}T^{*}M[2]$ given by $\bm{g}=[g;g]$. In
the conformal setting, the conformal metric can be used for raising and
lowering indices on tensor-valued densities.

A troublesome feature of conformal geometry is that, on the tangent bundle
$TM$, there is no connection that respects the conformal structure. Indeed,
even on densities, the naive exterior derivative only respects the conformal
structure on weight-0 densities. On the other hand, for any $g\in\bm{c}$,
there exists a true scale $0<\tau\in\Gamma(\mathcal{E}M[1])$ which is a
weight-1 positive density that determines $g$ via $g=\bm{g}/\tau^{2}$. Then,
for any true scale $\tau$, we can define a connection on weight-$w$ densities
via $\nabla^{\tau}:=\tau^{w}\circ d\circ\tau^{-w}$. A straightforward
calculation shows that for any two true scales $\tau$ and
$\tau^{\prime}=\Omega\tau$ and any $\varphi\in\Gamma(\mathcal{E}M[w])$, we
have that

$(\nabla^{\tau^{\prime}}-\nabla^{\tau})\varphi=-w(d\log\Omega)\varphi\,.$

Nontheless, this family of connections is insufficient for generating
invariants of a conformal structure, so we must turn instead to tractors. The
standard tractor bundle over $(M,\bm{c})$ is a rank $d+2$ vector bundle
$\mathcal{T}M$, which for any choice of metric $g\in\bm{c}$ is canonically
isomorphic to

$\mathcal{T}M\stackrel{{\scriptstyle g}}{{\cong}}\mathcal{E}M[1]\oplus
TM[-1]\oplus\mathcal{E}M[-1]\,.$

We call this isomorphism a choice of splitting. In a choice of splitting, we
may thus decompose a section of the standard tractor bundle
$T\in\Gamma(\mathcal{T}M)$ according to

$T^{A}\stackrel{{\scriptstyle g}}{{=}}(\tau^{+},\tau^{a},\tau^{-})\,.$

Any choice of splitting can be related to another according to

$T^{A}\stackrel{{\scriptstyle\Omega^{2}g}}{{=}}(\tau^{+},\tau^{a}+\Upsilon^{a}\tau^{+},\tau^{-}-\Upsilon_{a}\tau^{a}-\tfrac{1}{2}\Upsilon_{a}\Upsilon^{a}\tau^{+})\,,$

where $\Upsilon:=d\log\Omega$ and $\Upsilon^{a}:=\bm{g}^{ab}\Upsilon_{a}$.
Observe that $\tau^{+}$ is independent of choice of splitting and hence is
conformally invariant.

Note that for any $T,U\in\Gamma(\mathcal{T}M)$, there exists an inner product
that is independent of the choice of splitting given (in a choice of
splitting) by

$T\cdot U\stackrel{{\scriptstyle
g}}{{=}}\bm{g}_{ab}\tau^{a}\mu^{b}+\tau^{+}\mu^{-}+\tau^{-}\mu^{+}\,,$

where $U^{A}\stackrel{{\scriptstyle g}}{{=}}(\mu^{+},\mu^{a},\mu^{-})$.
Indeed, this inner product defines a tractor metric
$h\in\Gamma(\odot^{2}\mathcal{T}^{*}M)$ expressible in a choice of splitting
as

$h_{AB}\stackrel{{\scriptstyle g}}{{=}}\begin{pmatrix}0&0&1\\\
0&\bm{g}_{ab}&0\\\ 1&0&0\end{pmatrix}\,.$

Furthermore, this tractor metric provides an isomorphism between the standard
tractor bundle and its dual

$\mathcal{T}^{*}M\stackrel{{\scriptstyle g}}{{\cong}}\mathcal{E}M[1]\oplus
T^{*}M[1]\oplus\mathcal{E}M[-1]\,.$

The standard tractor bundle also appears in the construction of other useful
bundles. By tensoring the standard tractor bundle with the weight-$1$ density
bundle, we obtain $\mathcal{T}M[1]$, which has a canonically defined section
called the canonical tractor, given in any choice of splitting by
$X^{A}\stackrel{{\scriptstyle g}}{{=}}(0,0,1)\in\Gamma(\mathcal{T}M[1])$.

Of prime importance is that there exists a well-defined tractor connection on
the standard tractor bundle

$\displaystyle\nabla^{\mathcal{T}}:\Gamma(\mathcal{T}M)$
$\displaystyle\rightarrow\Gamma(T^{*}M\otimes\mathcal{T}M)$ $\displaystyle
T^{B}$
$\displaystyle\mapsto\nabla^{\mathcal{T}}_{a}T^{B}\stackrel{{\scriptstyle
g}}{{=}}\begin{pmatrix}\nabla_{a}\tau^{+}-\tau_{a}\\\
\nabla_{a}\tau^{b}+\bm{g}_{a}^{b}\tau^{-}+(P^{g})_{a}^{b}\tau^{+}\\\
\nabla_{a}\tau^{-}-P^{g}_{ab}\tau^{b}\end{pmatrix}\,.$

Indeed, this 1-form-valued tractor still behaves as expected when another
choice of splitting is selected. Note that this connection can be extended to
any tensor product of standard tractor bundles. However, this does not map
tractor bundles to tractor bundles. For that, we can use the tractor
connection to build a tractor-valued operator known as the Thomas-$D$
operator, given in a choice of splitting by

$\displaystyle D_{A}:\Gamma(\mathcal{T}^{\Phi}M[w])$
$\displaystyle\rightarrow\Gamma(\mathcal{T}^{*}M\otimes\mathcal{T}^{\Phi}M[w-1])$
$\displaystyle T^{\Phi}$ $\displaystyle\mapsto
D_{A}T^{\Phi}\stackrel{{\scriptstyle
g}}{{=}}\begin{pmatrix}(d+2w-2)wT^{\Phi}\\\
(d+2w-2)\nabla^{\mathcal{T}}_{a}T^{\Phi}\\\
-(\Delta^{\mathcal{T}}+wJ^{g})T^{\Phi}\end{pmatrix}\,,$

where
$\Delta^{\mathcal{T}}=\bm{g}^{ab}\nabla^{\mathcal{T}}_{a}\nabla^{\mathcal{T}}_{b}$
is the tractor Laplacian. From the Thomas-$D$ operator we can define another
useful operator on $\mathcal{T}^{\Phi}M[w]$—the hatted Thomas-$D$ operator

$\hat{D}_{A}=\tfrac{1}{d+2w-2}D_{A}\,,$

which is well-defined so long as $w\neq 1-\tfrac{d}{2}$. While the Thomas-$D$
operator is not a derivation, it has the useful feature that $D_{A}\circ
D^{A}=0$. Furthermore, its failure to be a derivation is well-controlled and
given by the following formula [18]:

$\hat{D}^{A}(T_{1}\cdots
T_{\ell})-\sum_{i=1}^{\ell}T_{1}\cdots(\hat{D}^{A}T_{i})\cdots T_{\ell}\\\
=-\frac{2X^{A}}{d+2\sum_{i=1}^{\ell}w_{i}-2}\sum_{1\leqslant
i<j\leqslant\ell}T_{1}\cdots(\hat{D}^{B}T_{i})\cdots(\hat{D}_{B}T_{j})\cdots
T_{\ell}\,,$

where for each $1\leqslant i\leqslant\ell$,
$T_{i}\in\Gamma(\mathcal{T}^{\Phi}M[w_{i}])$, and the weight constraints
$d+\sum_{i=1}^{\ell}w_{i}-2\neq 0$ and $\\{w_{i}\neq
1-\tfrac{d}{2}\\}_{i=1}^{\ell}$ are satisfied. While it is clear that this
operator is not a covariant derivative, it often plays the role of one.
Indeed, the commutator of two hatted Thomas-$D$ operators produces a
curvature-like tractor called the $W$-tractor, according to

$[\hat{D}_{A},\hat{D}_{B}]T^{D}=W_{AB}{}^{D}{}_{E}T^{E}+\tfrac{4}{d+2w-4}X_{[A}W_{B]C}{}^{D}{}_{E}\hat{D}^{C}T^{E}\,,$

where such a commutator is defined. The $W$-tractor has Riemann-like
symmetries, is trace-free, and satisfies

$X^{A}W_{ABCD}=0=\hat{D}^{A}W_{ABCD}\,.$

This tractor is composed of the Weyl tensor, the Cotton tensor, and a
generalized notion of the Bach tensor. See [14, 4] for more details.

Because all of the aforementioned tractor objects respect the conformal
structure of $(M,\bm{c})$, any combination of these tractors and tractor-
valued operators can be used to produce a conformally invariant quantity. It
is this feature that makes the tractor calculus analogous to the Ricci
calculus of Riemannian geometry, and we will utilize this advantage heavily in
the sections that follow.

##  6\. Orthonormal Frames for Normal Bundles of Conformal Submanifold
Embeddings

We now turn to the case of conformal submanifold embedings
$\Lambda^{d-k}\hookrightarrow(M^{d},\bm{c})$. As the tangent bundle
decomposition for a Riemannian submanifold embedding in Equation (3.1) only
depends on the orthogonality of vectors rather than their lengths, a conformal
structure admits an identical decomposition. Furthermore, given any
representative $g\in\bm{c}$ and an orthonormal frame
$\\{(n^{g})^{a}_{\alpha}\\}$ with respect to $g$ of $N\Lambda$, we can promote
such an orthonormal frame to an orthonormal frame-valued density (OFVD)
$\\{\bm{n}^{a}_{\alpha}\\}:=\\{[g;(n^{g})^{a}_{\alpha}]\\}$ with weight $-1$.
Then it follows that
$\bm{g}(\bm{n}_{\alpha},\bm{n}_{\beta})=\delta_{\alpha\beta}$.

Interestingly, normal fundamental forms are conformally-invariant. Indeed,
given an OFVD and a metric representative $g\in\bm{c}$, we find that
$\beta_{a\alpha\beta}\in\Gamma(T^{*}\Sigma[0]\times\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{1,1}}})$
and hence the normal curvature is also invariant. Thus all of the results of
Section 3 directly generalize to the equivalent conformal structures without
additional consideration. Unfortunately, the results of Section 4 do not
generalize directly in the conformal setting. Instead, additional obstructions
and other troublesome features arise.

######  Remark 6.1.

The conformal invariance of the normal fundamental form may seem to contradict
the observation that, for a curve $\Lambda^{1}\hookrightarrow\mathbb{R}^{3}$,
the normal fundamental form of the Frenet frame picks out the torsion.
However, this observation is naive. The normal fundamental form is conformally
invariant when the orthonormal frame vectors are density valued. However,
rescaling of the metric generically changes the plane of osculation for a
given curve and hence rotates the Frenet frame. Thus the failure of a
spacecurve’s torsion to be conformally-invariant is a consequence of the
changing orientation of the Frenet frame.

##  7\. Defining Densities for Parallelized Conformal Submanifold Embeddings

As in the Riemannian setting, we would like to better study the conformal
submanifold embedding $\Lambda^{d-k}\hookrightarrow(M^{d},\bm{c})$ by studying
the jets of geometrically-determined defining densities—densities of weight-1
that, in any metric representative, are represented by a defining map for
$\Lambda$. To do so, we will attempt to construct the conformal analog of
associated defining maps. We begin by picking any $g\in\bm{c}$ and
constructing the canonical defining map $s_{\alpha}^{g}$ for the provided OFVD
in that metric representative. We can then promote this associated defining
map to a defining density $\sigma_{\alpha}$ of weight 1. (Note that we are
abusing language by calling an element of
$\Gamma(\mathcal{E}M[1]\times\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{1}}})$
a defining density, but we do so regardless.) Had we picked another metric
conformally related to the first, the difference between the two densities
would be at least second order in a defining density. For that reason, the
initial choice of arbitrary metric to begin the construction is irrelevant for
what follows (as we will change higher order terms anyway). That is, for a
given OFVD, modulo terms of second order in a defining density, there exists a
unique defining density
$\sigma_{\alpha}=[g;s^{g}_{\alpha}]\in\Gamma(\mathcal{E}M[1])$ such that
$d\sigma_{\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}\bm{g}(\bm{n}_{\alpha},\cdot)$.
(Note that $d\sigma_{\alpha}$ is not density-valued away from $\Lambda$ but a
short computation shows that it is indeed density-valued along $\Lambda$.) Our
goal will be to maximally improve this density so that we can extract
invariants of the parallelized conformal submanifold embedding.

In general, the Gram matrix of a defining map is not conformally invariant
away from $\Lambda$. Even in the codimension-1 case, for two metric
representatives $g,\Omega^{2}g\in\bm{c}$, we find that

$|ds^{\Omega^{2}g}|^{2}_{\Omega^{2}g}=|d(\Omega
s^{g})|^{2}_{\Omega^{2}g}=|\Omega(n^{g}_{a}+s\Upsilon_{a})|^{2}_{\Omega^{2}g}=|n^{g}_{a}+s^{g}\Upsilon_{a}|^{2}_{g}\neq|ds^{g}|^{2}_{g}\,.$

From this computation, we have shown explicitly that $G^{\Omega^{2}g}\neq
G^{g}$ except on the submanifold $\Lambda$. Going forward, we will be dropping
the superscripts denoting metric representatives when the context is clear.

The natural analog of the (Riemannian) Gram matrix for a defining map to the
conformal case is the conformal Gram matrix of a defining density. Given a
defining density for a submanifold $\sigma_{\alpha}$, we can build a family of
scale tractors,

$\\{N^{A}_{\alpha}:=\hat{D}^{A}\sigma_{\alpha}\\}_{\alpha=1}^{k}\,.$

The conformal Gram matrix, then, is defined according to

$\bm{G}_{\alpha\beta}:=h_{AB}N^{A}_{\alpha}N^{B}_{\beta}\,.$

By construction, the conformal Gram matrix is conformally invariant.
Furthermore, upon restriction to the submanifold $\Lambda$, in any
representative $g\in\bm{c}$ the conformal Gram matrix agrees with the
Riemannian Gram matrix of the representative of $\sigma_{\alpha}$ given by
$[g;s_{\alpha}]$. Indeed, for $g\in\bm{c}$ any metric representative and
$\sigma_{\alpha}=[g;s_{\alpha}]$ a defining density for
$\Lambda^{d-k}\hookrightarrow(M,\bm{c})$, the components of
$\bm{G}_{\alpha\beta}$ are given by

$\bm{G}_{\alpha\beta}=[g;G_{\alpha\beta}]:=[g;(\nabla^{a}s_{\alpha})(\nabla_{a}s_{\beta})-\tfrac{2}{d}s_{(\alpha}(\Delta^{g}+J^{g})s_{\beta)}]\,.$

Note that we are overloading notation: going forward we use $G_{\alpha\beta}$
to refer to a representative of the conformal Gram matrix, rather than the
Riemannian Gram matrix. However note that every representative is the same, as
the conformal Gram matrix has weight 0—we merely mean that we refer not to the
double equivalence class $[g;G_{\alpha\beta}]$ but just $G_{\alpha\beta}$.

In the Riemannian setting, we found that, for a given parallelization of
$N\Lambda$, a unique family of canonical defining maps could be obtained to
arbitrary order. Further, when the background was flat and the the normal
curvature vanished, we found that this family consists entirely of unit
defining maps. We proceed now with a similar program in the conformal setting.

###  7.1. First Order

Given an initial choice of defining density built from an associated defining
map for $\Lambda\hookrightarrow(M,\bm{c})$, it is easy to see that any
representative of the conformal Gram matrix agrees with the Riemannian Gram
matrix for the same associated defining map along $\Lambda$. It follows then
that

$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+F^{(1)}_{\alpha\beta\gamma}\sigma_{\gamma}\,,$

for some
$F^{(1)}\in\Gamma(\mathcal{E}M[-1]\times(\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,1}}}\;\oplus\;\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{3}}}))$,
similar to the Riemannian setting. As in that setting, we will pick some
$F^{(1)}$ that extends $F^{(1)}|_{\Lambda}$, with the goal of showing that
this choice is immaterial. Note, however, both the trace and trace-free parts
of these representations will play a role in what follows.

We will now improve $\sigma$ at second order to cancel $F^{(1)}$ along
$\Lambda$. Let

$\tilde{\sigma}_{\alpha}=\sigma_{\alpha}+A^{(1)}_{\alpha\gamma_{1}\gamma_{2}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\,,$

where
$A^{(1)}\in\Gamma(\mathcal{E}M[-1]\times(\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,1}}}\;\oplus\;\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{3}}}))$.
Going forward, we will implicitly work in an arbitrary scale $g\in\bm{c}$. To
begin, we compute:

$\displaystyle\tilde{n}_{\alpha}=$ $\displaystyle
n_{\alpha}+2A^{(1)}_{\alpha(\gamma_{1}\gamma_{2})}n_{\gamma_{1}}s_{\gamma_{2}}+\mathcal{O}(s^{2})$
$\displaystyle\Delta\tilde{s}_{\alpha}=$ $\displaystyle\Delta
s_{\alpha}+2A^{(1)}_{\alpha\gamma\gamma}+\mathcal{O}(s)\,.$

Thus, we have that

$\displaystyle\tilde{G}_{\alpha\beta}$
$\displaystyle=\tilde{n}^{a}_{\alpha}\tilde{n}_{a\beta}-\tfrac{2}{d}\tilde{s}_{(\alpha}(\Delta^{g}+J^{g})\tilde{s}_{\beta)}$
$\displaystyle=n^{a}_{\alpha}n_{a\beta}+4n^{a}_{\gamma_{1}}n_{a(\alpha}A^{(1)}_{\beta)(\gamma_{1}\gamma_{2})}s_{\gamma_{2}}-\tfrac{2}{d}s_{(\alpha}(\Delta^{g}+J^{g})s_{\beta)}-\tfrac{4}{d}s_{(\alpha}A^{(1)}_{\beta)\gamma\gamma}+\mathcal{O}(s^{2})$
$\displaystyle=G_{\alpha\beta}+4A^{(1)}_{(\alpha\beta)\gamma}s_{\gamma}-\tfrac{4}{d}s_{(\alpha}A^{(1)}_{\beta)\gamma\gamma}+\mathcal{O}(s^{2})$
$\displaystyle=\delta_{\alpha\beta}+(F^{(1)}_{\alpha\beta\omega}+4A^{(1)}_{(\alpha\beta)\omega}-\tfrac{4}{d}\delta_{\omega(\alpha}A^{(1)}_{\beta)\gamma\gamma})s_{\omega}+\mathcal{O}(s^{2})\,.$

Indeed, we see that a trace of $A^{(1)}$ appears here. Thus we must be careful
to check that all irreducible components of $F^{(1)}$ can be cancelled. In
this case, we have that

$F^{(1)}=(\raisebox{3.0pt}{\scalebox{0.4}{\ydiagram{2,1}}}_{\,\circ}\oplus\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{1}}})\;\oplus\;(\scalebox{0.4}{\ydiagram{3}}_{\,\circ}\oplus\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{1}}})\,,$

and similarly for $A^{(1)}$. Note that the trace-free parts of $F^{(1)}$ can
be cancelled for the same reason they can be cancelled in the Riemannian case.
To force the traceful parts to vanish, we demand that

$\displaystyle 0$
$\displaystyle=F^{(1)}_{\alpha\alpha\omega}+4A^{(1)}_{\alpha\alpha\omega}-\tfrac{4}{d}A^{(1)}_{\omega\alpha\alpha}\;\;\text{
and }$ $\displaystyle 0$
$\displaystyle=F^{(1)}_{\omega\alpha\alpha}+2A^{(1)}_{\omega\alpha\alpha}+2A^{(1)}_{\alpha\alpha\omega}-\tfrac{2}{d}A^{(1)}_{\omega\alpha\alpha}-\tfrac{2k}{d}A^{(1)}_{\omega\alpha\alpha}$
$\displaystyle=F^{(1)}_{\omega\alpha\alpha}+2A^{(1)}_{\alpha\alpha\omega}+\tfrac{2(d-k-1)}{d}A^{(1)}_{\omega\alpha\alpha}\,.$

The determinant of this linear system (with $A^{(1)}_{\alpha\alpha\omega}$ and
$A^{(1)}_{\omega\alpha\alpha}$ as variables) is $8(d-k)/d$, and thus for all
$k\neq d$, we can solve for the traces of $A^{(1)}$ such that they cancel the
traces of $F^{(1)}$. We have thus shown the intermediate proposition:

######  Proposition 7.1.

Let $\Lambda^{d-k}\hookrightarrow(M^{d},\bm{c})$ be a conformal submanifold
embedding with a parallelization given in a choice of metric representative by
$\\{n_{\alpha}\\}$. Then, there exists a defining map
$\sigma_{\alpha}\in\Gamma(\mathcal{E}M[1]\times\raisebox{2.0pt}{\scalebox{0.2}{\ydiagram{1}}})$
such that $d\sigma_{\alpha}|_{\Lambda}=[g;n_{a\alpha}]$ and its conformal Gram
matrix satisfies

$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(\sigma^{2})\,.$

Indeed, all such defining maps differ by terms of order 3. Given such a
defining map that satisfies
$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(\sigma^{2})$, it will
be useful to evaluate $N^{A}_{\alpha}|_{\Lambda}$. Doing so requires the
evaluation of $\nabla\cdot n_{\alpha}|_{\Lambda}$ in a choice of metric
representative. First, note that, unlike in the Riemannian case, we have that

$n^{a}_{\alpha}n_{a\beta}=\delta_{\alpha\beta}+\tfrac{2}{d}s_{(\alpha}\Delta
s_{\beta)}+\mathcal{O}(s^{2})\,.$

Hence, we find that

$n^{b}_{\alpha}\nabla_{a}n_{b\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}\beta_{a\alpha\beta}+\tfrac{1}{d}n_{a\alpha}\nabla\cdot
n_{\beta}\,,$

in contrast with Equation (4.3). Consequently, we have that

(7.1)
$\displaystyle\nabla_{a}n_{b\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}{\rm
II}_{ab\beta}+n_{b\alpha}\beta_{a\alpha\beta}+n_{a\alpha}\beta_{b\alpha\beta}+\tfrac{1}{d}n_{a\alpha}n_{b\alpha}\nabla\cdot
n_{\beta}\,.$

It thus follows that

$\nabla\cdot n_{\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}dH_{\beta}\,,$

where $H_{\beta}=\tfrac{1}{d-k}{\rm II}^{a}_{a\beta}$ is the mean curvature,
and so

$N^{A}_{\alpha}\stackrel{{\scriptstyle\Lambda,g}}{{=}}\begin{pmatrix}0\\\
\,n^{a}_{\alpha}\\\ -H_{\alpha}\end{pmatrix}\,.$

We may also compute the tractor equivalent of the bulk formula for the normal
fundamental form, given by

$B^{A}_{\alpha\beta}:=N_{[\alpha}\cdot\hat{D}N^{A}_{\beta]}\in\Gamma(\mathcal{T}M[-1]\times\raisebox{2.5pt}{\scalebox{0.2}{\ydiagram{1,1}}})\,.$

By direct computation, this tractor has the property that

$X_{A}B^{A}_{\alpha\beta}=\hat{D}_{A}B^{A}_{\alpha\beta}=0\,.$

This tractor necessarily takes the form

$B^{A}_{\alpha\beta}\stackrel{{\scriptstyle g}}{{=}}\begin{pmatrix}0\\\
\,n^{b}_{[\alpha}\nabla_{b}n^{a}_{\beta]}-\tfrac{1}{d}n^{a}_{[\alpha}\nabla\cdot
n_{\beta]}+\mathcal{O}(s)\\\
\ast\end{pmatrix}\stackrel{{\scriptstyle\Lambda,g}}{{=}}\begin{pmatrix}0\\\
\,\beta^{a}_{\alpha\beta}\\\ \ast\end{pmatrix}\,.$

Finally, it is useful to express the projecting part of the tractor equivalent
of $\nabla_{a}n_{b}$:

$\displaystyle P_{AB\alpha}:=$ $\displaystyle\hat{D}_{A}N_{B\alpha}$
$\displaystyle=$ $\displaystyle\begin{pmatrix}0&0&0\\\
0&\nabla_{a}n_{b\alpha}+s_{\alpha}P_{ab}+g_{ab}\rho_{\alpha}&\ast\\\
0&\ast&\ast\end{pmatrix}\stackrel{{\scriptstyle\Lambda}}{{=}}\begin{pmatrix}0&0&0\\\
0&\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\alpha}+2n_{(a\beta}\beta_{b)\beta\alpha}&\ast\\\
0&\ast&\ast\end{pmatrix}\,,$

where $\rho_{\alpha}:=-\tfrac{1}{d}(\Delta s_{\alpha}+Js_{\alpha})$ and
$\hskip 1.13809pt\mathring{\rm{II}}{}_{\alpha}:={\rm
II}_{\alpha}-\bar{g}H_{\alpha}\in\Gamma(\odot^{2}_{\circ}T^{*}\Lambda[1])$ is
the trace-free component of the second fundamental form. With this tractor in
hand, we define a scalar-valued density called the submanifold rigidity
density $K_{\alpha\beta}:=P_{AB\alpha}P^{AB}_{\beta}$ which is ubiquitous. A
short calculation shows that

$K_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\beta}-2\beta_{b\gamma(\alpha}\beta^{b}_{\beta)\gamma}\,.$

We now proceed to higher orders.

write right-middle slot as d.Lap.s + P.n + Jn

###  7.2. Second Order

As in the Riemannian case, we now proceed to adjust $\sigma_{\alpha}$ at third
order to eliminate as much from the second order term $\bm{G}_{\alpha\beta}$
as possible. Pulling from the Riemannian case, we expect to fail here as well.
Suppose that $\sigma_{\alpha}$ has

$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\,.$

As a reminder, $F^{(2)}$ may depend on the initial choice made of $F^{(1)}$.
Now, we can decompose $F^{(2)}$ into irreducible representations so that

$F^{(2)}\in\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{2,2}}}_{\,\circ}\;\oplus\;\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{3,1}}}_{\,\circ}\;\oplus\;\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{4}}}_{\,\circ}\;\oplus\;3\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{2}}}_{\,\circ}\;\oplus\;\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{1,1}}}\;\oplus\;2.$

On the other hand, we can define
$\tilde{\sigma}_{\alpha}=\sigma_{\alpha}+A^{(2)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}$.
Then we can decompose $A^{(2)}$ in a similar fashion, yielding

$A^{(2)}\in\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{3,1}}}_{\,\circ}\;\oplus\;\raisebox{0.0pt}{\scalebox{0.4}{\ydiagram{4}}}_{\,\circ}\;\oplus\;2\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{2}}}_{\,\circ}\;\oplus\;\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{1,1}}}\;\oplus\;1.$

So naively one might predict there will be obstructions of the form

$F^{(2)}\in\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{2,2}}}_{\,\circ}\;\oplus\;\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{2}}}_{\,\circ}\;\oplus\;1.$

To check this naive understanding, we will explicitly compute for the last
time in this case. As before, we have that

$\displaystyle\tilde{n}_{\alpha}$
$\displaystyle=n_{\alpha}+3A^{(2)}_{\alpha(\gamma_{1}\gamma_{2}\gamma_{3})}s_{\gamma_{1}}s_{\gamma_{2}}n_{\gamma_{3}}+\mathcal{O}(s^{3})$
$\displaystyle\Delta\tilde{s}_{\alpha}$ $\displaystyle=\Delta
s_{\alpha}+6A^{(2)}_{\alpha\gamma_{1}\gamma\gamma}s_{\gamma_{1}}+\mathcal{O}(s^{2})\,.$

Plugging these into the formula for $G_{\alpha\beta}$, we have

$\displaystyle\tilde{G}_{\alpha\beta}$
$\displaystyle=\tilde{n}^{a}_{\alpha}\tilde{n}_{a\beta}-\tfrac{2}{d}\tilde{s}_{(\alpha}(\Delta^{g}+J^{g})\tilde{s}_{\beta)}$
$\displaystyle=n^{a}_{\alpha}n_{a\beta}-\tfrac{2}{d}s_{(\alpha}(\Delta^{g}+J^{g})s_{\beta)}+(6A^{(2)}_{(\alpha\beta)\omega_{1}\omega_{2}}-\tfrac{12}{d}\delta_{\omega_{1}(\alpha}A^{(2)}_{\beta)\gamma\gamma\omega_{2}})s_{\omega_{1}}s_{\omega_{2}}$
$\displaystyle=\delta_{\alpha\beta}+(F^{(2)}_{\alpha\beta\omega_{1}\omega_{2}}+6A^{(2)}_{(\alpha\beta)\omega_{1}\omega_{2}}-\tfrac{12}{d}\delta_{\omega_{1}(\alpha}A^{(2)}_{\beta)\gamma\gamma\omega_{2}})s_{\omega_{1}}s_{\omega_{2}}\,.$

Note that there is implicit symmetrization over $\omega_{1}$ and $\omega_{2}$
in these displays. Just as in the Riemannian case, it is straightforward to
see that the $\raisebox{2.5pt}{\scalebox{0.2}{\ydiagram{3,1}}}_{\,\circ}$ and
$\raisebox{1.5pt}{\scalebox{0.2}{\ydiagram{4}}}_{\,\circ}$ components of
$F^{(2)}$ can be cancelled directly, and it is impossible to cancel the
$\raisebox{2.5pt}{\scalebox{0.2}{\ydiagram{2,2}}}_{\,\circ}$ component of
$F^{(2)}$. We now consider the remaining components of $F^{(2)}$. If the
trace-free parts of this coefficient are to vanish, we must demand that:

$\displaystyle 0$
$\displaystyle=F^{(2)}_{\rho\rho(\omega_{1}\omega_{2})_{\circ}}+6A^{(2)}_{\rho\rho(\omega_{1}\omega_{2})_{\circ}}-\tfrac{12}{d}A^{(2)}_{(\omega_{1}\omega_{2})_{\circ}\rho\rho}$
$\displaystyle 0$
$\displaystyle=F^{(2)}_{(\omega_{1}\omega_{2})_{\circ}\rho\rho}+\tfrac{6(d-2)}{d}A^{(2)}_{(\omega_{1}\omega_{2})_{\circ}\rho\rho}$
$\displaystyle 0$
$\displaystyle=F^{(2)}_{\rho(\omega_{1}\omega_{2})_{\circ}\rho}+3A^{(2)}_{\rho\rho(\omega_{1}\omega_{2})_{\circ}}+\tfrac{3(d-k-2)}{d}A^{(2)}_{(\omega_{1}\omega_{2})_{\circ}\rho\rho}$
$\displaystyle 0$
$\displaystyle=F^{(2)}_{\rho[\omega_{1}\omega_{2}]\rho}+\tfrac{3(d-k-2)}{d}A^{(2)}_{[\omega_{1}\omega_{2}]\rho\rho}$
$\displaystyle 0$
$\displaystyle=F^{(2)}_{\rho\rho\gamma\gamma}+\tfrac{6(d-2)}{d}A^{(2)}_{\rho\rho\gamma\gamma}$
$\displaystyle 0$
$\displaystyle=F^{(2)}_{\rho\gamma\gamma\rho}+\tfrac{6(d-k-1)}{d}A^{(2)}_{\rho\rho\gamma\gamma}\,.$

To more easily visualize this system of equations, we denote
$x_{1}=A^{(2)}_{\rho\rho(\omega_{1}\omega_{2})_{\circ}}$,
$x_{2}=A^{(2)}_{(\omega_{1}\omega_{2})_{\circ}\rho\rho}$,
$x_{3}=A^{(2)}_{[\omega_{1}\omega_{2}]\rho\rho}$, and
$x_{4}=A^{(2)}_{\gamma\gamma\rho\rho}$. Then the matrix describing the linear
system of equations is given by

$\begin{pmatrix}6&-\tfrac{12}{d}&0&0\\\ 0&\tfrac{6(d-2)}{d}&0&0\\\
3&\tfrac{3(d-k-2)}{d}&0&0\\\ 0&0&\tfrac{3(d-k-2)}{d}&0\\\
0&0&0&\tfrac{6(d-2)}{d}\\\ 0&0&0&\tfrac{6(d-k-1)}{d}\end{pmatrix}\,.$

This matrix has rank at most 4. Furthermore, when $k=d-2$, this matrix has
rank 3. So as expected, we can remove at most 4 traced degrees of freedom from
$F^{(2)}$ using $A^{(2)}$, except when $k=d-2$, in which case we can only
remove 3 degrees of freedom. Indeed, for $k\neq d-2$, we have an obstruction
given by

$F^{(2)}=\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{2,2}}}_{\,\circ}\;\oplus\;\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{2}}}_{\,\circ}\;\oplus\;1.$

When $k=d-2$, we instead have an obstruction of the form

$F^{(2)}=\raisebox{2.5pt}{\scalebox{0.4}{\ydiagram{2,2}}}_{\,\circ}\;\oplus\;\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{2}}}_{\,\circ}\;\oplus\;\,\raisebox{1.0pt}{\scalebox{0.4}{\ydiagram{1,1}}}_{\,\circ}\;\oplus\;1.$

For the sake of uniformity, when $k\neq d-2$, we will choose to eliminate as
much of the components of $F^{(2)}$ in  3,1 and  4 as possible. Certainly, as
in the Riemannian case, we can easily remove the trace-free parts of these
components, so we examine the traces.

First, observe that

$\delta^{\alpha\beta}\delta^{\gamma\delta}F^{(2)}_{(\alpha\beta\gamma\delta)}=\tfrac{1}{3}(F^{(2)}_{\alpha\alpha\beta\beta}+2F^{(2)}_{\alpha\beta\beta\alpha})\,.$

So we demand that

$F^{(2)}_{\alpha\alpha\beta\beta}+2F^{(2)}_{\alpha\beta\beta\alpha}+\tfrac{6(3d-4-2k)}{d}A_{\alpha\alpha\beta\beta}^{(2)}=0\,.$

This has solutions so long as $(d,k)\neq(2,1)$, and thus we can always remove
the full trace of $F^{(2)}_{(\alpha\beta\gamma\delta)}$ in that setting. In
fact, when $(d,k)=(2,1)$, we may directly compute this full trace term (which
is in fact the only possibly obstruction, as $k=1$) and see that it vanishes.
However, as $A^{(2)}$ is not determined the total trace of $F^{(2)}$, we may
not proceed there.

Next we consider the trace-free symmetric part of the trace of the totally-
symmetric component. By the same considerations as before, this component is
given by

$\tfrac{1}{6}(F_{\gamma\gamma(\alpha\beta)_{\circ}}+4F_{\gamma(\alpha\beta)_{\circ}\gamma}+F_{(\alpha\beta)_{\circ}\gamma\gamma})\,.$

Thus, we demand that

(7.2) $\displaystyle
F_{\gamma\gamma(\alpha\beta)_{\circ}}+4F_{\gamma(\alpha\beta)_{\circ}\gamma}+F_{(\alpha\beta)_{\circ}\gamma\gamma}+18A^{(2)}_{\gamma\gamma(\alpha\beta)_{\circ}}+\tfrac{6(3d-2k-8)}{d}A^{(2)}_{(\alpha\beta)_{\circ}\gamma\gamma}=0\,.$

To fully determine both of the components of $A^{(2)}$ above, we can also
demand that the trace-free symmetric part of the trace of the  3,1 component
of $F^{(2)}$ vanishes. This component of $F^{(2)}$ can be expressed as

$\tfrac{1}{6}(F^{(2)}_{(\alpha\beta)_{\circ}\gamma\gamma}-F^{(2)}_{\gamma\gamma(\alpha\beta)_{\circ}})\,.$

So, we must have that

(7.3) $\displaystyle
F^{(2)}_{(\alpha\beta)_{\circ}\gamma\gamma}-F^{(2)}_{\gamma\gamma(\alpha\beta)_{\circ}}-6A^{(2)}_{\gamma\gamma(\alpha\beta)_{\circ}}+6A^{(2)}_{(\alpha\beta)_{\circ}\gamma\gamma}=0\,.$

Combining Equations (7.2) and (7.3), we find that the determinant of the
system is $\frac{72}{d}(3d-k-4)$, which never vanishes for $1\leqslant
k\leqslant d-1$. Thus, we can completely remove the trace-free symmetric
pieces of both  3,1 and  4 .

Finally, we wish to remove the antisymmetric part of the trace of the  3,1
component of $F^{(2)}$, which is given by
$F^{(2)}_{\gamma[\alpha\beta]\gamma}$. So long as $k\neq d-2$, this piece can
be removed as well. This accounts for all of the degrees of freedom in
$A^{(2)}$, and thus we are left with a completely undetermined  2,2 component
of $F^{(2)}$, precisely as in the Riemannian setting.

Having fixed $A^{(2)}$ and in so doing, forced
$F^{(2)}\in\Gamma(\mathcal{E}M[-2]\times\raisebox{2.5pt}{\scalebox{0.2}{\ydiagram{2,2}}})$,
it is now instructive to compute this obstruction. We begin by noting that

(7.4)
$\displaystyle\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+\left(P_{\scalebox{0.2}{\ydiagram{2,2}}}\,F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\right)\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\,.$

Furthermore, we have that

$P_{\scalebox{0.2}{\ydiagram{2,2}}}\,F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}=\tfrac{1}{3}\left(2F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}+2F^{(2)}_{\gamma_{1}\gamma_{2}\alpha\beta}-2F^{(2)}_{\gamma_{1}(\alpha\beta)\gamma_{2}}-2F^{(2)}_{\gamma_{2}(\alpha\beta)\gamma_{1}}\right)\,.$

Now consider the following projection:

$P_{\scalebox{0.2}{\ydiagram{2,2}}}\,N^{A}_{\gamma_{1}}N^{B}_{\gamma_{2}}\hat{D}_{A}\hat{D}_{B}\bm{G}_{\alpha\beta}\,.$

Using Equation (7.4) and simplifying, we find that

(7.5)
$\displaystyle\begin{split}P_{\scalebox{0.2}{\ydiagram{2,2}}}\,N^{A}_{\gamma_{1}}N^{B}_{\gamma_{2}}\hat{D}_{A}\hat{D}_{B}\bm{G}_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}2F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}&+\tfrac{2}{9(d-2)}\delta_{\alpha\beta}(F^{(2)}_{\rho\gamma_{1}\gamma_{2}\rho}+F^{(2)}_{\rho\gamma_{2}\gamma_{1}\rho}-F^{(2)}_{\gamma_{1}\gamma_{2}\rho\rho}-F^{(2)}_{\rho\rho\gamma_{1}\gamma_{2}})\\\
&+\tfrac{2}{9(d-2)}\delta_{\gamma_{1}\gamma_{2}}(F^{(2)}_{\rho\alpha\beta\rho}+F^{(2)}_{\rho\beta\alpha\rho}-F^{(2)}_{\alpha\beta\rho\rho}-F^{(2)}_{\rho\rho\alpha\beta})\\\
&+\tfrac{1}{9(d-2)}\delta_{\gamma_{1}\alpha}(F^{(2)}_{\gamma_{2}\beta\rho\rho}+F^{(2)}_{\rho\rho\gamma_{2}\beta}-F^{(2)}_{\rho\gamma_{2}\beta\rho}-F^{(2)}_{\rho\beta\gamma_{2}\rho})\\\
&+\tfrac{1}{9(d-2)}\delta_{\gamma_{2}\alpha}(F^{(2)}_{\gamma_{1}\beta\rho\rho}+F^{(2)}_{\rho\rho\gamma_{1}\beta}-F^{(2)}_{\rho\gamma_{1}\beta\rho}-F^{(2)}_{\rho\beta\gamma_{1}\rho})\\\
&+\tfrac{1}{9(d-2)}\delta_{\gamma_{1}\beta}(F^{(2)}_{\gamma_{2}\alpha\rho\rho}+F^{(2)}_{\rho\rho\gamma_{2}\alpha}-F^{(2)}_{\rho\gamma_{2}\alpha\rho}-F^{(2)}_{\rho\alpha\gamma_{2}\rho})\\\
&+\tfrac{1}{9(d-2)}\delta_{\gamma_{2}\beta}(F^{(2)}_{\gamma_{1}\alpha\rho\rho}+F^{(2)}_{\rho\rho\gamma_{1}\alpha}-F^{(2)}_{\rho\gamma_{1}\alpha\rho}-F^{(2)}_{\rho\alpha\gamma_{1}\rho})\,.\end{split}$

We may also compute the same projection using the fact that
$\bm{G}_{\alpha\beta}=N_{\alpha}\cdot N_{\beta}$. This computation is somewhat
involved, but results in

(7.6)
$\displaystyle\begin{split}P_{\scalebox{0.2}{\ydiagram{2,2}}}\,N^{A}_{\gamma_{1}}N^{B}_{\gamma_{2}}\hat{D}_{A}\hat{D}_{B}\bm{G}_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}&-2\beta_{a\gamma_{1}(\alpha}\beta^{a}_{\beta)\gamma_{2}}-\tfrac{2}{3}W_{\gamma_{1}(\alpha\beta)\gamma_{2}}\\\
&+\tfrac{2}{3(d-2)}\big{[}\delta_{\gamma_{1}\gamma_{2}}(-\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\beta}+2\beta_{a\alpha\rho}\beta^{a}_{\rho\beta})+\delta_{\alpha\beta}(-\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\gamma_{1}\gamma_{2}}+2\beta_{a\gamma_{1}\rho}\beta^{a}_{\rho\gamma_{2}})\\\
&\phantom{\tfrac{1}{3(d-2)}[}+\delta_{\gamma_{1}(\alpha}(\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\beta)\gamma_{2}}-2\beta^{a}_{\beta)\rho}\beta_{a\rho\gamma_{2}})+\delta_{\gamma_{2}(\alpha}(\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\beta)\gamma_{1}}-2\beta^{a}_{\beta)\rho}\beta_{a\rho\gamma_{1}})\big{]}\,.\end{split}$

Now, as traces are combined in non-trivial ways with the above expressions, it
is fruitful to compute the irreducible components of $F^{(2)}|_{\Lambda}$
independently. To do so, first observe that

$Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}:=\delta^{\alpha\beta}\delta^{\gamma\delta}P_{\scalebox{0.2}{\ydiagram{2,2}}}\,F^{(2)}_{\alpha\beta\gamma\delta}=\tfrac{2}{3}(F^{(2)}_{\alpha\alpha\beta\beta}-F^{(2)}_{\alpha\beta\beta\alpha})\,.$

Evaluating this trace combination on Equation (7.5), we have that

$Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}N^{A}_{\gamma_{1}}N^{B}_{\gamma_{2}}\hat{D}_{A}\hat{D}_{B}\bm{G}_{\alpha\beta}=\tfrac{4(3d-2k-4)}{9(d-2)}(F^{(2)}_{\alpha\alpha\beta\beta}-F^{(2)}_{\alpha\beta\beta\alpha})=\tfrac{2(3d-2k-4)}{3(d-2)}Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}\,.$

But we can also take the double trace of Equation (7.6). Doing so yields

$Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}\,N^{A}_{\gamma_{1}}N^{B}_{\gamma_{2}}\hat{D}_{A}\hat{D}_{B}\bm{G}_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}-\tfrac{4(k-1)}{3(d-2)}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\alpha}+\tfrac{2(3d-4k-2)}{3(d-2)}\beta_{a\alpha\beta}\beta^{a}_{\alpha\beta}-\tfrac{2}{3}W_{n_{\alpha}n_{\beta}n_{\beta}n_{\alpha}}\,.$

Thus, we have that

$\displaystyle
Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}\stackrel{{\scriptstyle\Lambda}}{{=}}-\tfrac{2(k-1)}{3d-2k-4}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\alpha}+\tfrac{3d-4k-2}{3d-2k-4}\beta_{a\alpha\beta}\beta^{a}_{\alpha\beta}-\tfrac{d-2}{3d-2k-4}W_{n_{\alpha}n_{\beta}n_{\beta}n_{\alpha}}\,.$

In general, this obstruction $Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}$
depends in a non-trivial way on the parallelization of $N\Lambda$ through the
appearance of $\beta$, however for special choices of $(d,k)$, we find that
this obstruction is a true submanifold invariant that does not depend on the
choice of orthonormal frame. This follows because $\hskip
1.13809pt\mathring{\rm{II}}{}_{\alpha}$ and $n_{\alpha}$ both occupy tensor
representations of $O(k)$, unlike $\beta$. This occurs when $3d-4k-2=0$, i.e.
for any integer $n\in\mathbb{N}$, then
$Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}$ is a true invariant so long
as $(d,k)=(4n+2,3n+1)$.

We next consider the tracefree part of the trace of $F^{(2)}$. As above, it is
useful to compute which component this is in terms of $F^{(2)}$ prior to
projection. There, we have that

$\left[\left(\operatorname{t.f.}\circ
Tr_{\scalebox{0.2}{\ydiagram{2,2}}}\right)F^{(2)}\right]_{\alpha\beta}:=\tfrac{1}{3}F^{(2)}_{(\alpha\beta)_{\circ}\rho\rho}+\tfrac{1}{3}F^{(2)}_{\rho\rho(\alpha\beta)_{\circ}}-\tfrac{2}{3}F^{(2)}_{\rho(\alpha\beta)_{\circ}\rho}\,,$

where $\operatorname{t.f.}$ is a projection operator to the trace-free part of
a tensor. Taking this trace of Equations (7.5) and (7.6), we find that

$\left[\left(\operatorname{t.f.}\circ
Tr_{\scalebox{0.2}{\ydiagram{2,2}}}\right)F^{(2)}\right]_{\alpha\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}-\tfrac{k-2}{3(3d-k-4)}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{(\alpha\beta)_{\circ}}-\tfrac{3d-2k-2}{3(3d-k-4)}\beta_{a\rho(\alpha}\beta^{a}_{\beta)_{\circ}\rho}-\tfrac{d-2}{3(3d-k-4)}W_{n_{\rho}n_{(\alpha}n_{\beta)_{\circ}}n_{\rho}}\,.$

Unlike the double trace obstruction, there are no pairs $(d,k)$ such that this
is a true extrinsic submanifold invariant. The final component of $F^{(2)}$,
the trace-free part, can be read off from Equations (7.5) and (7.6).

Now, given that $F^{(2)}$ occupies the window tensor representation, we have
that $F^{(2)}_{\alpha\beta\gamma\delta}=F^{(2)}_{\gamma\delta\alpha\beta}$ and
that $F^{(2)}_{(\alpha\beta\gamma)\delta}=0$. It follows that
$F^{(2)}_{\alpha\beta\gamma\delta}+2F^{(2)}_{\gamma(\alpha\beta)\delta}=0$,
and so

$Tr_{\scalebox{0.2}{\ydiagram{2,2}}}^{2}F^{(2)}=\tfrac{1}{3}F^{(2)}_{\alpha\alpha\beta\beta}\qquad\text{and}\qquad\left[\left(\operatorname{t.f.}\circ
Tr_{\scalebox{0.2}{\ydiagram{2,2}}}\right)F^{(2)}\right]_{\alpha\beta}=\tfrac{2}{3}F^{(2)}_{\rho\rho(\alpha\beta)_{\circ}}\,.$

We now summarize all of these results in the following proposition:

######  Proposition 7.2.

Let $\Lambda^{d-k}\hookrightarrow(M^{d},\bm{c})$ be a parallelized submanifold
embedding with $k\neq d-2$. Then there exists a family of defining densities
$\sigma_{\alpha}$ for $\Lambda$ that agree modulo terms of order $\sigma^{4}$
such that

$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\,,$

where $F^{(2)}\in
C^{\infty}M\times\raisebox{3.0pt}{\scalebox{0.2}{\ydiagram{2,2}}}$, and

$\operatorname{t.f.}F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\stackrel{{\scriptstyle\Lambda}}{{=}}\operatorname{t.f.}\left(-\beta_{a\gamma_{1}(\alpha}\beta^{a}_{\beta)\gamma_{2}}-\tfrac{1}{3}W_{\gamma_{1}(\alpha\beta)\gamma_{2}}\right)\,,$
$F^{(2)}_{\rho\rho(\alpha\beta)_{\circ}}\stackrel{{\scriptstyle\Lambda}}{{=}}-\tfrac{k-2}{2(3d-k-4)}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{(\alpha\beta)_{\circ}}-\tfrac{3d-2k-2}{2(3d-k-4)}\beta_{a\rho(\alpha}\beta^{a}_{\beta)_{\circ}\rho}-\tfrac{d-2}{2(3d-k-4)}W_{n_{\rho}n_{(\alpha}n_{\beta)_{\circ}}n_{\rho}}\,,$

and

$F^{(2)}_{\alpha\alpha\beta\beta}\stackrel{{\scriptstyle\Lambda}}{{=}}-\tfrac{6(k-1)}{3d-2k-4}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\alpha}+\tfrac{3(3d-4k-2)}{3d-2k-4}\beta_{a\alpha\beta}\beta^{a}_{\alpha\beta}-\tfrac{3(d-2)}{3d-2k-4}W_{n_{\alpha}n_{\beta}n_{\beta}n_{\alpha}}\,.$

We call these defining densities associated defining densities for
$\Lambda^{d-k}\hookrightarrow(M^{d},g)$.

We now consider the case where $k=d-2$. As observed above, we may not set
$F^{(2)}_{\gamma[\alpha\beta]\gamma}$ to zero, and hence this term obstructs
setting the Gram matrix to the identity. Indeed, we can compute this
obstruction and find that

$F^{(2)}_{\gamma[\alpha\beta]\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}\bar{\nabla}^{a}\beta_{a\alpha\beta}\,.$

As noted in Section 3.3, when $\Lambda$ is closed and $N\Lambda$ admits a
parallelization, there exists such a parallelization such that this
obstruction vanishes. Further, this is only conformally invariant for $k=d-2$:
the divergence of a weight-0 1-form is only conformally invariant on surfaces.
Regardless, we find that $\sigma_{\alpha}$ cannot be uniquely fixed to third
order, as there is no constraint equation controlling
$A^{(2)}_{[\alpha\beta]\rho\rho}$. To that end, when $k=d-2$, we define an
equivalence class of defining densities, given by the following relation:

$[\sigma_{\alpha}]\sim[\sigma_{\alpha}+A_{[\alpha\gamma]}\sigma_{\gamma}\sigma_{\rho}\sigma_{\rho}]\,,$

for any
$A_{[\alpha\beta]}\in\Gamma(\mathcal{E}M[-2]\times\raisebox{2.5pt}{\scalebox{0.2}{\ydiagram{1,1}}})$.

Note that despite the trouble that this sporadic obstruction provides, the
remainder of the obstruction still occupies a window tensor representation and
the projections to its components are unchanged from the $k\neq d-2$ case.

###  7.3. Third Order

While there is an obstruction at second order, we can still ask which
components at third order can be removed by a judicious choice of
$\sigma_{\alpha}$. As the case where $k=d-2$ is qualitatively different from
that of $k\neq d-2$, we consider these situations separately, beginning with
the case where $k\neq d-2$. We will follow the method applied in the
Riemannian case. To that end, our goal will be to show that there exists such
a defining density and then establish that this defining density is
independent (to sufficiently higher order) on the initial choice of $F^{(1)}$.

We can now proceed to extend $\sigma_{\alpha}$ as in the Riemannian setting.

$\sigma_{\alpha}\mapsto\tilde{\sigma}_{\alpha}:=\sigma_{\alpha}+A^{(3)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}\sigma_{\gamma_{4}}\,,$

so that the Gram matrix takes the form

(7.7)
$\displaystyle\begin{split}\tilde{\bm{G}}_{\alpha\beta}=\delta_{\alpha\beta}&+F^{(2)}_{\alpha\beta\gamma_{1}\gamma_{2}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\\\
&+\left[F^{(3)}_{\alpha\beta\gamma_{1}\gamma_{2}\gamma_{3}}+8A^{(3)}_{(\alpha\beta)\gamma_{1}\gamma_{2}\gamma_{3}}-\tfrac{24}{d}\delta_{\gamma_{1}(\alpha}A^{(3)}_{\beta)\gamma\gamma\gamma_{2}\gamma_{3}}\right]\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}\,.\end{split}$

Now we could, in principle, perform the same representation theory analysis as
in the second order case. Surely, we may force the trace-free part of
$\tilde{F}^{(3)}$ to occupy a generalized window symmetry, as that calculation
amounts to a rehashing of the Riemannian case. However, computations involving
the representation theory of the remaining components quickly get out of hand.
Instead, we are only interested when there are obstructions, special or
otherwise. Because $\tilde{F}^{(3)}$ has five Greek indices, there are no
total trace terms, so the terms we will examine are terms that have one Greek
index, i.e. are the result of a double trace.

First, observe that when $k=1$, there is only one maximal trace term,
$F^{(3)}_{11111}$, and only one term that contributes to the correction,
$A^{(3)}_{11111}$. As we are considering the case where $k\neq d-2$, this
means that $d\neq 3$. So in this setting, it is easy to see that we may remove
$F^{(3)}_{11111}$ in its entirety.

When $k\neq 1$, it is useful to consider the three distinct traces we may
obtain at third order that can appear in Equation (7.7). Tracing, we obtain:

$\displaystyle\tilde{F}^{(3)}_{\gamma\gamma\rho\rho\alpha}=F^{(3)}_{\gamma\gamma\rho\rho\alpha}$
$\displaystyle-\tfrac{8}{d}A^{(3)}_{\alpha\gamma\gamma\rho\rho}+\tfrac{8(d-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}$
$\displaystyle\tilde{F}^{(3)}_{\gamma\rho\rho\gamma\alpha}=F^{(3)}_{\gamma\rho\rho\gamma\alpha}$
$\displaystyle+\tfrac{8(d-k-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}$
$\displaystyle\tilde{F}^{(3)}_{\alpha\gamma\gamma\rho\rho}=F^{(3)}_{\alpha\gamma\gamma\rho\rho}$
$\displaystyle+\tfrac{4(d-k-3)}{d}A^{(3)}_{\alpha\gamma\gamma\rho\rho}+\tfrac{4(d-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}\,.$

The vanishing of these expressions clearly leads to an overdetermined system,
so we must choose at least one trace that is unresolved by even the most
judicious choice of $A^{(3)}$. In this case, we choose
$F^{(3)}_{\gamma\rho\rho\gamma\alpha}$ to always be unfixed for reasons that
will become clear later. In general, we can solve the remaining system.
Suppose we let $\sigma_{\alpha}\mapsto\tilde{\sigma}_{\alpha}$ such that
$\tilde{F}^{(3)}_{\gamma\gamma\rho\rho\alpha}$ vanishes (this is always
possible)—we do this by performing an arbitrary total-trace transformation of
$\sigma_{\alpha}$ and then fixing $A^{(3)}_{\alpha\gamma\gamma\rho\rho}$ to
depend on $F^{(3)}_{\gamma\gamma\rho\rho\alpha}$ and an arbitrary choice of
$A^{(3)}_{\gamma\gamma\rho\rho\alpha}$ such that

${F}^{(3)}_{\gamma\gamma\rho\rho\alpha}\mapsto\tilde{F}^{(3)}_{\gamma\gamma\rho\rho\alpha}=0\,.$

Under the same modification of $\sigma_{\alpha}$, we have that

$F^{(3)}_{\alpha\gamma\gamma\rho\rho}\mapsto\tilde{F}^{(3)}_{\alpha\gamma\gamma\rho\rho}=F^{(3)}_{\alpha\gamma\gamma\rho\rho}+\tfrac{d-k-3}{2}F^{(3)}_{\gamma\gamma\rho\rho\alpha}+\tfrac{4(d-2)(d-k-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}\,.$

Thus, for $k\neq d-2$, we can fix $A^{(3)}_{\gamma\gamma\rho\rho\alpha}$ so
that $\tilde{F}^{(3)}_{\alpha\gamma\gamma\rho\rho}=0$. It is entirely
expected, however, that we would not be able to remove all of the maximal
traces—we expect that at least an entire generalized window’s worth of
obstructions will appear at third order, and that includes precisely one
maximal trace term.

We now examine whether the construction provided so far is independent of the
choice of $F^{(1)}$ by mimicking the argument in Section 4.3. To do so,
suppose that we had chosen two distinct extensions of $F^{(1)}|_{\Lambda}$,
call them $F^{(1)}$ and $F^{(1)^{\prime}}$. When $k\neq d-2$, these choices
lead to two (naively) distinct defining densities $\sigma_{\alpha}$ and
$\sigma_{\alpha}^{\prime}$ that are determined modulo terms of fourth order.
However, as their differentials must agree in any choice of metric
representative along $\Lambda$, we must have that they differ at second order:

$\sigma_{\alpha}^{\prime}=\sigma_{\alpha}+M^{(1)}_{\alpha\gamma_{1}\gamma_{2}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\,.$

Notice that this procedure is just like modifying $\sigma_{\alpha}$ so that we
may cancel $F^{(1)}$. The only distinction here is that $F^{(1)}$ is already
established as vanishing, so we need only ensure that this uniquely fixes
$M^{(1)}|_{\Lambda}=0$. Indeed, so long as $k\neq d$, this is the case, by
considerations in Section 7.1.

Similarly, the same style of argument applies to check that $\sigma_{\alpha}$
and $\sigma_{\alpha}^{\prime}$ agree to third order. Again, we find that so
long as $k\neq d-2$, $\sigma_{\alpha}$ is independent of the choice of
$F^{(1)}$ modulo terms of order $\sigma^{4}$—the only pieces of $F^{(2)}$ that
may be affected by a change in $\sigma$ at third order are precisely those
terms that we forced to vanish in $F^{(2)}$. Hence, we must have that
$M^{(2)}|_{\Lambda}=0$.

While we do not proceed with the irreducible representation at third order (as
the calculations get very messy but remain uninteresting), it is still clear
that such a decomposition is possible. Thus, one may choose $A^{(3)}$ to
eliminate all but a preferred set of components of $F^{(3)}$. As described
above, one preferred component (when $k\neq d-2$) might be
$F^{(3)}_{\gamma\rho\rho\gamma\alpha}$. Regardless of the choices made, so
long as $\sigma_{\alpha}$ and $\sigma_{\alpha}^{\prime}$ are constructed so
that $F^{(3)}$ and $F^{(3)^{\prime}}$ have the same vanishing components,
$M^{(3)}$ is necessarily fixed to zero. This implies that the non-vanishing
components of $F^{(3)}|_{\Lambda}$ are independent of the choice of $F^{(1)}$
and $\sigma_{\alpha}$ is fixed modulo terms of order $\sigma^{5}$.

We now turn our attention to the case where $k=d-2$. Rather than dealing with
a unique $\sigma_{\alpha}$ for a given choice of $F^{(1)}$, we instead deal
with a unique equivalence class $[\sigma_{\alpha}]$. We will attempt to fix
the behavior of our equivalence class at fourth order, according to

$[\sigma_{\alpha}]\mapsto[\sigma_{\alpha}+A^{(3)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}\sigma_{\gamma_{4}}]\,.$

To enable us to work concretely with this equivalence class, we will pick an
arbitrary representative $\sigma_{\alpha}$ and then parametrize the
equivalence class by the arbitrary set of functions $A$. That is, we consider

$\tilde{\sigma}_{\alpha}:=\sigma_{\alpha}+A_{[\alpha\gamma]}\sigma_{\gamma}\sigma_{\rho}\sigma_{\rho}+A^{(3)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{3}\gamma_{4}}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}\sigma_{\gamma_{4}}\,,$

where $A$ is treated as a free (function-valued) set of parameters. Then, we
have that

$\tilde{\bm{G}}_{\alpha\beta}=\bm{G}_{\alpha\beta}+\Big{(}8A^{(3)}_{(\alpha\beta)\gamma_{1}\gamma_{2}\gamma_{3}}-\tfrac{24}{d}\delta_{\gamma_{1}(\alpha}A^{(3)}_{\beta)\rho\rho\gamma_{2}\gamma_{3}}\\\
+2\delta_{\gamma_{2}\gamma_{3}}N_{(\alpha}\cdot\hat{D}A_{\beta)\gamma_{1}}+\tfrac{8}{d}N_{\gamma_{3}}\cdot\hat{D}A_{\gamma_{2}(\alpha}\delta_{\beta)\gamma_{1}}+\tfrac{4}{d}N_{\rho}\cdot\hat{D}A_{\rho(\alpha}\delta_{\beta)\gamma_{1}}\delta_{\gamma_{2}\gamma_{3}}\Big{)}\sigma_{\gamma_{1}}\sigma_{\gamma_{2}}\sigma_{\gamma_{3}}$

We wish to examine the maximal trace terms again. These take the form

$\displaystyle F^{(3)}_{\gamma\gamma\rho\rho\alpha}$
$\displaystyle-\tfrac{8}{d}A^{(3)}_{\alpha\gamma\gamma\rho\rho}+\tfrac{8(d-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}-\tfrac{2(d+2)}{3}N_{\gamma}\cdot\hat{D}A_{\alpha\gamma}$
$\displaystyle F^{(3)}_{\gamma\rho\rho\gamma\alpha}$ $\displaystyle
F^{(3)}_{\alpha\gamma\gamma\rho\rho}$
$\displaystyle-\tfrac{4}{d}A^{(3)}_{\alpha\gamma\gamma\rho\rho}+\tfrac{4(d-2)}{d}A^{(3)}_{\gamma\gamma\rho\rho\alpha}-\tfrac{d+2}{3}N_{\gamma}\cdot\hat{D}A_{\alpha\gamma}\,.$

Evidently, there is no choice of $A^{(3)}$ that can eliminate
$F^{(3)}_{\gamma\rho\rho\gamma\alpha}$, regardless of the representative that
was chosen. Furthermore, so long as $F^{(3)}_{\gamma\gamma\rho\rho\alpha}\neq
2F^{(3)}_{\alpha\gamma\gamma\rho\rho}$, this system is inconsistent, and hence
we may only choose to set one of these two maximal trace terms to zero. For
consistency with the case where $k\neq d-2$, we can always fix $A^{(3)}$ so
that
$F^{(3)}_{\gamma\gamma\rho\rho\alpha}\mapsto\tilde{F}^{(3)}_{\gamma\gamma\rho\rho\alpha}=0$—this
choice is made so that we may compute a Willmore-like obstruction explicitly,
as will be shown in the next section. In that case, we have that

$\tilde{F}^{(3)}_{\alpha\gamma\gamma\rho\rho}=F^{(3)}_{\alpha\gamma\gamma\rho\rho}-\tfrac{1}{2}F^{(3)}_{\gamma\gamma\rho\rho\alpha}\,.$

Importantly, $\tilde{F}^{(3)}_{\alpha\gamma\gamma\rho\rho}$ is independent of
the choice of representative. Furthermore (as in the argument for the case
where $k\neq d-2$), for different choices of extensions of $F^{(1)}$, if both
$[\sigma_{\alpha}]$ and $[\sigma_{\alpha}^{\prime}]$ are constructed so that
$F^{(3)}_{\gamma\gamma\rho\rho\alpha}$ and
$F^{(3)^{\prime}}_{\gamma\gamma\rho\rho\alpha}$ vanish, then $M^{(3)}$ cannot
affect $F^{(3)}_{\alpha\gamma\gamma\rho\rho}|_{\Lambda}$. It thus follows that
this obstruction is independent of the choice of $F^{(1)}$, and thereby is a
true invariant of the parallelized submanifold embedding.

Observe that this part of $F^{(3)}$ is only an obstruction when $k=d-2$. This
observation is key and suggests that this component of $F^{(3)}$ is a special
invariant of the system. Furthermore, the uniqueness of this obstruction is
fortuitous, as it is precisely this term that mimicks the obstruction density
at order $d$ that occurs when this same procedure is applied to hypersurfaces
[18]. In fact, for $d=3$ and $k=1$, this is precisely the celebrated Willmore
invariant, as we will see in the following section. This suggests that for all
$\Lambda^{2}\hookrightarrow(M^{d},\bm{c})$, we have that
$F^{(3)}_{\alpha\gamma\gamma\rho\rho}$ represents a parallelized submanifold
Willmore invariant.

##  8\. A Holographic Submanifold Willmore Invariant

As noted above, when $d-k=2$, $F^{(3)}$ has a special codimension-dependent
obstruction that can be forced into one of its maximal traces, namely
$F^{(3)}_{\alpha\gamma\gamma\rho\rho}$. In this section, we provide a
holographic formula for this obstruction and compute it explicitly.

Drawing inspiration from [18], we can construct an extrinsically-coupled
conformal Laplacian on $\Lambda$. Indeed, this conformal Laplacian exists on
all tractor bundles with weight $\tfrac{2+k-d}{2}$. This is captured by the
following straightforward result:

######  Theorem 8.1.

Let $\Lambda^{d-k}\hookrightarrow(M^{d},\bm{c})$ be a conformal submanifold
embedding with some conformal defining map that satisfies
$\bm{G}_{\alpha\beta}=\delta_{\alpha\beta}+\mathcal{O}(\sigma^{2})$. Then, the
operator

$P_{2}:=N_{\alpha}\cdot\hat{D}N_{\alpha}\cdot
D:\Gamma\left(\mathcal{T}^{\Phi}M\left[\tfrac{2+k-d}{2}\right]\right)|_{\Lambda}\rightarrow\Gamma\left(\mathcal{T}^{\Phi}M\left[\tfrac{-2+k-d}{2}\right]\right)|_{\Lambda}$

is tangential and has principal symbol $-k\Delta^{\top}$, where
$\Delta^{\top}:=\bm{g}^{ab}\nabla^{\top}_{a}\nabla^{\top}_{b}$ and
$\nabla^{\top}$ is the tangential tractor-coupled Levi-Civita connection.

###### Proof.

Explicit computation in a choice of metric representative $g\in\bm{c}$ yields

$\displaystyle N_{\alpha}\cdot\hat{D}N_{\alpha}\cdot D$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\left[\nabla_{n_{\alpha}}+(w-1)\rho_{\alpha}\right]\circ\left[(d+2w-2)(\nabla_{n_{\alpha}}+w\rho_{\alpha})-\sigma_{\alpha}(\Delta+wJ)\right]$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}-k\left(\Delta^{\top}+\tfrac{d-k-2}{2}(\nabla_{n_{\alpha}}\rho_{\alpha})-\tfrac{d-k-2}{2}J-\tfrac{(d-k)(d-k-2)}{4}H_{\alpha}H_{\alpha}\right)\,.$

Evidently, this operator is tangential. ∎

######  Remark 8.2.

Note that this operator $P_{2}$ is a submanifold generalization of the
extrinsic conformal Laplacian described in [18].

We are now equipped to provide a holographic formula for
$F^{(3)}_{\alpha\gamma\gamma\rho\rho}$. From a straightforward computation,
one finds that

$\displaystyle\begin{split}N_{\gamma}\cdot\hat{D}N_{\gamma}\cdot\hat{D}N_{\alpha}^{A}\stackrel{{\scriptstyle\Lambda}}{{=}}&-\tfrac{3}{d-2}X^{A}F^{(3)}_{\alpha\gamma_{1}\gamma_{1}\gamma_{2}\gamma_{2}}+\tfrac{1}{d-2}X^{A}N_{\gamma}\cdot\hat{D}K_{\gamma\alpha}-N_{\gamma}\cdot\hat{D}B^{A}_{\alpha\gamma}\\\
&+\tfrac{1}{d-2}N^{A}_{\gamma}K_{\gamma\alpha}+N^{A}_{\gamma_{1}}F^{(2)}_{\alpha\gamma_{2}\gamma_{2}\gamma_{1}}-\tfrac{1}{d-2}N^{A}_{\gamma_{1}}F^{(2)}_{\alpha\gamma_{1}\gamma_{2}\gamma_{2}}\,.\end{split}$

Because we are interested in $F^{(3)}$, it is sensible to only consider the
projection of the above display to the tractor bundle on $\Lambda$. As such,
we have

(8.1) $\displaystyle\begin{split}\top
P_{2}N_{\alpha}^{A}\stackrel{{\scriptstyle\Lambda}}{{=}}&-3X^{A}F^{(3)}_{\alpha\gamma_{1}\gamma_{1}\gamma_{2}\gamma_{2}}+X^{A}N_{\gamma}\cdot\hat{D}K_{\gamma\alpha}-(d-2)\top(N_{\gamma}\cdot\hat{D}B^{A}_{\alpha\gamma})\,,\end{split}$

where $\top$ also refers to the projection to the submanifold tractor bundle,
such that $N^{A}_{\alpha}T^{\top}_{A}\stackrel{{\scriptstyle\Lambda}}{{=}}0$
for any tractor $T$, i.e. $\top T^{A}:=T^{A}-N^{A}_{\alpha}N_{\alpha}\cdot T$.
From the principal symbol calculation of $P_{2}$ and the fact that
$F^{(3)}_{\alpha\gamma_{1}\gamma_{1}\gamma_{2}\gamma_{2}}$ appears as a
coefficient for $X_{A}$, we have that the leading structure of this maximal-
trace obstruction is $\bar{\Delta}H_{\alpha}$—as expected for a Willmore
invariant. So, we may indeed use this obstruction to define a holographic
Willmore invariant for a parallelized surface embedded in $d$ dimensions.

To explicitly compute this Willmore invariant requires more care. Indeed, if
one were to calculate naively, one would realize that this computation
involves tractor expressions such as

$N_{\gamma}\cdot\hat{D}F^{(2)}_{\omega[\gamma\alpha]\omega}\,.$

For a given choice of $F^{(1)}$, these terms are uniquely determined, as
$F^{(2)}$ is determined to all orders by $F^{(1)}$ and $A^{(1)}$ (which itself
is fixed in terms of $F^{(1)}$). Furthermore, as $P_{2}$ is a tangential
operator and $F^{(3)}_{\alpha\gamma\gamma\rho\rho}$ is a parallelized
submanifold invariant, the dependence of such terms on the choice of $F^{(1)}$
must either cancel with one another or themselves vanish. Thus, such terms may
simply be absorbed into $F^{(3)}$ by writing schematically

$F^{(2)}=F^{(2c)}+F^{(2,1)}\sigma\,,$

where

$F^{(2c)}:=F^{(2)}-\sigma_{\alpha}N^{A}_{\alpha}\hat{D}_{A}F^{(2)}\,.$

Then, $N_{\alpha}\cdot\hat{D}F^{(2c)}\stackrel{{\scriptstyle\Lambda}}{{=}}0$,
and the terms involving $F^{(2,1)}$ are simply absorbed into $F^{(3)}$. While
this may seem like we are changing the obstruction density, this is equivalent
to simply choosing another $F^{(1)}$, as doing so simply reshuffles pieces of
$F^{(2)}$ off $\Lambda$ into $F^{(3)}$. However, as
$F^{(3)}_{\alpha\gamma\gamma\rho\rho}|_{\Lambda}$ is independent of $F^{(1)}$,
this choice may be made with no ill-effects. (Note that this is the beginning
of a solution to an extension problem which will be discussed in greater
detail in Section 9.) Thus, for the sake of the computation that follows, we
may drop terms involving $N_{\alpha}\cdot\hat{D}F^{(2)}$.

To explicitly compute $F^{(3)}_{\alpha\gamma\gamma\rho\rho}$, we must first
handle $\top N_{\gamma}\cdot\hat{D}B^{A}_{\alpha\gamma}$.

$\displaystyle 2\top(N_{\gamma}\cdot\hat{D}B^{A}_{\alpha\gamma})$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left[N_{\gamma}\cdot\hat{D}(N_{\alpha}\cdot\hat{D}N^{A}_{\gamma}-N_{\gamma}\cdot\hat{D}N^{A}_{\alpha})\right]$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}+[N_{\gamma}\cdot\hat{D},N_{\alpha}\cdot\hat{D}]N^{A}_{\gamma}+N_{\alpha}\cdot\hat{D}N_{\gamma}\cdot\hat{D}N^{A}_{\gamma}\right)$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}+W_{\gamma\alpha}{}^{A}{}_{\gamma}+2B_{B\gamma\alpha}P^{BA}_{\gamma}+N_{\alpha}\cdot\hat{D}N_{B\gamma}P^{AB}_{\gamma}\right)$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}+W_{\gamma\alpha}{}^{A}{}_{\gamma}+2B_{B\gamma\alpha}P^{BA}_{\gamma}+N_{\alpha}\cdot\hat{D}(B^{A}_{\gamma\gamma}+\tfrac{1}{2}\hat{D}^{A}G_{\gamma\gamma}+\tfrac{1}{d-2}X^{A}K_{\gamma\gamma})\right)$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}+W_{\gamma\alpha}{}^{A}{}_{\gamma}+2B_{B\gamma\alpha}P^{BA}_{\gamma}+N_{\alpha}\cdot\hat{D}(\tfrac{1}{2}\hat{D}^{A}G_{\gamma\gamma}+\tfrac{1}{d-2}X^{A}K_{\gamma\gamma})\right)$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}+W_{\gamma\alpha}{}^{A}{}_{\gamma}+2B_{B\gamma\alpha}P^{BA}_{\gamma}+\tfrac{1}{2}N_{\alpha}\cdot\hat{D}\hat{D}^{A}G_{\gamma\gamma}+\tfrac{1}{d-2}X^{A}N_{\alpha}\cdot\hat{D}K_{\gamma\gamma}\right)\,.$

In the above computations, we have used the fact that
$\hat{D}_{A}X_{B}=h_{AB}$. Now, to complete this calculation, we examine the
term $\top N_{\alpha}\cdot\hat{D}\hat{D}^{A}G_{\gamma\gamma}$. To do so, we
recall that we may drop terms of the form $N_{\alpha}\cdot\hat{D}F^{(2)}$
along $\Lambda$. Thus, we have:

$\displaystyle\top
N_{\alpha}\cdot\hat{D}\hat{D}^{A}G_{\gamma\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle\top
N_{\alpha}\cdot\hat{D}\hat{D}^{A}(F^{(2)}_{\gamma\gamma\beta_{1}\beta_{2}}\sigma_{\beta_{1}}\sigma_{\beta_{2}}+F^{(3)}_{\gamma\gamma\beta_{1}\beta_{2}\beta_{3}}\sigma_{\beta_{1}}\sigma_{\beta_{2}}\sigma_{\beta_{3}})$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle\top
N_{\alpha}\cdot\hat{D}\left(2\sigma_{(\beta_{1}}N^{A}_{\beta_{2})}F^{(2)}_{\gamma\gamma\beta_{1}\beta_{2}}-\tfrac{2}{d-2}X^{A}\left[G_{\beta_{1}\beta_{2}}F^{(2)}_{\gamma\gamma\beta_{1}\beta_{2}}+2\sigma_{(\beta_{1}}N_{\beta_{2})}\cdot\hat{D}F^{(2)}_{\gamma\gamma\beta_{1}\beta_{2}}\right]\right)$
$\displaystyle+\top
N_{\alpha}\cdot\hat{D}\left(-\tfrac{6}{d-2}X^{A}\sigma_{(\beta_{1}}\bm{G}_{\beta_{2}\beta_{3})}F^{(3)}_{\gamma\gamma\beta_{1}\beta_{2}\beta_{3}}+\mathcal{O}(\sigma^{2})\right)$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-\tfrac{6}{d-2}X^{A}\bm{G}_{\alpha(\beta_{1}}\bm{G}_{\beta_{2}\beta_{3})}F^{(3)}_{\gamma\gamma\beta_{1}\beta_{2}\beta_{3}}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-\tfrac{6}{d-2}X^{A}F^{(3)}_{\gamma\gamma\alpha\rho\rho}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle 0\,.$

In this computation we used the fact that
$F^{(3)}_{\gamma\gamma\alpha\rho\rho}|_{\Lambda}$ is the only maximal trace
that we have chosen to remove in general.

define the top notation

Combining the above two results with the formula for
$\top(N_{\gamma}\cdot\hat{D})^{2}N^{A}_{\alpha}$, we have thus obtained a
formula for the obstruction in terms of a tangential operator and lower-order
terms:

(8.2) $\displaystyle
X^{A}F^{(3)}_{\alpha\gamma\gamma\rho\rho}\stackrel{{\scriptstyle\Lambda}}{{=}}\top\left(-\tfrac{1}{6}P_{2}N^{A}_{\alpha}-\tfrac{d-2}{6}W_{\gamma\alpha}{}^{A}{}_{\gamma}-\tfrac{d-2}{3}B_{B\gamma\alpha}P^{BA}_{\gamma}+\tfrac{1}{3}X^{A}N_{\gamma}\cdot\hat{D}K_{\gamma\alpha}-\tfrac{1}{6}X^{A}N_{\alpha}\cdot\hat{D}K_{\gamma\gamma}\right)\,.$

To express the obstruction in terms of Riemannian quantities, we require a few
more calculations. Note that both
$P_{2}N^{A}_{\alpha}\stackrel{{\scriptstyle\Lambda}}{{=}}\nabla^{\top
b}\nabla^{\top}_{b}N^{A}_{\alpha}$ and
$B_{B\gamma\alpha}P^{BA}_{\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}\beta^{b}_{\gamma\alpha}\nabla^{\top}_{b}N^{A}_{\gamma}$
both contain $\nabla^{\top}N_{\alpha}$, so we compute:

(8.3)
$\displaystyle\nabla^{\top}_{a}N_{B\alpha}\stackrel{{\scriptstyle\Lambda,g}}{{=}}\begin{pmatrix}0\\\
\nabla^{\top}_{a}n_{b\alpha}-\bar{g}_{ab}H_{\alpha}\\\
-\bar{\nabla}_{a}H_{\alpha}-\bar{g}_{ab}P^{bc}n^{c}_{\alpha}\end{pmatrix}\,.$

Further computation of $P_{2}N$ is tedious but straightforward: a simple
application of the projected tractor connection suffices.

Furthermore, computation of the last two terms can also be performed
explicitly by noting that

$N_{\alpha}\cdot\hat{D}K_{\beta\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}(\nabla_{n_{\alpha}}+2H_{\alpha})\left[(\nabla_{a}n_{b\beta})(\nabla^{a}n^{b}_{\gamma})-d\rho_{\beta}\rho_{\gamma}+2s_{(\beta}P^{ab}\nabla_{a}n_{b\gamma)}\right]\,.$

We may thus calculate:

(8.4) $\displaystyle
N_{\alpha}\cdot\hat{D}K_{\beta\gamma}\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle
2(\nabla^{a}n^{b}_{(\beta})\nabla_{|\alpha|}\nabla_{a}n_{b\gamma)}+2dH_{(\beta}\nabla_{|\alpha|}\rho_{\gamma)}+2\delta_{\alpha(\beta}P^{ab}\nabla_{a}n_{b\gamma)}+2H_{\alpha}K_{\beta\gamma}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle 2({\rm
II}^{ab}_{(\beta}+n^{a}_{\rho}\beta^{b}_{\rho(\beta}+n^{b}_{\rho}\beta^{a}_{\rho(\beta}+n^{a}_{\rho}n^{b}_{\rho}H_{(\beta})\nabla_{|\alpha|}\nabla_{a}n_{b\gamma)}+2dH_{(\beta}\nabla_{|\alpha|}\rho_{\gamma)}$
$\displaystyle+2\delta_{\alpha(\beta}P^{ab}({\rm
II}_{ab\gamma)}+n_{a|\rho}\beta_{b\rho|\gamma)}+n_{b|\rho}\beta_{a\rho|\gamma)}+n_{a|\rho}n_{b\rho|}H_{\gamma)})+2H_{\alpha}K_{\beta\gamma}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle 2({\rm
II}^{ab}_{(\beta}+n^{a}_{\rho}\beta^{b}_{\rho(\beta}+n^{b}_{\rho}\beta^{a}_{\rho(\beta}+(g^{ab}-\bar{g}^{ab})H_{(\beta})\nabla_{|\alpha|}\nabla_{a}n_{b\gamma)}-2H_{(\beta}\nabla_{|\alpha|}(\Delta
s_{\gamma)}+Js_{\gamma)})$ $\displaystyle+2\delta_{\alpha(\beta}P^{ab}({\rm
II}_{ab\gamma)}+n_{a|\rho}\beta_{b\rho|\gamma)}+n_{b|\rho}\beta_{a\rho|\gamma)}+n_{a|\rho}n_{b\rho|}H_{\gamma)})+2H_{\alpha}K_{\beta\gamma}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle 2(\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{(\beta}+n^{a}_{\rho}\beta^{b}_{\rho(\beta}+n^{b}_{\rho}\beta^{a}_{\rho(\beta})n^{c}_{|\alpha|}\nabla_{c}\nabla_{a}n_{b\gamma)}$
$\displaystyle+2\delta_{\alpha(\beta}P^{ab}(\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma)}+n_{a|\rho}\beta_{b\rho|\gamma)}+n_{b|\rho}\beta_{a\rho|\gamma)})+2H_{\alpha}K_{\beta\gamma}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle
2n^{c}_{\alpha}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{(\beta}\nabla_{c}\nabla_{a}n_{b\gamma)}+2n^{c}_{\alpha}n^{a}_{\rho}\beta^{b}_{\rho(\beta}\nabla_{c}\nabla_{b}n_{a\gamma)}+2n^{c}_{\alpha}n^{b}_{\rho}\beta^{a}_{\rho(\alpha}\nabla_{c}\nabla_{a}n_{b\gamma)}$
$\displaystyle+2\delta_{\alpha(\beta}P^{ab}(\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma)}+n_{a|\rho}\beta_{b\rho|\gamma)}+n_{b|\rho}\beta_{a\rho|\gamma)})+2H_{\alpha}K_{\beta\gamma}$
$\displaystyle\stackrel{{\scriptstyle\Lambda}}{{=}}$ $\displaystyle
2n^{c}_{\alpha}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{(\beta}(\nabla_{a}^{\top}\nabla_{c}n_{b\gamma)}+R_{cabd}n^{d}_{\gamma)})+2n^{c}_{\alpha}n^{a}_{\rho}\beta^{b}_{\rho(\beta}(\nabla_{b}^{\top}\nabla_{c}n_{a\gamma)}+R_{cbad}n^{d}_{\gamma)})$
$\displaystyle+2n^{c}_{\alpha}n^{b}_{\rho}\beta^{a}_{\rho(\alpha}(\nabla^{\top}_{a}\nabla_{c}n_{b\gamma)}+R_{cabd}n^{d}_{\gamma)})+2\delta_{\alpha(\beta}P^{ab}(\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma)}+n_{a|\rho}\beta_{b\rho|\gamma)}+n_{b|\rho}\beta_{a\rho|\gamma)})$
$\displaystyle+2H_{\alpha}K_{\beta\gamma}\,.$

Further evaluating both Equations 8.3 and 8.4 requires a few non-standard
submanifold identities. Tracing the Gauss equation (3.4) by applying
$\bar{g}=g-n_{\alpha}\otimes n_{\alpha}$ to both sides yields a Fialkow–Gauss-
like equation, namely

$\displaystyle(d-k-2)\left(P^{\top}_{ab}-\bar{P}_{ab}+H_{\alpha}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\alpha}+\tfrac{1}{2}\bar{g}_{ab}H_{\alpha}^{2}\right)=\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{(ab)\alpha\alpha}-W_{\alpha
ab\alpha}^{\top}-\tfrac{1}{2(d-k-1)}\bar{g}_{ab}(\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\alpha}+W_{\alpha\beta\beta\alpha})\,.$

Similarly (when $d-k>2$), tracing again with $\bar{g}$ yields a generalization
of the theorema egregium:

$J-P_{n_{\alpha}n_{\alpha}}=\bar{J}-\tfrac{d-k}{2}H_{\alpha}H_{\alpha}+\tfrac{1}{2(d-k-1)}(\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\alpha}-\bar{g}^{cd}W_{\alpha
cd\alpha})\,.$

Similarly, tracing the Codazzi–Mainardi equation (3.5) leaves us with

$(d-k-1)P_{an_{\beta}}^{\top}=\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}_{a\beta}-(d-k-1)(\bar{\nabla}_{a}H_{\beta}+\beta_{a\beta\alpha}H_{\alpha})+\hskip
1.13809pt\mathring{\rm{II}}{}^{b}_{a\alpha}\beta_{b\beta\alpha}+W_{\gamma\beta\gamma}{}^{b}\bar{g}_{ab}\,.$

Conversely, we can take the tracefree part to obtain

$W_{abcn_{\alpha}}^{\top}-\tfrac{2}{d-k-1}(W_{\beta\alpha\beta[a}^{\top}\bar{g}_{b]c})=2\bar{\nabla}_{[a}\hskip
1.13809pt\mathring{\rm{II}}{}_{b]c\alpha}+2\beta_{[a\alpha\beta}\hskip
1.13809pt\mathring{\rm{II}}{}_{b]c\beta}-\tfrac{2}{d-k-1}\bar{g}_{c[a}\left(\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}_{b]\alpha}+\hskip
1.13809pt\mathring{\rm{II}}{}_{b]c\beta}\beta^{c}_{\alpha\beta}\right).$

We are now equipped to evaluate Equations 8.3 and 8.4. When $k=d-2$, we find
that

(8.5)
$\displaystyle\nabla^{\top}_{a}N_{B\alpha}\stackrel{{\scriptstyle\Lambda,g}}{{=}}\begin{pmatrix}0\\\
\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\alpha}+n_{b\beta}\beta_{a\beta\alpha}\\\
-\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}_{a\alpha}-\beta^{b}_{\alpha\beta}\hskip
1.13809pt\mathring{\rm{II}}{}_{ba\beta}+H_{\beta}\beta_{a\alpha\beta}-W_{\alpha\beta
a\beta}^{\top}\end{pmatrix}\,.$

and

$\displaystyle\tfrac{1}{3}N_{\gamma}\cdot\hat{D}K_{\gamma\alpha}-\tfrac{1}{6}N_{\alpha}\cdot\hat{D}K_{\gamma\gamma}\stackrel{{\scriptstyle\Lambda,g}}{{=}}$
$\displaystyle-\tfrac{2}{3}(\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\beta}\bar{\nabla}_{a}\beta_{b\alpha\beta}+2\beta_{b\alpha\beta}\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}^{b}_{\beta})-\tfrac{1}{3}\hskip
1.13809pt\mathring{\rm{II}}{}^{3}_{\alpha\beta\beta}+\tfrac{1}{3}\beta^{a}_{\beta\gamma}\beta^{b}_{\beta\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\alpha}$
$\displaystyle-\tfrac{4}{3}\beta^{a}_{\alpha\beta}\beta^{b}_{\beta\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma}+\tfrac{2}{3}\beta^{a}_{\alpha\beta}W_{\beta\delta\delta
a}+\tfrac{2}{3}\beta^{a}_{\beta\gamma}W_{\alpha\beta\gamma
a}+\tfrac{1}{3}\hskip 1.13809pt\mathring{\rm{II}}{}^{ab}_{\alpha}W_{\beta
ab\beta}\,.$

Finally, we compute $\Delta^{\top}N_{B\alpha}$:

(8.6)
$\displaystyle\Delta^{\top}N_{B\alpha}\stackrel{{\scriptstyle\Lambda,g}}{{=}}\begin{pmatrix}0\\\
\ast\\\ \scalebox{0.75}{$\begin{aligned}
-\left(\bar{\nabla}_{a}\bar{\nabla}_{b}+P^{\top}_{ab}+H_{\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma}\right)\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\alpha}-(\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\bar{\nabla}_{a}\beta_{b\alpha\gamma}+2\beta_{b\alpha\gamma}\bar{\nabla}_{a}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma})\\\
+\beta_{a\alpha\beta}\beta_{b\beta\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}+2\beta_{a\alpha\beta}\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}^{a}_{\beta}+\beta^{a}_{\alpha\beta}W_{\beta\gamma\gamma
a}-\bar{\nabla}^{a}W_{a\beta\beta\alpha}^{\top}\end{aligned}$}\end{pmatrix}-[\beta^{a}_{\alpha\beta}\beta_{a\gamma\beta}+\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\alpha\gamma}-\bar{\nabla}\cdot\beta_{\alpha\gamma}]N_{B\gamma}$

Note that in the above computation, one must be careful to notice that the
tractor in Equation (8.5) is a bulk tractor. After differentiation to obtain
Equation (8.6), one must then project the resulting tractor to
$\mathcal{T}\Lambda$. For example, differentiating the middle slot of Equation
(8.5) yields a term in the middle slot of the form $\nabla^{\top a}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\alpha}=\bar{\nabla}\cdot\hskip
1.13809pt\mathring{\rm{II}}{}_{\alpha}-n_{b\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ac\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}^{ac}_{\alpha}$. Projecting the resulting tractor
yields a term proportional to $H_{\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}^{2}_{\gamma\alpha}$ in the bottom slot, found in
Equation (8.6).

With these identities in tow, we can explicitly compute, in terms of more
familiar tensors, the obstruction $F^{(3)}_{\alpha\gamma\gamma\rho\rho}$
present in Equation (8.2). Indeed, in a choice of scale $g\in\bm{c}$, we find
that

$\displaystyle
F^{(3)}_{\alpha\gamma\gamma\rho\rho}\stackrel{{\scriptstyle\Lambda}}{{=}}$
$\displaystyle-\tfrac{d-2}{6}\left(\bar{\nabla}_{a}\bar{\nabla}_{b}+P^{\top}_{ab}+H_{\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma}\right)\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\alpha}-\tfrac{1}{3}\hskip
1.13809pt\mathring{\rm{II}}{}^{bc}_{\alpha}\bar{g}^{ad}W_{abcd}-\tfrac{1}{3}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{b\gamma}^{c}\hskip
1.13809pt\mathring{\rm{II}}{}_{ac\alpha}-\tfrac{d-2}{6}{{\bf\rm IV}{\hskip
0.56905pt}}{}_{\alpha}$ $\displaystyle-\tfrac{d+2}{6}\left[\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\bar{\nabla}_{a}\beta_{b\alpha\gamma}+2\beta_{b\alpha\gamma}\bar{\nabla}_{a}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\right]+\tfrac{1}{3}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\alpha}\beta_{a\beta\gamma}\beta_{b\beta\gamma}-\tfrac{d+6}{6}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\beta_{a\alpha\beta}\beta_{b\beta\gamma}$
$\displaystyle+\tfrac{2}{3}W_{\alpha\beta\gamma
a}\beta^{a}_{\beta\gamma}+\tfrac{d+2}{6}W_{\beta\gamma\gamma
a}\beta^{a}_{\alpha\beta}\,,$

where

${{\bf\rm IV}{\hskip
0.56905pt}}{}_{\alpha}:=C_{\alpha\beta\beta}+H_{\rho}W_{\alpha\beta\beta\rho}+\tfrac{1}{d-k-3}\bar{\nabla}^{c}W_{c\beta\beta\alpha}^{\top}\in\Gamma(\mathcal{E}\Lambda[-3]\times\raisebox{1.0pt}{\scalebox{0.2}{\ydiagram{1}}})$

is a novel conformal submanifold invariant for every $d-k\neq 3$. This can be
verified by explicitly computing the conformal transformation
$g\mapsto\Omega^{2}g$. Furthermore, the combination

$\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\bar{\nabla}_{a}\beta_{b\alpha\gamma}+\tfrac{2}{d-k-1}\beta_{b\alpha\gamma}\bar{\nabla}_{a}\hskip
1.13809pt\mathring{\rm{II}}{}^{ab}_{\gamma}\in\Gamma(\mathcal{E}\Lambda[-3]\times\raisebox{1.0pt}{\scalebox{0.2}{\ydiagram{1}}})\,,$

is conformally invariant for all $d-k\neq 1$, and can be constructed from an
appropriate combination of the tractor second fundamental form [18], the
tractor projection of $B$ to the submanifold, and the submanifold Thomas-$D$
operator. Finally, the operator

$L_{ab}:=\bar{\nabla}_{a}\bar{\nabla}_{b}+P^{\top}_{ab}+H_{\gamma}\hskip
1.13809pt\mathring{\rm{II}}{}_{ab\gamma}:\Gamma(\odot_{\circ}^{2}T\Lambda[-3])\rightarrow\Gamma(\mathcal{E}\Lambda[-3])$