# Lie algebroid invariants for subgeometry

Anthony D. Blaom  E-mail<EMAIL_ADDRESS>
###### Abstract.

We investigate the infinitesimal invariants of an immersed submanifold
$\Sigma$ of a Klein geometry $M\cong G/H$, and in particular an invariant
filtration of Lie algebroids over $\Sigma$. The invariants are derived from
the logarithmic derivative of the immersion of $\Sigma$ into $M$, a complete
invariant introduced in the companion article, A characterization of smooth
maps into a homogeneous space. Applications of the Lie algebroid approach to
subgeometry include a new interpretation of Cartan’s method of moving frames
and a novel proof of the fundamental theorem of hypersurfaces in Euclidean,
elliptic and hyperbolic geometry.

##  1\. Introduction

We initiate a general analysis of subgeometry invariants using the language of
Lie algebroids, which is well suited to the purpose, according to [6]. By a
subgeometry we mean an immersed submanifold $\Sigma$ of a Klein geometry,
which shall refer to any smooth manifold $M$ on which a Lie group $G$ is
acting smoothly and transitively. The geometry will be required to be simple
in the sense that its isotropy groups are weakly connected (Definition 2.1
below). For example, the Riemannian space forms ${\mathbb{R}}^{n}$, $S^{n}$
and ${\mathbb{H}}^{n}$ are all simple Klein geometries if we take $G$ to be
the group of orientation-preserving isometries, or the group of arbitrary
isometries. Informally, an object associated with $\Sigma$ is an invariant if
it is unchanged when $\Sigma$ is replaced by its image under a symmetry of
$M$. The group of symmetries of $M$, as we shall define it in §2.2, is
generally larger than the transformations defined by $G$; however, in most
cases of interest to geometers this transformation group is extended by a
factor of, at most, finite order.

###  1.1. Logarithmic derivatives

According to [6], and, in particular Theorem 1.3 quoted below, a complete
infinitesimal invariant of an immersion $f\colon\Sigma\rightarrow M$ is its
logarithmic derivative $\delta f\colon A(f)\rightarrow{\mathfrak{g}}$. Here
${\mathfrak{g}}$ is the Lie algebra of $G$, $A(f)$ is the pullback of the
action algebroid ${\mathfrak{g}}\times M$ under $f$, and $\delta f$ the
composite of the natural map $A(f)\rightarrow{\mathfrak{g}}\times M$ with the
projection ${\mathfrak{g}}\times M\rightarrow{\mathfrak{g}}$, both of which
are Lie algebroid morphisms. In saying that $\delta f$ is complete, we mean
that $\delta f$ determines $f$ up to symmetry.

The logarithmic derivative of $f\colon\Sigma\rightarrow M$ generalizes Élie
Cartan’s logarithmic derivative of a smooth map $f\colon\Sigma\rightarrow G$
into a Lie group, and the theory laid down here is closely related to Cartan’s
method of moving frames, as explained in the introduction to [6]. Indeed,
Cartan’s method, which cannot always be applied globally, can be reinterpreted
within the new framework (Section 6) but the new theory can also be applied
globally and without fixing coordinates or frames, as we shall demonstrate.

The main contribution of the present article is to ‘deconstruct’ logarithmic
derivatives sufficiently that known invariants may be recovered in some
familiar but non-trivial examples, and to show how subgeometries in these
examples may be reconstructed from their invariants by applying the general
theory. The analysis of the finer structure of logarithmic derivatives given
here is by no means exhaustive.

Specifically, we shall recover the well-known fundamental theorem of
hypersurfaces, or Bonnet theorem, for Euclidean, elliptic and hyperbolic
geometry. In principle, Bonnet-type theorems for other Klein geometries could
be tackled using the present framework, although this is not attempted here.

Introductions to the theory of Lie groupoids and Lie algebroids are to be
found in [13, 10, 14, 17]. In particular, the reader should be acquainted with
the representations of a Lie algebroid or groupoid, which will be ubiquitous.
Some of the ideas presented here and in [6] are also sketched in [7], which
may be regarded as an invitation to Lie algebroids for the geometer who is
unfamiliar with (or skeptical of) these objects.

###  1.2. Invariants

We now briefly describe the two types of invariants to be derived from the
logarithmic derivative of an immersion $f\colon\Sigma\rightarrow M$, when $M$
is a simple Klein geometry. Details are given in Section 3. The first kind of
invariant is an ${\mathbb{R}}$-valued (or ${\mathbb{C}}$-valued) function on
$A(f)$ obtained, very simply, by composing the logarithmic derivative $\delta
f\colon A(f)\rightarrow{\mathfrak{g}}$ with a polynomial on the Lie algebra
${\mathfrak{g}}$ invariant under the adjoint action. Many familiar invariants
of submanifold geometry can be derived with the help of such invariants.

To describe the second kind of invariant, note that because
$f\colon\Sigma\rightarrow M$ is an immersion, we may view $A(f)$ as a
subbundle of the trivial bundle
$E={\mathfrak{g}}\times\Sigma\rightarrow\Sigma$:

$A(f)=\\{(\xi,x)\in{\mathfrak{g}}\times\Sigma\,\left|\right.\,\xi^{\dagger}(f(x))\in
T_{f(x)}\Sigma\\}.$

Here $\xi^{\dagger}$ denotes the infinitesimal generator of
$\xi\in{\mathfrak{g}}$. Unless $\Sigma$ enjoys a lot of symmetry (see §1.5
below) the canonical flat connection $\nabla$ on $E$ does not restrict to a
connection on $A(f)$. However, under a suitable regularity assumption, there
will be a largest subbundle $A(f)^{2}\subset A(f)$ such that for any vector
field $U$ on $\Sigma$, $\nabla_{U}X$ will be a section of $A(f)$ whenever $X$
is a section of $A(f)^{2}$. The new bundle $A(f)^{2}$ need not be
$\nabla$-invariant either, so we repeat the process and obtain an invariant
filtration

$A(f)\supset A(f)^{2}\supset A(f)^{3}\cdots$

by what turn out to be subalgebroids of $A(f)$.

By definition, $A(f)$ is the Lie algebroid of the pullback ${\mathcal{G}}(f)$
of the action groupoid $G\times M$ to $\Sigma$:

(1) ${\mathcal{G}}(f)=\\{(y,g,x)\in\Sigma\times
G\times\Sigma\,\left|\right.\,\text{$f(x)\in\Sigma$ and $f(y)=g\cdot
f(x)$}\\}.$

Tangent-lifting the action of $G$ on $M$, we obtain an action of the groupoid
${\mathcal{G}}(f)$ on $T_{\Sigma}M$, the vector bundle pullback of $TM$. If we
define ${\mathcal{G}}(f)^{2}\subset{\mathcal{G}}(f)$ to be the isotropy of
$T\Sigma\subset T_{\Sigma}M$ with respect to this representation, then its Lie
algebroid is in fact $A(f)^{2}$. In particular, $A(f)^{2}$ is the isotropy
subalgebroid under the corresponding infinitesimal representation.

In fact, all the Lie algebroids $A(f)^{j}$ in the filtration that are
transitive can be characterized as isotropy subalgebroids, under
representations built out of one fundamental representation of $A(f)$ on $E$
that we define in §3.4. Corresponding interpretations of the globalisations
${\mathcal{G}}(f)^{j}$, for $j\geqslant 3$, are not offered here, but nor are
they needed for making computations.

There are evidently many ways to combine polynomial invariants with the
invariant filtration to construct new invariants. For example, if
$A(f)^{k}\cong T\Sigma$ for sufficiently large $k$, then a polynomial
invariant restricts to an ordinary symmetric tensor on $\Sigma$ that must also
be an invariant. The curvature of a curve in the Euclidean, hyperbolic,
elliptic or equi-affine plane may be understood in this way. For codimension-
one submanifolds in higher dimensional elliptic or hyperbolic geometry (see
Section 5) the Killing form on ${\mathfrak{g}}$ encodes the first fundamental
form (inherited metric) of an immersion $f\colon\Sigma\rightarrow M$, and also
allows one to define a normal to $A(f)$ in $E={\mathfrak{g}}\times\Sigma$
whose ordinary derivative, as a ${\mathfrak{g}}$-valued function, determines
the second fundamental form of the immersion.111The normal can also be
constructed up to scale by considering certain representations associated with
the invariant filtration, i.e., without consideration of polynomial
invariants. In Euclidean geometry, summarized below, a similar role is played
by an appropriate $\operatorname{Ad}$-invariant quadratic form on
$\operatorname{\operatorname{rad}}{\mathfrak{g}}\subset{\mathfrak{g}}$, the
subalgebra of constant vector fields.

Finally, for each $j\geqslant 2$, the Lie algebroid $A(f)^{j}$ in the
filtration acts on the tangent bundle $T\Sigma$, which may give $\Sigma$ an
intrinsic ‘infinitesimal geometric structure’ [2]. In the Riemannian
geometries mentioned above the inherited metric (up to scale) can also be
understood in this way.

###  1.3. Primitives and their existence and uniqueness

Abstracting the properties of the logarithmic derivatives of smooth maps
$f\colon\Sigma\rightarrow M$, we obtain generalized Mauer-Cartan forms [6]. If
we restrict attention to the logarithmic derivatives of immersions, then as a
special case we obtain infinitesimal immersions, formally defined in §2.3. An
infinitesimal immersion is a Lie algebroid morphism $\omega\colon
A\rightarrow{\mathfrak{g}}$, where $A$ is a Lie algebroid over $\Sigma$, which
is, in particular, injective on fibres. The construction of polynomial
invariants and the invariant filtration given above generalizes to
infinitesimal immersions.

A smooth map $f\colon\Sigma\rightarrow M$ is a primitive of an infinitesimal
immersion $\omega\colon A\rightarrow{\mathfrak{g}}$ if $\omega$ ‘is’ the
logarithmic derivative of $f$ in the following sense: there exists a Lie
algebroid morphism $\lambda\colon A\rightarrow A(f)$ covering the identity on
$\Sigma$, and an element $l\in G$, such that the following diagram commutes:

(1) $\begin{CD}A@>{\omega}>{}>{\mathfrak{g}}\\\
@V{\lambda}V{}V@V{}V{\operatorname{Ad}_{l}}V\\\ A(f)@>{\delta
f}>{}>{\mathfrak{g}}\end{CD}\qquad.$

The primitive is principal if we can take $l=1_{G}$. Primitives of
infinitesimal immersions are necessarily immersions, and the morphism
$\lambda$ automatically an isomorphism. Specialising [6, Theorem 2.8] to
infinitesimal immersions, we have:

######  Theorem.

Let $M$ be a Klein geometry with transitively acting group $G$. Then an
infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$ admits a
primitive $f\colon\Sigma\rightarrow M$ if and only if it has trivial
monodromy, in which case $A$ is an integrable Lie algebroid. Assuming $M$ is a
simple Klein geometry, the primitive $f$ is unique up to symmetry, and can be
chosen to be principal.

The monodromy is a map $\pi_{1}(\Sigma,x_{0})\rightarrow M$, whose definition
is recalled in §2.5. The monodromy is trivial if it is constant, which is
automatically true if $\Sigma$ is simply-connected (always the case in the
illustrations to be given here).

###  1.4. Riemannian subgeometry and the Bonnet theorem

As a concrete illustration of the general theory to be developed, we consider,
in Sections 4 and 5, the case that $M$ is Euclidean space ${\mathbb{R}}^{n}$,
the sphere $S^{n}$, or hyperbolic space ${\mathbb{H}}^{n}$. The problem is to
classify all oriented codimension-one immersed submanifolds $\Sigma\subset M$,
up to actions of the group $G$ of orientation-preserving isometries. We
identify the Lie algebra ${\mathfrak{g}}$ of $G$ with the space of Killing
fields on $M$. Our purpose is not demonstrate anything new about Riemannian
geometry, but to show how to apply and interpret the general theory in
familiar cases. The following is a synopsis of the case $M={\mathbb{R}}^{n}$.

It may be observed that the pullback ${\mathcal{G}}(f)$ of the action groupoid
$G\times M$ by an immersion $f\colon\Sigma\rightarrow M$ acts faithfully on
$T_{\Sigma}M$ — that is, a rigid motion mapping a point $x\in f(\Sigma)$ to
another point of $f(\Sigma)$ is completely determined by how it acts on the
ambient tangent space at $x$ — allowing us to identify ${\mathcal{G}}(f)$ with
the groupoid $\mathrm{SO}[T_{\Sigma}M]$ of orientation-preserving orthonormal
relative frames of $T_{\Sigma}M$. By a relative frame of a vector bundle we
mean an isomorphism between two fibres over possibly different basepoints. As
$\Sigma$ is oriented, it has a well-defined unit normal, allowing us to
identify $T_{\Sigma}M$ with the ‘thickened’ bundle
$T^{+}\Sigma:=T\Sigma\oplus({\mathbb{R}}\times\Sigma)$, equipped with the
obvious extension of the metric on $T\Sigma$ to an inner product. So
${\mathcal{G}}(f)\cong\mathrm{SO}[T^{+}\Sigma]$. Under this identification we
see that the isotropy ${\mathcal{G}}(f)^{2}$ of $T\Sigma\subset T_{\Sigma}M$
is identified with $\mathrm{SO}[T\Sigma]\subset\mathrm{SO}[T^{+}\Sigma]$, the
fundamental Lie groupoid associated with $\Sigma$ as a Riemannian manifold in
its own right. Infinitesimalizing, we can identify the first two Lie
algebroids in the invariant filtration $A(f)\supset A(f)^{2}\supset
A(f)^{3}\supset\cdots$ with the Lie algebroids
$\mathfrak{so}[T^{+}\Sigma]\supset\mathfrak{so}[T\Sigma]$. The same conclusion
is drawn in Section 4 by purely infinitesimal arguments.

Infinitesimal computations are straightforward, once one has the right model
of the action algebroid ${\mathfrak{g}}\times M$ and a corresponding
description of its flat connection $\nabla$. As the action of $G\times M$ on
$TM$ is faithful and preserves the metric, we have $G\times
M\cong\mathrm{SO}[TM]$ and consequently

(1) ${\mathfrak{g}}\times M\cong\mathfrak{so}[TM]\cong
TM\oplus\mathfrak{so}(TM).$

Here $\mathfrak{so}(TM)$ is the $\mathfrak{so}(n)$-bundle of skew-symmetric
endomorphisms of the tangent bundle $TM$ (the kernel of the anchor of
$\mathfrak{so}[TM]$) and the second isomorphism is obtained with the help of
the Levi-Cevita connection; see §4.1. In this model the connection $\nabla$
coincides with the canonical Cartan connection on $\mathfrak{so}[TM]$, in the
sense of [1], constructed for arbitrary Riemannian manifolds in [2].

Readers familiar with tractor bundles will recognise the model on the right of
(1) as the adjoint tractor bundle associated with $M$ [11]. Those familiar
with our work on infinitesimal geometric structures will recognise a model of
the isotropy subalgebroid of the metric on $M$ associated with a canonical Lie
algebroid representation of the jet bundle $J^{1}(TM)$ on $S^{2}\,T^{*}\\!M$
[1, 2]. Similar models are known for other Klein geometries, and in particular
for parabolic geometries [12], although the Lie algebroid structure of these
models has been mostly ignored.

The definitions of $\mathfrak{so}[T^{+}\Sigma]$ and $\mathfrak{so}[T\Sigma]$
depend on (and encode, up to scale) the metric that $\Sigma$ inherits from $M$
but on no other aspect of the immersion $f$. The second fundamental form
$\mathrm{I\\!I}$ of the immersion enters into the present picture in two ways.
Firstly, it enters indirectly in the form of $A(f)^{3}$, which we show is the
isotropy subalgebroid of $\mathrm{I\\!I}$, in the canonical representation of
the Lie algebroid $A(f)^{2}\cong\mathfrak{so}[T\Sigma]$ on the vector bundle
$S^{2}\,T^{*}\Sigma$, of which $\mathrm{I\\!I}$ is a section. The well-known
fact, that $\Sigma\subset M$ has maximal symmetry when $\mathrm{I\\!I}$ is a
constant multiple of the inherited metric, follows immediately from this
observation and the general characterisation of symmetries given in §1.5
below. More generally, $A(f)^{3}$ is an intransitive Lie algebroid (and
consequently does not appear in any formulation based on principal bundles,
such as $G$-structures).

Secondly, we may recover $\mathrm{I\\!I}$ directly as follows. The radical
$\operatorname{\operatorname{rad}}{\mathfrak{g}}$ of ${\mathfrak{g}}$ consists
of all constant vector fields on $M={\mathbb{R}}^{n}$, so that
$\operatorname{\operatorname{rad}}{\mathfrak{g}}\cong{\mathbb{R}}^{n}$. This
identification transfers the standard inner product on ${\mathbb{R}}^{n}$ to
an inner product $Q$ on $\operatorname{\operatorname{rad}}{\mathfrak{g}}$ that
is in fact $\operatorname{Ad}$-invariant, and we obtain an inner product on
the trivial vector bundle
$\operatorname{\operatorname{rad}}E=\operatorname{\operatorname{rad}}{\mathfrak{g}}\times\Sigma$
that we also denote by $Q$. It is not hard to see that $A(f)$, which we may
identify with a subbundle of $E={\mathfrak{g}}\times\Sigma$, intersects
$\operatorname{\operatorname{rad}}E$ in a subbundle of corank one. The normal
for the immersion $f$ is the section $\xi$ of
$\operatorname{\operatorname{rad}}E$ defined, up to sign, by requiring $\xi$
to have $Q$-length one and be $Q$-orthogonal to
$A(f)\cap\operatorname{\operatorname{rad}}E$. The ambiguity in sign is
resolved by requiring that the image ${\mathbf{n}}$ of $\xi$ under the natural
projection $(\xi,x)\mapsto\xi(x)\colon E\rightarrow T_{\Sigma}M$ — which is a
unit length vector field orthogonal to $T\Sigma$ with respect to the metric on
$M$ — have a direction consistent with the orientations on $\Sigma$ and $M$.
Explicitly, we have at each point $x\in\Sigma$, $\xi(x)=(\xi^{\prime}(x),x)$,
where $\xi^{\prime}(x)\in{\mathfrak{g}}$ is the constant vector field on $M$
with $\xi^{\prime}(x)(x)={\mathbf{n}}(x)$. Furthermore, $\nabla_{u}\xi\in
A(f)$ for all $u\in T\Sigma$, where $\nabla$ is the canonical flat connection
on $E={\mathfrak{g}}\times\Sigma$, and

$\mathrm{I\\!I}(u,v)=-\langle\\!\langle\\#\nabla_{u}\xi,v\rangle\\!\rangle,$

where $\\#$ denotes anchor and
$\langle\\!\langle\,\cdot\,,\,\cdot\,\rangle\\!\rangle$ is the metric on
$\Sigma$.

Conversely, with ${\mathfrak{g}}$ continuing to denote the Lie algebra of
Killing fields on ${\mathbb{R}}^{n}$ as above, suppose we are given an
infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$, for some
Lie algebroid $A$ over an oriented $(n-1)$-dimensional manifold $\Sigma$.
Since $\omega$ is injective on fibres, it is an elementary observation that
$T\Sigma$ can be regarded as a subbundle of $E/{\mathfrak{h}}$, where
$E={\mathfrak{g}}\times\Sigma$ and ${\mathfrak{h}}$ is the kernel of the
anchor of $A$. We define an inner product $Q$ on
$\operatorname{\operatorname{rad}}E$ as before and can show that the natural
projection $E\rightarrow E/{\mathfrak{h}}$ restricts to an isomorphism
$\operatorname{\operatorname{rad}}E\rightarrow E/{\mathfrak{h}}$, pushing the
inner product $Q$ to one on $E/{\mathfrak{h}}$. The restriction of this form
to $T\Sigma\subset E/{\mathfrak{h}}$ is the first fundamental form of
$\omega$, denoted
$\langle\\!\langle\,\cdot\,,\,\cdot\,,\rangle\\!\rangle_{\omega}$, coinciding
with the inherited metric on $\Sigma$ when $\omega$ is the logarithmic
derivative of an immersion. After showing that $A$ must intersect
$\operatorname{\operatorname{rad}}E$ in a corank-one subbundle, and explaining
how $E/{\mathfrak{h}}$ can be naturally oriented, we are able to define the
normal $\xi\in\Gamma(\operatorname{\operatorname{rad}}E)$ of $\omega$ just as
we did for immersions. The second fundamental form of $\omega$, denoted
$\mathrm{I\\!I}_{\omega}$, is defined by

$\mathrm{I\\!I}_{\omega}(u,v)=-\langle\\!\langle\\#\nabla_{u}\xi,v\rangle\\!\rangle_{\omega},$

and this coincides with the usual second fundamental form when $\omega$ is the
logarithmic derivative of an immersion. Applying Theorem 1.3, we will obtain:

######  Theorem (The abstract Bonnet theorem; proven in §5.4).

Assuming $\Sigma$ is simply-connected and $\operatorname{rank}A=(n+2)(n-1)/2$,
any infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$ of
$\Sigma$ into $M={\mathbb{R}}^{n}$ has, as primitive, an immersion
$f\colon\Sigma\rightarrow{\mathbb{R}}^{n}$ with first and second fundamental
forms $\langle\\!\langle\,\cdot\,,\,\cdot\,,\rangle\\!\rangle_{\omega}$ and
$\mathrm{I\\!I}_{\omega}$. The immersion $f$ is unique up to orientation-
preserving isometries of ${\mathbb{R}}^{n}$.

The classical Bonnet theorem for isometric immersions is obtained as a
corollary. In principle, Theorem 1.3 could also be used to study monodromy
obstructions to realising hypersurfaces in non-simply-connected cases but this
is not attempted here.

A simpler illustration of the general theory, to planar curves, appears in
§3.11. An application to curves in the equi-affine plane is offered in Section
6.

###  1.5. Symmetries of a subgeometry

Under our assumptions of constant rank, an invariant filtration $A(f)\supset
A(f)^{2}\supset A(f)^{3}\supset\cdots$ must terminate at some Lie algebroid
$A(f)^{k}$, $k\geqslant 1$. By construction, the flat connection $\nabla$
drops to a connection on $A(f)^{k}$ that is actually a Cartan connection. In
particular, the space ${\mathfrak{s}}$ of $\nabla$-parallel sections of
$A(f)^{k}$ is a Lie subalgebra of all sections acting infinitesimally on the
submanifold $\Sigma$. We may view ${\mathfrak{s}}$ as a subalgebra of
${\mathfrak{g}}$ and ${\mathfrak{s}}$ has the dimension of the fibres of
$A(f)^{k}$ because $\nabla$ is flat. In §3.6 we show that the infinitesimal
action integrates to a pseudogroup of transformations of $\Sigma$ consisting
of restrictions to open subsets of $f(\Sigma)\subset M$ of symmetries of $M$.

###  1.6. Concluding remarks

The results of [6] constitute a natural generalization of an elegant and
widely applied result of Élie Cartan, which here delivers a conceptual
simplification in one of its main applications, the study of subgeometry.
Evidently there is an additional overhead, in the form of abstraction, in
recovering known results in those relatively simple cases considered here. We
expect the additional abstraction will be justified in more sophisticated
applications.

A well-regarded and practical alternative to Cartan’s moving frames which must
be mentioned here, with diverse applications in mathematics and elsewhere, is
due to Peter Olver and Mark Fels [15, 16]; see also the survey [19].

There are far simpler, coordinate-free proofs of the global Bonnet theorem for
Euclidean geometry; see, e.g., [8, Theorem 1.1]. However, the simpler methods
do not generalize to arbitrary Klein geometries. In the case of the class of
parabolic geometries there is a tractor bundle approach to subgeometry,
involving a comparable degree of abstraction, due to Burstall and Calderbank
[8]. In particular, this approach has been successfully applied to conformal
geometry [9].

Possible extensions of the analysis initiated here include:

#### Illustrations to other concrete Klein geometries.

Preliminary investigations suggest that hypersurfaces in affine geometry and
conformal geometry are quite tractable, but it would be nice to obtain Bonnet-
type theorems in novel cases also.

#### Detailed descriptions of monodromy obstructions to hypersurface
reconstruction in Riemannian and other geometries.

As we explain in Remark 2.3, there are generally two monodromy-type
obstructions.

#### Illustrations or novel applications to submanifolds of other codimension.

Even codimension zero (local diffeomorphisms) could be interesting to revisit,
in particular with regard to monodromy obstructions.

#### Descriptions of globalisations ${\mathcal{G}}(f)^{j}$ of the Lie
algebroids $A(f)^{j}$, for $j\geqslant 3$.

It is not difficult to guess likely candidates for these (considering jets of
immersions). Nice geometric interpretations of our infinitesimal analyses
would likely follow. For example, why, from the present viewpoint, is the
curvature of a curve in the Euclidean plane the radius of the circle of
second-order tangency to the curve (see Remark §3.11)?

#### Unified description of curves (or loops) in planar geometries.

The examples investigated so far by the author, including the Euclidean and
affine cases described here, suggest that in generic cases the last transitive
Lie algebroid $A(f)^{k}$ in the invariant filtration is just a copy of the
tangent bundle of the interval (or circle), delivering a right-inverse $s$ for
the anchor of $A(f)\supset A(f)^{k}$. The ‘intrinsic’ geometry of the curve is
then (generically) a parallelism, equipping the curve with a canonical
reparameterisation $\tilde{f}$, up to scale; each polynomial invariant $Q$ on
${\mathfrak{g}}$ defines an ordinary function
$Q(\delta\tilde{f}(s(\partial/\partial t)))$ on the curve. Presumably these
invariants suffice to describe the curve up to symmetry.

#### Generalizations to submanifolds of curved geometries.

Suppose $M$ is a a Cartan geometry, i.e., a Klein geometry deformed by
curvature [20], and $f\colon\Sigma\rightarrow M$ an immersed submanifold. Then
there exists a canonical Cartan connection $\nabla$ on an Atiyah Lie algebroid
$A_{M}$ associated with the geometry [1, 4], a connection that generalizes the
canonical flat connection on the action algebroids considered here. (A more
direct description of the pair $(A_{M},\nabla)$ might also be possible, as we
recall in the case of Riemannian geometry in §4.1; see also [2].) One can use
$f$ to pull back $A_{M}$, in the category of Lie algebroids, to a subbundle
$A(f)$ of $E$, where $E$ is the pullback of $A_{M}$ in the category of vector
bundles. The connection $\nabla$ pulls back to a connection on $E$ and we may
mimic the construction of the invariant filtration $A(f)\supset
A(f)^{2}\supset A(f)^{3}\cdots$ given above. This is still a filtration of Lie
algebroids, and the fundamental representation of $A(f)$ on $E$ persists, as
does the isotropy interpretation of transitive members of the filtration. For
example, one can show that a knowledge of this filtration in the case of a
hypersurface of a Riemannian manifold suffices to describe its inherited
metric and second fundamental form, up to a choice of scale.

### Acknowledgements

We thank Yuri Vyatkin for helpful discussions.

### Bracket convention

Throughout this article, brackets on Lie algebras and Lie algebroids are
defined using right-invariant vector fields.

### Notation

We reserve the symbol $G$ for Lie groups and ${\mathfrak{g}}$ for Lie
algebras. The symbols ${\mathcal{G}}$, ${\mathcal{G}}(f)$, and
${\mathcal{G}}(f)^{2}$ denote a Lie groupoids; $A$, $A^{2}$, $A(f)$,
$A(f)^{2}$, etc., denote Lie algebroids. The kernel of the anchor $\\#$ of $A$
or $A(f)$ is a Lie algebra bundle denoted ${\mathfrak{h}}$, that of $A^{2}$ or
$A^{2}(f)$ is denoted by ${\mathfrak{h}}^{2}$, etc. We reserve $E$ for the
trivial ${\mathfrak{g}}$-bundle ${\mathfrak{g}}\times\Sigma$ over $\Sigma$.

The frame groupoid of a vector bundle $B$ will be denoted $\mathrm{GL}[B]$ and
the corresponding subgroupoid of automorphisms of fibres of $B$ by
$\mathrm{GL}(B)$. So, an element of $\mathrm{GL}[B]$ is an isomorphism
(relative frame) $B|_{m_{1}}\rightarrow B|_{m_{2}}$, becoming an element of
$\mathrm{GL}(B)$ if $m_{1}=m_{2}$. Similar notation applies to the
subgroupoids and Lie subalgebroids of frame groupoids.

##  2\. Symmetries, infinitesimal immersions and monodromy

In formulating Theorem 1.3, several notions were introduced on which we now
elaborate.

###  2.1. Simple Klein geometries

Let $M$ be a Klein geometry with transitively acting Lie group $G$. Let
$G_{m_{0}}^{\circ}$ denote the connected component of the isotropy $G_{m_{0}}$
at $m_{0}\in M$. As $G_{m_{0}}^{\circ}$ is path-connected,
$N_{G}(G_{m_{0}})\subset N_{G}(G_{m_{0}}^{\circ})$. Here $N_{G}(H)$ is the
normaliser of $H\subset G$ in $G$.

######  Definition.

We say the isotropy groups of the $G$ action are weakly connected if for some
(and hence any) $m_{0}\in M$, we have
$N_{G}(G_{m_{0}})=N_{G}(G_{m_{0}}^{\circ})$.

As already mentioned, $M$ is called a simple Klein geometry in this case.

###  2.2. Symmetries of a Klein geometry

For reasons explained in [6], we allow symmetries of Klein geometries to
include more than the left translations $m\mapsto g\cdot m$, $g\in G$:

######  Definition.

Let $M$ be a simple Klein geometry and $G$ the transitively acting Lie group.
Then a symmetry of $M$ is any diffeomorphism $\phi\colon M\rightarrow M$ for
which there exists some $l\in G$ such that $\phi(g\cdot
m)=lgl^{-1}\cdot\phi(m)$ for all $g\in G,\,m\in M$.

The symmetries of $M$ form a Lie group, henceforth denoted
$\operatorname{Aut}(M)$. Fixing arbitrarily some $m_{0}\in M$, let
$W=N_{G}(G_{m_{0}})/G_{m_{0}}$. Then $W$ acts from the left on $M$ according
to

$rG_{m_{0}}\cdot(g\cdot{m_{0}})=(gr^{-1})\cdot{m_{0}}$

and this action commutes with that of $G$. We have:

######  Proposition ([6]).

The left translations, together with the transformations of $M$ defined by the
above action of $W$, generate $\operatorname{Aut}(M)$.

In contrast to the special case in which $G$ acts freely,
$\operatorname{Aut}(M)$ is frequently not much larger than the group of left
translations, in applications of interest to geometers. For example, to the
group $G$ of orientation-preserving isometries of a Riemannian space form
${\mathbb{R}}^{n}$, $S^{n}$ or ${\mathbb{H}}^{n}$, one has only to add the
orientation-reversing isometries to obtain the full symmetry group
$\operatorname{Aut}(M)$, and then only in the case of even-dimensional
spheres. If, in these examples, we instead take $G$ to be the full group of
isometries (in which case $M$ is still a simple Klein geometry) then
$\operatorname{Aut}(M)=G$. For further examples, see [6, §2.1].

###  2.3. Infinitesimal immersions and their synthesis

Let $M$ be a Klein geometry, $G$ the transitively acting Lie group, and
$\Sigma$ a fixed connected smooth manifold. Then a Lie algebroid morphism
$\omega\colon A\rightarrow{\mathfrak{g}}$ is an infinitesimal immersion of
$\Sigma$ into $M$ if $A$ has base $\Sigma$ and $\omega$ satisfies the
following properties, satisfied by the logarithmic derivatives of bona fide
immersions $f\colon\Sigma\rightarrow M$:

  1. I1.

$A$ is transitive.

  2. I2.

$\omega$ is injective on fibres.

  3. I3.

For some (and hence any [6, §2.2]) point $x_{0}\in\Sigma$ there exists
$m_{0}\in M$ such that $x_{0}\xrightarrow{\omega}m_{0}$.

  4. I4.

$\dim{\mathfrak{g}}-\operatorname{rank}A=\operatorname{dim}M-\operatorname{dim}\Sigma$.

Here $x_{0}\xrightarrow{\omega}m_{0}$ is shorthand for the condition
$\omega(A_{x_{0}})\subset{\mathfrak{g}}_{m_{0}}$, were $A_{x_{0}}$ denotes the
isotropy algebra of $A$ at ${x_{0}}\in\Sigma$, i.e., the kernel of the
restriction of the anchor to the fibre $A|_{x_{0}}$, while
${\mathfrak{g}}_{m_{0}}$ denotes the usual isotropy (stabiliser) at $m_{0}$ of
the infinitesimal action of ${\mathfrak{g}}$ on $M$. Axiom I4 ensures that $A$
has the largest rank consistent with the Axioms I1 and I2, in which case Axiom
I3 actually implies

(1) $\omega(A_{x_{0}})={\mathfrak{g}}_{m_{0}}.$

In the language of [6], $\omega$ is an infinitesimal immersion if it is a
maximal generalized Mauer-Cartan form, injective on fibres.

Now suppose $\omega\colon A\rightarrow{\mathfrak{g}}$ is an infinitesimal
immersion of $\Sigma$ into $M$. By virtue of Axiom I2, we may use $\omega$ to
identify $A$ with a subbundle of the trivial bundle
$E:={\mathfrak{g}}\times\Sigma\rightarrow\Sigma$. This bundle comes equipped
with a canonical flat connection $\nabla$ and a canonical $\nabla$-parallel
Lie bracket $\\{\,\cdot\,,\,\cdot\,\\}$ on its space of sections making it
into a ${\mathfrak{g}}$-bundle: $\\{X,Y\\}(x):=[X(x),Y(x)]_{\mathfrak{g}}$.
Applying the characterisation of Lie algebroid morphisms in terms of
connections (see, e.g., [18, Proposition 4.1.9, p. 154]) we deduce the
(generalized) Mauer-Cartan equation,

(2) $[X,Y]=\nabla_{\\#X}Y-\nabla_{\\#Y}X+\\{X,Y\\},$

holding for all sections $X,Y$ of $A$. Implicit in this formula is that the
right-hand side is a section of $A\subset E$.

The preceding observations have the following readily established converse:

######  Proposition.

Let $E$ be a ${\mathfrak{g}}$-bundle, over a simply-connected manifold
$\Sigma$, equipped with a flat connection $\nabla$, with respect to which the
bracket on $E$ is parallel. Then:

  * The space ${\mathfrak{g}}^{\prime}$ of $\nabla$-parallel sections of $E$ is a Lie subalgebra of $\Gamma(E)$ isomorphic to ${\mathfrak{g}}$ and the map $(\xi,x)\mapsto\xi(m)\colon{\mathfrak{g}}^{\prime}\times\Sigma\rightarrow E$ is a vector bundle isomorphism.

For any Lie algebroid $A$ over $\Sigma$, realised as subbundle of $E$ in such
a way that the Mauer-Cartan equation (2) holds, the composite
$A\hookrightarrow E\rightarrow{\mathfrak{g}}^{\prime}\cong{\mathfrak{g}}$ is
an injective Lie algebroid morphism $\omega\colon A\rightarrow{\mathfrak{g}}$.
Here $E\rightarrow{\mathfrak{g}}^{\prime}$ is the projection determined by the
isomorphism $E\cong{\mathfrak{g}}^{\prime}\times\Sigma$ in (Proposition).

Equipped with the preceding proposition, we posit the following general
principle, basic to later illustrations of the general theory:

######  Construction Principle.

To construct an infinitesimal immersion $\omega\colon
A\rightarrow{\mathfrak{g}}$, attempt to realize the Lie algebroid $A$ as a
subbundle of a $\mathfrak{g}$-bundle $E$ equipped with a flat connection
$\nabla$, with respect to which the bracket on $E$ is parallel, and chosen
such that the Mauer-Cartan equation (2) holds. Candidates for $A$, $E$ and
$\nabla$ are suggested by computing the logarithmic derivatives of smooth
immersions.

######  Remark.

If we drop the proposition’s simple connectivity hypothesis then there is a
global obstruction to synthesising a Lie algebroid morphism $\omega\colon
A\rightarrow{\mathfrak{g}}$ in this way, namely the monodromy of $\nabla$. So,
in general, there are two monodromy obstructions to reconstructing immersions
from infinitesimal data: the monodromy of a connection — necessary for
constructing an infinitesimal immersion $\omega$ — and the monodromy of
$\omega$ itself (to be defined in §2.5 below).

For a first illustration of the Construction Principle, see the section on
planar curves, §3.11.

###  2.4. Classical Mauer-Cartan forms

Theorem 1.3 generalises a classical result of Cartan which is convenient to
quote explicitly here.

A Lie algebroid morphism $\omega\colon T\Sigma\rightarrow{\mathfrak{g}}$ is
called a Mauer-Cartan form on $\Sigma$. A primitive of $\omega$ is a smooth
map $f\colon\Sigma\rightarrow G$ such that $f^{*}\omega_{G}=\omega$, where
$\omega_{G}$ is the canonical right-invariant Mauer-Cartan form on the Lie
group $G$. The following is very well-known:

######  Theorem (Cartan).

Every Mauer-Cartan form $\omega$ on a simply-connected manifold $\Sigma$
admits a primitive $f\colon\Sigma\rightarrow G$. If
$f^{\prime}\colon\Sigma\rightarrow G$ is a second map with logarithmic
derivative $\omega$, then there exists a unique $g\in G$ such that
$f^{\prime}(x)=f(x)g$.

###  2.5. $A$-paths and monodromy

Let $A$ be a Lie algebroid over a manifold $\Sigma$. Then a smooth map
$a\colon[0,1]\rightarrow A$, covering an ordinary path
$\gamma\colon[0,1]\rightarrow\Sigma$, is an $A$-path if
$\\#a(t)=\dot{\gamma}(t)$, for all $t\in[0,1]$. Here $\\#\colon A\rightarrow
T\Sigma$ denotes the anchor of $A$. Any smooth map
$\xi\colon[0,1]\rightarrow{\mathfrak{g}}$ is an ${\mathfrak{g}}$-path,
defining a Mauer-Cartan form $\hat{\xi}$ on $[0,1]$ by
$\hat{\xi}(\partial/\partial t)=\xi$. By Theorem 2.4, $\hat{\xi}$ admits a
unique primitive $g\colon[0,1]\rightarrow G$, such that $g(0)=1_{G}$, and we
will write

$\int_{0}^{t}\xi:=g(t).$

Now suppose $\omega\colon A\rightarrow{\mathfrak{g}}$ is a infinitesimal
immersion and that $x_{0}\xrightarrow{\omega}m_{0}$. We define a map
$\Omega_{\mathrm{pt}}\colon\pi_{1}(\Sigma,x_{0})\rightarrow M$ called the
monodromy of $\omega$ (pointed form) as follows: Given an element
$[\gamma]\in\pi_{1}(\Sigma,x_{0})$ with $\gamma\colon[0,1]\rightarrow\Sigma$
smooth, let $a\colon[0,1]\rightarrow A$ be any $A$-path covering $\gamma$, and
define

$\Omega_{\mathrm{pt}}([\gamma])=\left(\int_{0}^{1}\omega\circ a\right)\cdot
m_{0}.$

We say that $\omega$ has trivial monodromy if $\Omega_{\mathrm{pt}}$ takes
constant value $m_{0}$.

######  Theorem ([6]).

The monodromy is well-defined, i.e., independent of the choice of $\gamma$ and
$a$ above.

##  3\. Invariants

###  3.1. Morphisms between infinitesimal immersions

Fixing a Klein geometry $M$ with transitively acting group $G$, and a
connected manifold $\Sigma$, we will collect all infinitesimal immersions of
$\Sigma$ into $M$ into the objects of a category $\mathbf{Inf}$. In this
category a morphism $\omega_{1}\rightarrow\omega_{2}$ between objects
$\omega_{1}\colon A_{1}\rightarrow{\mathfrak{g}}$ and $\omega_{2}\colon
A_{2}\rightarrow{\mathfrak{g}}$ consists of a Lie algebroid morphism
$\lambda\colon A_{1}\rightarrow A_{2}$ covering the identity on $\Sigma$ and
an element $l\in G$ such that the following diagram commutes:

(1) $\begin{CD}A_{1}@>{\omega_{1}}>{}>{\mathfrak{g}}\\\
@V{\lambda}V{}V@V{}V{\operatorname{Ad}_{l}}V\\\
<EMAIL_ADDRESS>
Axioms I2 and I4 ensure that all morphisms in $\mathbf{Inf}$ are, in fact,
isomorphisms. In this language, a smooth map $f\colon\Sigma\rightarrow M$ is a
primitive of an infinitesimal immersion $\omega$ if and only if $\omega$ and
$\delta f$ are isomorphic in $\mathbf{Inf}$. These abstractions are further
justified by the following:

######  Theorem ([6]).

Suppose $M$ is a simple Klein geometry and let
$f_{1},f_{2}\colon\Sigma\rightarrow M$ be smooth maps. Then there exists an
isomorphism $\delta f_{1}\cong\delta f_{2}$ in $\mathbf{Inf}$ if and only if
there exists $\phi\in\operatorname{Aut}(M)$ such that $f_{2}=\phi\,\circ
f_{1}$.

In particular, smooth immersions $f_{1},f_{2}\colon\Sigma\rightarrow M$
agreeing up to a symmetry of $M$ have isomorphic logarithmic derivatives.

###  3.2. Invariants defined

While the magnanimous reader will generally understand the term ‘invariant’
without further comment, to mitigate possible confusion we state here formal
definitions sufficient for the sequel. Stronger definitions are possible, but
only at the cost of further abstraction we prefer to avoid.

Suppose ${\mathcal{C}}$ is a collection of smooth immersions
$f\colon\Sigma\rightarrow M$ closed under the action of
$\operatorname{Aut}(M)$. Then a map $f\mapsto Q_{f}$ from ${\mathcal{C}}$ to
some category will be called an invariant for ${\mathcal{C}}$ if, for all
$\phi\in\operatorname{Aut}(M)$, there exists an isomorphism $Q_{\phi\circ
f}\cong Q_{f}$. An invariant for subcategory ${\mathcal{D}}$ of $\mathbf{Inf}$
will be any functor $\omega\mapsto Q_{\omega}$ into another category. A
trivial corollary of the preceding theorem is:

######  Proposition.

Every invariant for infinitesimal immersions delivers an invariant for smooth
immersions. Specifically, with ${\mathcal{C}}$ and ${\mathcal{D}}$ as above,
and assuming $\delta({\mathcal{C}})\subset{\mathcal{D}}$, every invariant
$\omega\mapsto Q_{\omega}$ for ${\mathcal{D}}\subset\mathbf{Inf}$ delivers an
invariant $f\mapsto Q_{\delta f}$ for the collection of smooth immersions
${\mathcal{C}}$.

###  3.3. Invariants defined by polynomials

Let $M$ be a homogeneous $G$-space and
$Q\colon{\mathfrak{g}}\rightarrow{\mathbb{K}}$ a polynomial invariant under
the adjoint representation of $G$; here ${\mathbb{K}}$ is ${\mathbb{R}}$ or
${\mathbb{C}}$. For each infinitesimal immersion $\omega\colon
A\rightarrow{\mathfrak{g}}$ we define

$Q_{\omega}=Q\circ\omega\colon A(f)\rightarrow{\mathbb{K}}$

and immediately obtain:

######  Proposition.

Let $\omega\colon A\rightarrow{\mathfrak{g}}$ and $\omega^{\prime}\colon
A^{\prime}\rightarrow{\mathfrak{g}}$ be isomorphic infinitesimal immersions
and $\lambda\colon A\rightarrow A^{\prime}$ a corresponding isomorphism. Then
the following diagram commutes

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In particular, $\omega\mapsto Q_{\omega}$ is an invariant for $\mathbf{Inf}$.

Similarly, for any $\operatorname{Ad}$-invariant subalgebra
${\mathfrak{g}}^{\prime}\subset{\mathfrak{g}}$, such the commutator
$[{\mathfrak{g}},{\mathfrak{g}}]$ or radical of ${\mathfrak{g}}$, the pre-
image $A^{\prime}:=\omega^{-1}({\mathfrak{g}}^{\prime})$ is Lie subalgebroid
of $A$, assuming it has constant rank (because $\omega$ is a Lie algebroid
morphism), and is an invariant for $\mathbf{Inf}$. Any
$\operatorname{Ad}$-invariant polynomial on ${\mathfrak{g}}^{\prime}$
furnishes yet another invariant.

######  Remark.

In applying the Construction Principle of §2.3 it is important to distinguish
between polynomials on ${\mathfrak{g}}$ that are merely invariant under the
adjoint action (inner invariants) and those invariant under arbitrary Lie
algebra automorphisms (outer invariants) for the following reason. In
Proposition 2.3(Proposition) the infinitesimal immersion $\omega\colon
A\rightarrow{\mathfrak{g}}$ is only determined up to an automorphism of
${\mathfrak{g}}$, because of the choice of isomorphism
${\mathfrak{g}}^{\prime}\cong{\mathfrak{g}}$. Therefore, given an
$\operatorname{Ad}$-invariant polynomial $Q$ on ${\mathfrak{g}}$, the
corresponding map $Q_{\omega}\colon A\rightarrow{\mathbb{K}}$ depends on the
choice made, unless $Q$ is an outer invariant (e.g., if $Q$ is the Killing
form). Concrete illustrations given later will make this point clearer.

Note that an $\operatorname{Ad}$-invariant polynomial
$Q\colon{\mathfrak{g}}\rightarrow{\mathbb{K}}$ defines, in a trivial way, a
map $Q\colon E\rightarrow{\mathbb{K}}$ on the trivial bundle
$E={\mathfrak{g}}\times\Sigma$, which will be called an invariant polynomial
on $E$ and denoted $Q$ also. Then, when we use an infinitesimal immersion
$\omega\colon A\rightarrow{\mathfrak{g}}$ to identify $A$ with a subbundle of
$E={\mathfrak{g}}\times\Sigma$, $Q_{\omega}$ is the composite of the inclusion
$A\hookrightarrow E$ with $Q\colon E\rightarrow{\mathbb{K}}$.

###  3.4. The fundamental representation

Continuing with the notation of §2.3, one has the following representation
$\bar{\nabla}$ of the Lie algebroid $A$ on $E$:

(1) $\bar{\nabla}_{X}\xi=\nabla_{\\#X}\xi+\\{X,\xi\\}.$

That $\bar{\nabla}$ is indeed a representation follows from the Mauer-Cartan
equation 2.3(2), the fact that $\\{\,\cdot\,,\,\cdot\,\\}$ is
$\nabla$-parallel, and the Jacobi identity for $\\{\,\cdot\,,\,\cdot\,\\}$. We
shall call this representation the fundamental representation. We note that
the Mauer-Cartan equation also allows us to write

(2) $\bar{\nabla}_{X}Y=\nabla_{\\#Y}X+[X,Y],$

in the special case that $X$ and $Y$ are both sections of $A$.

######  Lemma.

Any $\operatorname{Ad}$-invariant polynomial $Q$ on $E$ is invariant with
respect to the fundamental representation. The radical
$\operatorname{\operatorname{rad}}E=\operatorname{\operatorname{rad}}{\mathfrak{g}}\times\Sigma$,
the commutator $\\{E,E\\}$, and elements of the upper or lower derived series
of $E$, are all invariant under the fundamental representation.

###### Proof.

For the first statement, suppose $Q$ is quadratic, the other cases being
similar. View $Q$ as a symmetric bilinear form on $E$. Then, because
$\operatorname{Ad}$-invariance implies $\operatorname{ad}$-invariance, we have
$Q(\\{X,\xi_{1}\\},\xi_{2})+Q(\xi_{1},\\{X,\xi_{2}\\})$ for any
$X,\xi_{1},\xi_{2}\in\Gamma(E)$. It is trivial that $Q$ is $\nabla$-parallel,
and so not hard to see from (1) that $\bar{\nabla}_{X}Q=0$ for any
$X\in\Gamma(A)$. The second statement follows from the fact that the bracket
$\\{\,\cdot\,,\,\cdot\,\\}$ is invariant with respect to the fundamental
representation, which follows from its $\nabla$-invariance and the Jacobi
identity. ∎

The fundamental representation gives an alternative characterization of the
invariant filtration to be described next.

###  3.5. The invariant filtration

In general $A\subset E$ is not $\nabla$-invariant (or invariant under the
fundamental representation $\bar{\nabla}$). We are therefore led to define,
for each $x\in\Sigma$,

$A^{2}|_{x}=\\{X(x)\,\left|\right.\,\text{$X\in\Gamma(A)$ and $\nabla_{u}X\in
A$ for all $u\in T_{x}\Sigma$}\\},$

and, assuming the dimension of $A^{2}|_{x}$ is independent of $x$, obtain a
vector subbundle $A^{2}:=\amalg_{x\in\Sigma}A^{2}|_{x}\subset A$. Tacitly
assuming rank-constancy at each stage, we inductively define, for each
$j\geqslant 3$, a subbundle $A^{j}\subset A^{j-1}$ with fibres given by

$A^{j}|_{x}=\\{X(x)\,\left|\right.\,\text{$X\in\Gamma(A^{j-1})$ and
$\nabla_{u}X\in A^{j-1}$ for all $u\in T_{x}\Sigma$}\\},$

and obtain a filtration

$A\supset A^{2}\supset A^{3}\supset A^{4}\supset\cdots$

of vector subbundles, which are in fact subalgebroids, by the proposition
below. This filtration, determined by the infinitesimal immersion $\omega$,
will be called the associated invariant filtration, for one readily proves the
following:

######  Lemma.

Let $\omega$ and $\omega^{\prime}$ be isomorphic Mauer-Cartan forms and
$L\colon A\rightarrow A^{\prime}$ an isomorphism, as in Proposition 3.3. Then
$L(A^{j})=(A^{\prime})^{j}$ for all $j$.

In particular, the restriction to $A(f)^{j}$ of a invariant polynomial
$Q\colon E\rightarrow{\mathbb{K}}$ of a smooth map $f\colon\Sigma\rightarrow
M$ is an invariant of $f$, generally ‘finer’ than its restriction to $A(f)$.

######  Proposition.

For all $j\geqslant 1$, $A^{j}\subset A$ is a subalgebroid. In the case that
$A^{j}$ is transitive, $A^{j+1}$ is the isotropy subalgebroid of $A^{j}\subset
E$ under the restriction of the fundamental representation to a representation
of $A^{j}$.

###### Proof.

For a proof of the first statement by an induction on $j$, suppose
$A^{j}\subset A$ is a subalgebroid and let $X$ and $Y$ be sections of
$A^{j+1}$, so that $\nabla_{U}X$ and $\nabla_{U}Y$ are sections of $A^{j}$ for
all vector fields $U$ on $\Sigma$. We must show $\nabla_{U}[X,Y]$ is a section
of $A^{j}$ also. To this end, replace $X$ in the Mauer-Cartan equation 2.3(2)
with $\nabla_{U}X$ to conclude

(1) $\nabla_{\\#Y}\nabla_{U}X-\\{\nabla_{U}X,Y\\}\text{~{}is a section of
$A^{j}$}.$

Similarly, replace $Y$ in 2.3(2) with $\nabla_{U}Y$ to conclude

(2) $\nabla_{\\#X}\nabla_{U}Y+\\{X,\nabla_{U}Y\\}\text{~{}is a section of
$A^{j}$}.$

Finally, apply $\nabla_{U}$ to both sides of 2.3(2) and, appealing to the
flatness of $\nabla$ and the $\nabla$-invariance of
$\\{\,\cdot\,,\,\cdot\,\\}$, show that

$\displaystyle\nabla_{U}[X,Y]=$
$\displaystyle(\nabla_{\\#X}\nabla_{U}+\\{X,\nabla_{U}Y\\})-(\nabla_{\\#Y}\nabla_{U}X-\\{\nabla_{U}X,Y\\})$
$\displaystyle+\nabla_{[U,\\#Y]}X-\nabla_{[U,\\#X]}Y.$

With the help of (1) and (2) we now see that $\nabla_{U}[X,Y]$ must be a
section of $A^{j}$.

The second statement in the lemma follows easily enough from 3.4(2). ∎

###  3.6. Symmetries of infinitesimal immersions and subgeometries

A connection $\nabla$ on an arbitrary Lie algebroid $A$ is a Cartan connection
if it suitably respects the Lie algebroid structure of $A$. The canonical flat
connection $\nabla$ on the action algebroid ${\mathfrak{g}}\times M$ is such a
connection. Conversely, any Lie algebroid equipped with a flat Cartan
connection is called by us a twisted Lie algebra action, for it is locally an
action algebroid [1], and globally an action algebroid with ‘monodromy twist’
[3]. In the same way that a Lie algebra action integrates to a local Lie group
action, so, more generally, every twisted Lie algebra action integrates to a
pseudoaction, a geometric object encoding the pseudogroup of transformations
generated by the flows of the (locally defined) infinitesimal generators [5].

Let $\\{A^{j}\\}_{j\geqslant 1}$ be the invariant filtration defined by an
infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$. Notice that
if $A^{k+1}=A^{k}$, for some $k\geqslant 1$, then $A^{j}=A^{k}$ for all
$j\geqslant k$ — i.e., the filtration has stabilised at $A^{k}$. In our
finite-dimensional setting all invariant filtrations must stabilise eventually
(under our constant rank assumption) although $A^{k}=0$ is typical.

######  Theorem.

Suppose $A^{k+1}=A^{k}$. Then the canonical flat connection $\nabla$ on $E$
restricts to a flat Cartan connection on $A^{k}$, i.e., defines a twisted Lie
algebra action on $\Sigma$ which is, in fact, an ordinary action by a
subalgebra ${\mathfrak{s}}\subset{\mathfrak{g}}$. In the case that $\omega$ is
a logarithmic derivative of a smooth immersion $f\colon\Sigma\rightarrow M$,
elements of the corresponding pseudogroup of transformations on $\Sigma$
correspond to restrictions to open subsets of $f(\Sigma)\subset M$ of left
translations $m\mapsto g\cdot m$ in $M$, $g\in G$.

###### Proof.

If $A^{k+1}=A^{k}$, then $\nabla$ restricts to a connection on $A^{k}$ by the
definition of $A^{k+1}$. Now the restricted connection — also denoted $\nabla$
— is Cartan if its cocurvature vanishes. But, from the definition of
cocurvature (see [1]) follows the formula

$\operatorname{cocurv}\nabla(X,Y)U=\nabla_{U}\operatorname{tor}\bar{\nabla}\,(X,Y)+\operatorname{curv}\nabla(U,\\#X)Y-\operatorname{curv}\nabla(U,\\#Y)X,$

where
$\operatorname{tor}\bar{\nabla}(X,Y)=\nabla_{\\#Y}X-\nabla_{\\#X}Y+[X,Y]$. In
the present case $\operatorname{curv}\nabla=0$, while
$\operatorname{tor}\bar{\nabla}(X,Y)=\\{X,Y\\}$, on account of the Mauer-
Cartan equation 2.3(2). So the cocurvature vanishes.

Since $\nabla$ is a flat Cartan connection and $\Sigma$ is simply-connected,
the subspace ${\mathfrak{s}}\subset\Gamma(A^{k})$ of $\nabla$-parallel
sections is a Lie subalgebra acting infinitesimally on $\Sigma$ according to
$\xi^{\dagger}(x)=\\#(\xi(x))$ [1, Theorem A]. Trivially, every such section
is also $\nabla$-parallel as a section of $E={\mathfrak{g}}\times\Sigma$, and
so is nothing but the restriction to $\Sigma$ of an infinitesimal generator of
the action of ${\mathfrak{g}}$ on $M$. In particular, its local flows are
restrictions to open subsets of $\Sigma$ of left translations $m\mapsto g\cdot
m$ in $M$, $g\in G$. ∎

###  3.7. The invariant filtration in terms of isotropy

The isotropy characterization of of Proposition 3.5 of the Lie algebroids
appearing in an invariant filtration can be described more concisely. While
$A\subset E$ is not generally invariant with respect to the fundamental
representation, the kernel ${\mathfrak{h}}\subset A$ of its anchor is
invariant, by the identity 3.4(2). It follows that the fundamental
representation drops to a representation of $A$ on $E/{\mathfrak{h}}$. Under
the canonical identification $A/{\mathfrak{h}}\cong T\Sigma$, which we have by
virtue of the transitivity of $A$, we obtain a canonical inclusion of
$T\Sigma$ into $E/{\mathfrak{h}}$. Evidently, $A^{2}\subset A$ must coincide
with the isotropy of $T\Sigma\subset E/{\mathfrak{h}}$ with respect to this
representation. One continues in a similar fashion for the rest of the
filtration and obtains, with $A^{0}=E$ and $A^{1}=A$:

######  Proposition.

Assume $A^{j}$ is transitive, $j\geqslant 1$, and let
${\mathfrak{h}}^{j}\subset A^{j}$ denote the kernel of its anchor. Then the
fundamental representation restricts to a representation of $A^{j}$ that
leaves $A^{j-1}\subset E$ and ${\mathfrak{h}}^{j}\subset A^{j-1}$ invariant,
dropping to a representation of $A^{j}$ on $A^{j-1}/{\mathfrak{h}}^{j}$.
Moreover, $A^{j+1}$ is the isotropy of $T\Sigma\subset
A^{j-1}/{\mathfrak{h}}^{j}$ when one views $T\Sigma$ as a subbundle of
$A^{j-1}/{\mathfrak{h}}^{j}$ using the isomorphism $T\Sigma\cong
A^{j}/{\mathfrak{h}}^{j}$ determined by the anchor of $A_{j}$. In particular,
$A^{j+1}$ acts canonically on $T\Sigma$.

######  Remark.

In the special case that $\omega$ is the logarithmic derivative of an
immersion $f\colon\Sigma\rightarrow M$, $E/{\mathfrak{h}}$ has a canonical
identification with the pullback $T_{\Sigma}M$ of $TM$ because the morphism
$E\rightarrow T_{\Sigma}M$ sending $(\xi,m)$ to $\xi^{\dagger}(m)$ is
surjective. So the fundamental representation induces a representation of
$A(f)$ on $T_{\Sigma}M$ and $A(f)^{2}$ is the isotropy of $T\Sigma\subset
T_{\Sigma}M$ under this representation.

###  3.8. Orientability of the abstract model of $T_{\Sigma}M$

Of course if $M$ is orientable, then so is $T_{\Sigma}M$. In general, however,
the orientability of $E/{\mathfrak{h}}$ is a subtle question. The issue is
settled in the simplest scenario as follows:

######  Proposition.

Let $\omega\colon A\rightarrow{\mathfrak{g}}$ be an infinitesimal immersion
and ${\mathfrak{h}}$ the kernel of the anchor $\\#\colon A\rightarrow
T\Sigma$. Assume that $M$ is oriented and that the action of $G$ on $M$ is
orientation-preserving. Suppose, moreover, that for some $m_{0}\in M$, we have
$N_{G}(G_{m_{0}})=G_{m_{0}}$ (cf. Proposition 2.2) and that $G_{m_{0}}$ is
connected. Then the orientation on $M$ determines a natural orientation of
$E/{\mathfrak{h}}$ and this orientation is an invariant of $\omega$. If
$\omega$ is the logarithmic derivative of an infinitesimal immersion, then the
orientation of $E/{\mathfrak{h}}$ coincides with that of $T_{\Sigma}M$ under
the canonical identification $E/{\mathfrak{h}}\cong T_{\Sigma}M$.

###### Proof.

Let $x\in\Sigma$. Then by Axiom I3 and its consequence 2.3(1), there exists
$m\in M$ such that the fibre ${\mathfrak{h}}|_{x}$ coincides with
${\mathfrak{g}}_{m}\times\\{x\\}\subset E$, under our tacit identification of
${\mathfrak{h}}$ with a subbundle of $E={\mathfrak{g}}\times\Sigma$. So
$(E/{\mathfrak{h}})|_{x}\cong{\mathfrak{g}}/{\mathfrak{g}}_{m}$. For some
$g\in G$ we have
${\mathfrak{g}}_{m}=\operatorname{Ad}_{g}({\mathfrak{g}}_{m_{0}})$ (pick $g$
to satisfy $m=g\cdot m_{0}$). The morphism
$\operatorname{Ad}_{g}\colon{\mathfrak{g}}\rightarrow{\mathfrak{g}}$ drops to
an isomorphism
${\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}\rightarrow{\mathfrak{g}}/{\mathfrak{g}}_{m}$,
also denoted $\operatorname{Ad}_{g}$ below. Now the action of $G$ determines
an canonical isomorphism
$T_{m_{0}}M\cong{\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}$, so we obtain a
sequence of isomorphisms

(1)
$T_{m_{0}}M\cong{\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}\xrightarrow{\operatorname{Ad}_{g}}{\mathfrak{g}}/{\mathfrak{g}}_{m}\cong(E/{\mathfrak{h}})|_{x}.$

We use this isomorphism to transfer the orientation of $M$ to an orientation
of $(E/{\mathfrak{h}})|_{x}$. It remains to show that this orientation is
independent of the choices of $m\in M$ and $g\in G$. Indeed, suppose that
${\mathfrak{g}}_{m}={\mathfrak{g}}_{m^{\prime}}$ and
$\operatorname{Ad}_{g^{\prime}}({\mathfrak{g}}_{m_{0}})={\mathfrak{g}}_{m^{\prime}}$,
for some $m^{\prime}\in M$ and $g^{\prime}\in G$. Then
$\operatorname{Ad}_{g^{\prime}}({\mathfrak{g}}_{m_{0}})={\mathfrak{g}}_{m}$
and so $\operatorname{Ad}_{k}({\mathfrak{g}}_{m_{0}})={\mathfrak{g}}_{m_{0}}$,
where $k=g^{\prime}g^{-1}$. Since $G_{m_{0}}$ is connected, this implies
$kG_{m_{0}}k^{-1}=G_{m_{0}}$, i.e., $k\in N_{G}(G_{m_{0}})=G_{m_{0}}$. With
our new choices the isomorphism in (1) is replaced by the composite

$T_{m_{0}}M\cong{\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}\xrightarrow{\operatorname{Ad}_{k}}{\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}\xrightarrow{\operatorname{Ad}_{g}}{\mathfrak{g}}/{\mathfrak{g}}_{m}\cong(E/{\mathfrak{h}})|_{x}.$

Because $k\in G_{m_{0}}$, this composite coincides with the composite

(2)
$T_{m_{0}}M\xrightarrow{TL_{k}}T_{m_{0}}M\cong{\mathfrak{g}}/{\mathfrak{g}}_{m_{0}}\xrightarrow{\operatorname{Ad}_{g}}{\mathfrak{g}}/{\mathfrak{g}}_{m}\cong(E/{\mathfrak{h}})|_{x},$

where $TL_{k}$ is the tangent lift of $L_{k}(n):=k\cdot n$, $n\in M$. As the
action of $G$ on $M$ is orientation-preserving, whether we declare an
orientation on $(E/{\mathfrak{h}})|_{x}$ using the isomorphism (1) or (2)
therefore makes no difference. The claims made regarding this orientation
follow by construction. ∎

###  3.9. The globalisation of $A(f)^{2}$

We now inject some observations which are not essential to the computation of
invariant filtrations, but which help to anticipate their outcomes in the case
of a logarithmic derivative. Let $f\colon\Sigma\rightarrow M$ be an injective
immersion. By definition, we may view $A(f)$ as the Lie algebroid of the
pullback ${\mathcal{G}}(f)$ by $f$ of the action groupoid $G\times M$, given
by 1.2(1). The tangent-lifted action of $G$ on $TM$ determines an action of
the Lie groupoid $G\times M$ on $TM$, which pulls back to an action of
${\mathcal{G}}(f)$ on $T_{\Sigma}M$.

######  Proposition.

Let ${\mathcal{G}}(f)^{2}\subset{\mathcal{G}}(f)$ denote the isotropy of
$T\Sigma\subset T_{\Sigma}M$ under the representation of ${\mathcal{G}}(f)$ on
$T_{\Sigma}M$. Then $A(f)^{2}\subset A(f)$ is the Lie algebroid of
${\mathcal{G}}(f)^{2}$.

###### Proof.

By Remark 3.7, it suffices to show that the representation of $A(f)$ on
$T_{\Sigma}M$ determined by the fundamental representation is nothing more
than the infinitesimalization of the representation of ${\mathcal{G}}(f)$ on
$T_{\Sigma}M$ described above. To this end, note first that the tangent action
of $G\times M$ on $TM$ infinitesimalizes to a representation $\bar{\nabla}$ of
${\mathfrak{g}}\times M$ on $TM$ given by

(1) $\bar{\nabla}_{X}U=\\#\nabla_{U}X+[\\#X,U],$

where $\nabla$ is the canonical flat connection on ${\mathfrak{g}}\times M$
and $\\#$ the anchor map, $\\#(\xi,m)=\xi^{\dagger}(m)$. If $U=\\#Y$ for some
section $Y$ of ${\mathfrak{g}}\times M$ then we may rewrite (1) as

$\bar{\nabla}_{X}U=\\#(\nabla_{\\#X}Y+\\{X,Y\\}),$

where $\\{X,Y\\}:=\nabla_{\\#Y}X-\nabla_{\\#X}Y+[X,Y]$. Evidently this
representation pulls back, under $f$, to the representation of $A(f)$ on
$T_{\Sigma}M\cong E/{\mathfrak{h}}$ induced by the fundamental representation
3.4(1). ∎

Globalisations of $A(f)^{j}$ for $j\geqslant 3$ will be described elsewhere.

###  3.10. Intrinsic geometry

Suppose $j\geqslant 1$ and that the Lie algebroid $A^{j}$ in an invariant
filtration is transitive. Then, according to Proposition 3.7, $A^{j+1}$ acts
canonically on $T\Sigma$. Such a representation equips $\Sigma$ with an
‘intrinsic infinitesimal geometry’ in the following way. View the
representation as a Lie algebroid morphism
$A^{j+1}\rightarrow\mathfrak{gl}[T\Sigma]$, where $\mathfrak{gl}[T\Sigma]$ is
the Lie algebroid of derivations on $T\Sigma$ (the Lie algebroid of the frame
groupoid of $T\Sigma$). The adjoint representation $\operatorname{ad}\colon
J^{1}(T\Sigma)\rightarrow\mathfrak{gl}[T\Sigma]$, defined by
$\operatorname{ad}_{J^{1}X}Y=[X,Y]$, is an isomorphism, allowing us to regard
the image of $A^{j+1}\rightarrow\mathfrak{gl}[T\Sigma]$ as a subalgebroid of
$J^{1}(T\Sigma)$, i.e., as in infinitesimal geometric structure, in the sense
of [2]. For example, if $A^{j+1}\cong T\Sigma$ then its representation on
$T\Sigma$ amounts to an infinitesimal parallelism on $\Sigma$. In the case of
a hypersurface $f\colon\Sigma\hookrightarrow M$ in a Riemannian space form,
the representation of $A^{2}(f)$ on $T\Sigma$ determines a representation on
$S^{2}\,T^{*}\Sigma$ whose sole invariant sections are constant multiples of
the inherited metric; see §4.8.

###  3.11. Planar curves

For a simple illustration of the theory now developed, let us recover the
following well-known fact: A unit-speed planar curve $f\colon
I\rightarrow{\mathbb{R}}^{2}$, defined on some interval $I$, is completely
characterized, up to orientation-preserving rigid motions, by its curvature
$\kappa$, an invariant under such motions.

Let $\xi,\eta$ denote the standard coordinate functions on ${\mathbb{R}}^{2}$
and $x,y\colon I\rightarrow{\mathbb{R}}^{2}$ their pullbacks under a regular
curve $f\colon I\rightarrow{\mathbb{R}}^{2}$. The plane is a homogeneous
$G$-space if $G$ is the group of orientation-preserving rigid motions of the
plane, motions that preserve the speed of curves. The corresponding Lie
algebra ${\mathfrak{g}}$ of planar vector fields admits the basis
$\\{e_{1},e_{2},e_{3}\\}$, where

(1) $e_{1}=\frac{\partial}{\partial\xi},\qquad
e_{2}=\frac{\partial}{\partial\eta},\qquad
e_{3}=-\eta\frac{\partial}{\partial\xi}+\xi\frac{\partial}{\partial\eta}\,.$

We have $[e_{1},e_{2}]=0$, $[e_{3},e_{1}]=-e_{2}$, and $[e_{3},e_{2}]=e_{1}$.

Anticipating that polynomial invariants on ${\mathfrak{g}}$ are going to play
a role, we let $Q$ denote the positive definite quadratic form on the
commutator
$[{\mathfrak{g}},{\mathfrak{g}}]=\operatorname{span}\\{e_{1},e_{2}\\}$, with
respect to which $e_{1}$ and $e_{2}$ form an orthonormal basis, and let
$\mu\colon{\mathfrak{g}}\rightarrow{\mathbb{R}}$ be the linear form vanishing
on $[{\mathfrak{g}},{\mathfrak{g}}]$ with $\mu(e_{3})=1$. Both $Q$ and $\mu$
are $\operatorname{Ad}$-invariant, and these invariants encode the structure
of ${\mathfrak{g}}$: Agreeing to give $[{\mathfrak{g}},{\mathfrak{g}}]$ the
orientation defined by the ordered basis $(e_{1},e_{2})$, then for any other
positively-oriented, $Q$-orthogonal basis $(x,y)$, where $x$ and $y$ have the
same $Q$-length, we have $[x,y]=0$ and, for all $z\in{\mathfrak{g}}$,

(2) $[z,x]=-\mu(z)y,\qquad[z,y]=\mu(z)x.$

Viewing some $z\in{\mathfrak{g}}$ concretely as a Killing field, $\mu(z)$ is
the curl of $z$. Of course $[{\mathfrak{g}},{\mathfrak{g}}]$ is just the space
of constant vector fields on ${\mathbb{R}}^{2}$. Bearing Remark 3.3 in mind,
we record for later use:

######  Lemma A.

Given $\lambda>0$ there exists an automorphism of ${\mathfrak{g}}$ pulling $Q$
back to $\lambda Q$.

###### Proof.

Consider the automorphism $e_{1}\mapsto\lambda e_{1}$, $e_{2}\mapsto\lambda
e_{2}$, $e_{3}\mapsto e_{3}$. ∎

Since $f$ is regular (an immersion) we may identify $A(f)$ with the subbundle
of $E:={\mathfrak{g}}\times I$ consisting of those $(\zeta,t)$ for which the
Killing field $\zeta(t)$ is tangent to $f$ at $f(t)$. If
$\\{E,E\\}:=[{\mathfrak{g}},{\mathfrak{g}}]\times I\subset E$, then the
invariants $Q$ and $\mu$ define ${\mathbb{R}}$-valued maps on $\\{E,E\\}$ and
$E$ we denote by the same symbols. Evidently, $A(f)$ has rank two and there
exists a unique section $X$ of $A(f)\cap\\{E,E\\}$ such that $X$ has constant
$Q$-length one, and such that $\\#X=\frac{1}{s}\frac{\partial}{\partial t}$
for some positive function $s\colon I\rightarrow(0,\infty)$, namely the speed
of $f$:

$X=\frac{\dot{x}}{s}\frac{\partial}{\partial\xi}+\frac{\dot{y}}{s}\frac{\partial}{\partial\eta}.$

Next, choose $Y\in\Gamma(\\{E,E\\})$ such that $(X,Y)$ forms a positively
oriented $Q$-orthonormal basis of each fibre of $\\{E,E\\}$:

$Y=-\frac{\dot{y}}{s}\frac{\partial}{\partial\xi}+\frac{\dot{x}}{s}\frac{\partial}{\partial\eta}.$

The map $Q\colon\\{E,E\\}\rightarrow{\mathbb{R}}$ is tautologically invariant
with respect to the canonical flat connection $\nabla$. We now differentiate
$Q(X,X)=1$ to obtain $Q(X,\nabla_{\partial/\partial t}X)=0$, implying that

(3) $\nabla_{\partial/\partial t}X=s\kappa Y,$

for some function $\kappa\colon I\rightarrow{\mathbb{R}}$, namely the
curvature. Differentiating the identities $Q(X,Y)=0$ and $Q(Y,Y)=1$, we deduce
that

(4) $\nabla_{\partial/\partial t}Y=-s\kappa X.$

Finally, a section $Z$ of the rank-one kernel of the anchor $\\#\colon
A(f)\rightarrow TI$ is fixed by requiring $\mu(Z)=1$. Since the Killing field
$Z(t)$ must vanish at $f(t)=(x(t),y(t))$, we obtain

$Z=(y-\eta)\frac{\partial}{\partial\xi}-(x-\xi)\frac{\partial}{\partial\eta}.$

Since $\partial/\partial\xi$ and $\partial/\partial\eta$ are constant sections
of ${\mathfrak{g}}\times I$, we obtain, on differentiating,

(5) $\nabla_{\partial/\partial
t}Z=\dot{y}\frac{\partial}{\partial\eta}-\dot{x}\frac{\partial}{\partial\eta}=-sY.$

The sections $X,Y,Z$ furnish a basis for each fibre of $E={\mathfrak{g}}\times
I$ and $A(f)$ is spanned by $X$ and $Z$. The algebraic bracket on $E$ is given
by

(6) $\\{X,Y\\}=0,\qquad\\{Z,X\\}=-Y,\qquad\\{Z,Y\\}=X,$

where the last two relations follow from (2) or direct computation. Since
$\\#X=\frac{1}{s}\frac{\partial}{\partial t}$ and $\\#Z=0$, the Mauer-Cartan
equation 2.3(2) shows that $[X,Z]=0$, which completes the description of the
Lie algebroid structure of $A(f)$.

From (3)–(5), and our definition of the invariant filtration, we see that
$A(f)^{2}$ is generated by the single section $X^{2}:=s(X+\kappa Z)$, which
the anchor maps to $\partial/\partial t$, giving us an identification
$A(f)^{2}\cong TI$.

######  Remark.

Notice that $X^{2}$ has the following geometric interpretation: Evaluated at
$t\in I$, it is the planar Killing field whose integral curve through $f(t)$
is the circle best approximating the curve $f$ at $f(t)$ (Figure 1).

Figure 1. The vector field $X^{2}(t)\in{\mathfrak{g}}$, where
$X^{2}=s(X+\kappa Z$).

One computes $\nabla_{\partial/\partial t}X^{2}=\dot{s}X+(s\kappa)\dot{~{}}Z$,
from which it follows that $A(f)^{3}=A(f)^{2}$ when the the speed and
curvature are constant, and $A(f)^{3}=0$ otherwise. So, according to Theorem
3.6, we recover unit speed circular arcs and straight line segments as the
curves with non-trivial symmetry.

With the help of (3)–(6) we can compute the fundamental representation; it is
given by

$\displaystyle\bar{\nabla}_{X}X$ $\displaystyle=\kappa Y,\qquad$
$\displaystyle\bar{\nabla}_{X}Y$ $\displaystyle=-\kappa X,\qquad$
$\displaystyle\bar{\nabla}_{X}Z$ $\displaystyle=0$
$\displaystyle\bar{\nabla}_{Z}X$ $\displaystyle=-Y$
$\displaystyle\bar{\nabla}_{Z}Y$ $\displaystyle=X$
$\displaystyle\bar{\nabla}_{Z}Z$ $\displaystyle=0.$

We leave the reader to show that the canonical representation of
$A(f)^{2}\cong{T}{I}$ on $TI$ described in Proposition 3.7 is given by

$\bar{\nabla}_{\partial/\partial t}\,\frac{\partial}{\partial
t}=\frac{\dot{s}}{s}\frac{\partial}{\partial t}.$

In particular, $\partial/\partial t$ is parallel in the intrinsic geometry
inherited by $I$ à la §3.10 precisely when the speed is constant.

Our next task is to construct invariants of arbitrary infinitesimal immersions
$\omega\colon A\rightarrow{\mathfrak{g}}$. To this end, we state:

######  Lemma B.

Let $\omega\colon A\rightarrow{\mathfrak{g}}$ be any infinitesimal immersion
of $I$ into ${\mathbb{R}}^{2}$ and use $\omega$ to identify $A$ with a
subbundle of $E={\mathfrak{g}}\times I$. Then $A\cap\\{E,E\\}$ has rank one
and is mapped by the anchor $\\#\colon A\rightarrow TI$ onto $TI$.

###### Proof.

Since $I$ is simply-connected, $\omega$ admits a primitive $f\colon
I\rightarrow{\mathbb{R}}^{2}$, by Theorem 1.3. Since $\omega$ and $\delta f$
are isomorphic, the lemma follows because it is already true for the
logarithmic derivatives of curves. ∎

For an infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$ we
can accordingly find a unique section $X$ of $A\cap\\{E,E\\}$ with constant
$Q$-length one such that $\\#X=\frac{1}{s}\frac{\partial}{\partial t}$ for
some positive function $s\colon I\rightarrow(0,\infty)$ we call the speed of
$\omega$. We choose a second section $Y$ of $\\{E,E\\}$ as we did for curves
and, arguing as before, obtain a function $\kappa\colon
I\rightarrow{\mathbb{R}}$ defined by (3) that we call the curvature of
$\omega$. By construction, the speed and curvature of the logarithmic
derivative $\delta f$ of a curve $f$ coincide with the usual speed and
curvature of $f$. It is clear that speed and curvature are invariants of an
infinitesimal immersion.

We remark that the curvature of $\omega$ can be equivalently defined by
$\kappa=\mu(X^{2})/s$ where $X^{2}$ is the unique section of $A^{2}$ such that
$\\#X^{2}=\partial/\partial t$.

We now apply the Construction Principle of §2.3 to obtain:

######  Theorem.

For every smooth function $\kappa\colon I\rightarrow{\mathbb{R}}$ there exists
a unit-speed curve $f\colon I\rightarrow{\mathbb{R}}^{2}$ with curvature
$\kappa$, unique up to orientation-preserving isometries.

###### Proof.

Let $X,Y,Z$ denote the constant sections of the trivial bundle
$E:={\mathbb{R}}^{3}\times I$ and define an algebraic bracket
$\\{\,\cdot\,,\,\cdot\,\\}$ on $E$ by the relations (6). This makes $E$ into a
${\mathfrak{g}}$-bundle, with ${\mathfrak{g}}$ defined as above. Define a
connection $\nabla$ on $E$ by (3)—(5), with $s=1$, and verify that the
algebraic bracket is $\nabla$-parallel. Make the subbundle $A\subset E$
spanned by $X$ and $Z$ into a Lie algebroid by declaring $[X,Z]=0$,
$\\#X=\partial/\partial t$ and $\\#Z=0$. Then $A$ is integrable and the Mauer-
Cartan equation 2.3(2) holds. Applying Proposition 2.3, we obtain an
isomorphism $E\cong{\mathfrak{g}}\times I$ such that the composite of
$A\hookrightarrow E$ with the projection $E\rightarrow{\mathfrak{g}}$ is Lie
algebroid morphism $\omega\colon A\rightarrow{\mathfrak{g}}$. It is easy to
see that $\omega\colon A\rightarrow{\mathfrak{g}}$ must be an infinitesimal
immersion. Since $\\{E,E\\}\subset E$ must be $\nabla$-invariant, we have
$\nabla_{\partial/\partial t}X\in\Gamma(\\{E,E\\})$. Since $Q$, viewed as an
inner product on $E\cong{\mathfrak{g}}\times I$, is $\nabla$-parallel, we
compute

$\frac{\partial}{\partial t}Q(X,X)=2Q(X,\nabla_{\partial/\partial
t}X)=2\kappa\,Q(X,Y).$

On the other hand, since $Q$ is symmetric and $\operatorname{ad}$-invariant,
we have

$0=Q(X,\\{Z,X\\})=Q(X,Y).$

Combining these two equations, we see that $Q(X,X)$ is constant. Applying
Lemma A, we may arrange, by composing $\omega\colon
A\rightarrow{\mathfrak{g}}$ with an outer automorphism of ${\mathfrak{g}}$ if
necessary, that $Q(X,X)=1$.

By construction, $\omega$ has speed one and curvature $\kappa$. By Theorem
1.3, $\omega$ has a primitive $f\colon I\rightarrow{\mathbb{R}}^{2}$, unique
up to orientation-preserving isometries of ${\mathbb{R}}^{2}$. Since $\omega$
and $\delta f$ are isomorphic, they must have the same invariants, so that $f$
is a unit-speed curve with curvature $\kappa$. ∎

###  3.12. Curves in the elliptic plane

We now sketch a similar analysis of curves on $M=S^{2}$ with
$G=\operatorname{SO}(3)$. In this case all polynomial invariants are generated
by the Killing form of ${\mathfrak{g}}$, which we scale by a factor of $-1/2$
to obtain an inner product $Q$ on ${\mathfrak{g}}$, and on the trivial bundle
$E={\mathfrak{g}}\times I$.

Given an infinitesimal immersion $\omega\colon A\rightarrow{\mathfrak{g}}$ we
define a section $X$ of the rank-two Lie algebroid $A$ by requiring it to be
$Q$-orthogonal to the kernel ${\mathfrak{h}}$ of the anchor, to have constant
$Q$-length one, and satisfy $\\#X=\frac{1}{s}\frac{\partial}{\partial t}$, for
some positive function $s\colon I\rightarrow{\mathbb{R}}$ called the speed of
$\omega$. This speed coincides with the usual speed of a curve $f\colon
I\rightarrow S^{2}$ when we take $\omega$ to be the logarithmic derivative of
a curve $f$ (with our choice of scaling factor, $-1/2$). Let $Y$ be a section
of $E$ that is $Q$-orthogonal to $A$ and has constant $Q$-length one. Such a
$Y$ is determined uniquely up to sign. Differentiating $Q(Y,Y)=1$, one shows
that $\nabla_{\partial/\partial t}Y$ is a section of $A$ and can therefore
define $\kappa\colon I\rightarrow{\mathbb{R}}$ by

$\\#\nabla_{\partial/\partial t}Y=-\kappa\frac{\partial}{\partial t}\,.$

This, in turn, defines the curvature $|\kappa|$ of $\omega$, which is
independent of the choice of $Y$ above, and hence an invariant, and which
coincides with the absolute value of the curvature of a curve $f\colon
I\rightarrow S^{2}$ when we take $\omega$ to be the logarithmic derivative of
$f$. Note that the usual (signed) curvature of a curve is not an invariant in
our theory because symmetries of $S^{2}$ include the orientation-reversing
isometries.

For a curve $f$ one goes on to define $Z=\\{X,Y\\}$, a section of
${\mathfrak{h}}$, and then obtains a complete description of the logarithmic
derivative of $f$ readily enough. One then proves an analogue of Theorem 3.11
by applying the Construction Principle of §2.3. Details are left to the
reader.

For an analysis of curves in the equi-affine plane, see Section 6.

##  4\. The invariant filtration for Riemannian subgeometry

In this section we study oriented codimension-one immersed submanifolds
$\Sigma$ of the Riemannian manifold $M=S^{n}$, ${\mathbb{R}}^{n}$, or
${\mathbb{H}}^{n}$.

###  4.1. The Lie algebroid associated with a Riemannian manifold

The basic groupoid associated with an oriented Riemannian manifold $M$ is the
Lie groupoid $\mathrm{SO}[TM]$ of orientation-preserving orthonormal relative
frames. Being a frame groupoid, the Lie algebroid of $\mathrm{SO}[TM]$ can be
viewed as a subalgebroid $A$ of the Lie algebroid $\mathfrak{gl}[TM]$ of
derivations of $TM$ (defined in, e.g., [18, §3.3]). However, the Levi-Cevita
connection $\nabla^{M}$ furnishes a splitting $u\mapsto\nabla_{u}^{M}$ of the
canonical exact sequence

$0\rightarrow\mathfrak{so}(TM)\rightarrow A\xrightarrow{\\#}TM\rightarrow 0,$

allowing us to identify $A$ with

$\mathfrak{so}[TM]:={TM}\oplus\mathfrak{so}({TM}).$

Recall here that $\mathfrak{so}({TM})\subset T^{*}\\!M\otimes TM$ denotes the
subbundle of tangent space endomorphisms that are skew-symmetric with respect
to the metric — a Lie algebra bundle of type $\mathfrak{so}(n)$, $n=\dim M$.

The anchor of $\mathfrak{so}[TM]$ is projection onto the first summand.
Writing elements and sections of
$\mathfrak{so}[TM]={TM}\oplus\mathfrak{so}({TM})$ vertically, the Lie bracket
is given by

$\left[\begin{pmatrix}V_{1}\\\ \Phi_{1}\end{pmatrix},\begin{pmatrix}V_{2}\\\
\Phi_{2}\end{pmatrix}\right]=\begin{pmatrix}[V_{1},V_{2}]\\\
[\Phi_{1},\Phi_{2}]+\nabla^{M}_{V_{1}}\Phi_{2}-\nabla^{M}_{V_{2}}\Phi_{1}+\operatorname{curv}\nabla^{M}(V_{1},V_{2})\end{pmatrix}\enspace,$

as the reader is invited to check. The canonical representation of
$\mathfrak{so}[TM]$ on $TM$ is given in the present model by

(1) $\bar{\nabla}_{\begin{pmatrix}U\\\
\Phi\end{pmatrix}}V=\nabla^{M}_{U}V+\Phi V.$

This representation induces a representation of $\mathfrak{so}[TM]$ on the
bundle $S^{2}\,T^{*}\\!M$ of which the metric on $M$ is a section. By
construction, the metric is invariant under this representation and is, up to
scale, the unique such section.

As explained in §1.4, we are also interested in the Lie groupoid
$\mathrm{SO}[T^{+}M]$ of orientation-preserving orthonormal relative frames of
$T^{+}M:=TM\oplus({\mathbb{R}}\times M)$, equipped with the inner product

(2) $\langle\\!\langle V_{1}\oplus a_{1},V_{2}\oplus
a_{2}\rangle\\!\rangle:=\langle\\!\langle
V_{1},V_{2}\rangle\\!\rangle+a_{1}a_{2}.$

This product is $\nabla^{M+}$-parallel if we define $\nabla^{M+}_{u}(V\oplus
a)=\nabla^{M}_{u}V\oplus da(u)$ and, by a similar argument, we obtain a split
exact sequence for the Lie algebroid of $\mathrm{SO}[T^{+}M]$, identifying it
with $TM\oplus\mathfrak{so}(T^{+}M)$, which in turn has a natural
identification with

$\mathfrak{so}[T^{+}M]:=TM\oplus\mathfrak{so}(TM)\oplus
TM=\mathfrak{so}[TM]\oplus TM.$

The anchor of $\mathfrak{so}[T^{+}M]$ is given by $\\#(X,w)=\\#X$ and the
bracket by

$[X_{1}\oplus W_{1},X_{2}\oplus W_{2}]=\left([X_{1},X_{2}]+\begin{pmatrix}0\\\
W_{1}^{\flat}\otimes W_{2}-W_{2}^{\flat}\otimes
W_{1}\end{pmatrix}\right)\oplus(\bar{\nabla}_{X_{1}}W_{2}-\bar{\nabla}_{X_{2}}W_{1}),$

where $\bar{\nabla}$ denotes the canonical representation of
$\mathfrak{so}[TM]$ on $TM$ defined in (1). The Lie algebroid
$\mathfrak{so}[T^{+}M]$ itself acts on $T^{+}M$ according to

(3) $\bar{\nabla}_{X\oplus W}(V\oplus
a)=(\bar{\nabla}_{X}V+aW)\oplus(da(\\#X)-\langle\\!\langle
W,V\rangle\\!\rangle).$

By construction, this action preserves the inner product (2) which, up to
scale, is the unique such product.

###  4.2. The Cartan connection for Riemannian geometry

There is a canonical Cartan connection $\nabla$ on $\mathfrak{so}[TM]$ [1, 2],
given by

(1) $\nabla_{U}\begin{pmatrix}V\\\
\Phi\end{pmatrix}=\begin{pmatrix}\nabla^{M}_{U}V+\Phi U\\\
\nabla^{M}_{U}\Phi+\operatorname{curv}\nabla^{M}(U,V)\end{pmatrix}\,.$

Since $\nabla$ is Cartan the space ${\mathfrak{g}}_{\nabla}$ of all
$\nabla$-parallel sections of $\mathfrak{so}[TM]$ forms a Lie subalgebra of
all sections.

To recall the significance of $\nabla$, remember that a vector field $\xi$ on
$M$ is a Killing field (infinitesimal isometry) precisely when the
corresponding section $\nabla^{M}\xi$ of $T^{*}\\!M\otimes{TM}$ is in fact a
section of $\mathfrak{so}({TM})$. The following is proven in [2]:

######  Theorem.

For every Killing field $\xi$ the section

$X_{\xi}=\begin{pmatrix}\xi\\\ -\nabla^{M}\xi\end{pmatrix}$

of $\mathfrak{so}[TM]$ is $\nabla$-parallel and all parallel sections are of
this form. In particular, the anchor $\\#\colon\mathfrak{so}[TM]\rightarrow
TM$, when viewed as a map of corresponding section spaces
$\Gamma(\mathfrak{so}[TM])\rightarrow\Gamma({TM})$, maps the Lie algebra
${\mathfrak{g}}_{\nabla}$ of $\nabla$-parallel sections of $\mathfrak{so}[TM]$
isomorphically onto the Lie algebra of Killing fields of $M$.

By this result the curvature of $\nabla$ (see below) is the local obstruction
to the existence of Killing fields.

The bracket on ${\mathfrak{g}}_{\nabla}$ can be expressed in terms of the
torsion of $\nabla$ — an algebraic bracket defined on any Lie algebroid
equipped with a connection by

$\\{X,Y\\}:=\nabla_{\\#Y}X-\nabla_{\\#X}Y+[X,Y].$

For if $\xi,\eta\in{\mathfrak{g}}_{\nabla}$ then $[\xi,\eta]=\\{\xi,\eta\\}$.
In the present case one computes

(2) $\left\\{\begin{pmatrix}V_{1}\\\
\Phi_{1}\end{pmatrix},\begin{pmatrix}V_{2}\\\
\Phi_{2}\end{pmatrix}\right\\}=\begin{pmatrix}\Phi_{1}V_{2}-\Phi_{2}V_{1}\\\
[\Phi_{1},\Phi_{2}]-\operatorname{curv}\nabla^{M}(V_{1},V_{2})\end{pmatrix}\,.$

The curvature of the connection $\nabla$ on $\mathfrak{so}[TM]$ is readily
computed and seen to vanish precisely when $M$ has constant sectional
curvature $s$, i.e., when the curvature of the Levi-Cevita connection is given
by

(3) $\operatorname{curv}\nabla^{M}(V_{1},V_{2})=s(V_{2}^{\flat}\otimes
V_{1}-V_{1}^{\flat}\otimes V_{2}).$

Here $V\mapsto V^{\flat}\colon TM\rightarrow T^{*}\\!M$ is the canonical
isomorphism determined by the metric. Assuming $M$ is simply-connected, the
Lie algebra ${\mathfrak{g}}_{\nabla}$ assumes its maximal possible dimension
$k$ in this case; $k=\operatorname{rank}\mathfrak{so}[TM]=n(n+1)/2$, where
$n=\dim M$.

###  4.3. The ambient space of embeddings

In the remainder of this section $M$ denotes either the sphere $S^{n}$,
Euclidean space ${\mathbb{R}}^{n}$, or hyperbolic space ${\mathbb{H}}^{n}$.
Then $M$ is a homogeneous $G$-space where $G$ is respectively
$\operatorname{\operatorname{SO}}(n+1)$, the semidirect product
$\operatorname{\operatorname{SO}}(n)\ltimes{\mathbb{R}}^{n}$, or
$\operatorname{\operatorname{SO}}^{+}(n,1)$, each acting in the usual way by
isometries of $M$. Note that $M$ has constant sectional curvature $s=1,0,-1$,
respectively. According to [6, §2.1], $\operatorname{Aut}(M)=G$ in all cases
except the even-dimensional spheres, in which case $\operatorname{Aut}(M)$ is
the group of all isometries, both orientation-preserving and orientation-
reversing. The Lie algebra of $G$ will be identified with the Lie algebra
${\mathfrak{g}}\subset\Gamma(TM)$ of all Killing fields (infinitesimal
isometries), which is possible because $G$ acts faithfully and
$\dim(G)=n(n+1)/2$.

######  Proposition.

The action algebroid ${\mathfrak{g}}\times M$ is isomorphic to the Lie
algebroid $\mathfrak{so}[TM]$ defined above. An explicit isomorphism is given
by

$\displaystyle(\xi,m)\mapsto X_{\xi}(m):=\begin{pmatrix}\xi(m)\\\
-(\nabla^{M}\xi)(m)\end{pmatrix}\colon{\mathfrak{g}}\times
M\rightarrow\mathfrak{so}[TM].$

Under this isomorphism the canonical flat connection on ${\mathfrak{g}}\times
M$ coincides with the Cartan connection $\nabla$.

###### Proof.

It is readily checked that the map is a Lie algebroid morphism. To show the
map is an isomorphism it suffices, by a dimension count, to establish
surjectivity. But every element of $\mathfrak{so}[TM]$ can be extended to a
$\nabla$-parallel section, because $\nabla$ is flat ($M$ has constant scalar
curvature) and $M$ is simply-connected. By the preceding theorem this section
is of the form $X_{\xi}$ for some $\xi\in{\mathfrak{g}}$. The last statement
in the theorem is now obvious. ∎

###  4.4. The logarithmic derivative of an immersion

Now let $\Sigma$ denote an immersed, oriented, codimension-one submanifold,
and $f\colon\Sigma\rightarrow M$ the immersion. The logarithmic derivative of
$f$ is a map $\delta f\colon A(f)\rightarrow{\mathfrak{g}}$, where $A(f)$ is
the pullback of the action algebroid ${\mathfrak{g}}\times M$ by $f$. In view
of the preceding proposition, we may identity ${\mathfrak{g}}\times M$ with
$\mathfrak{so}[TM]={TM}\oplus\mathfrak{so}({TM})$. Then, as the anchor of
$\mathfrak{so}[TM]$ is just projection onto the first summand, we have

$A(f)=T\Sigma\oplus\mathfrak{so}({T_{\Sigma}M}),$

where $T_{\Sigma}M$ denotes the vector bundle pullback of ${TM}$, and
$\mathfrak{so}({T_{\Sigma}M})$ is the bundle of skew-symmetric endomorphisms
thereof. In §4.5 below we shall show that the Lie bracket on $A(f)$ is given
by

(1) $\left[\begin{pmatrix}v_{1}\\\
\Phi_{1}\end{pmatrix},\begin{pmatrix}v_{2}\\\
\Phi_{2}\end{pmatrix}\right]=\begin{pmatrix}[v_{1},v_{2}]\\\
[\Phi_{1},\Phi_{2}]+\nabla^{M}_{v_{1}}\Phi_{2}-\nabla^{M}_{v_{2}}\Phi_{1}+s(v_{2}^{\flat}\otimes
v_{1}-v_{1}^{\flat}\otimes v_{2})\end{pmatrix}\,.$

In this case we should interpret $\nabla^{M}$ as the pullback of the Levi-
Cevita connection on $TM$ to a connection on $T_{\Sigma}M$ (and a
corresponding connection on $\mathfrak{so}(T_{\Sigma}M)$). We use lowercase
letters to distinguish elements of $T\Sigma$ from elements of $TM$. Under our
identifications, the logarithmic derivative $\delta f$ of $f$ is just the
composite of the inclusion

$T\Sigma\oplus\mathfrak{so}(T_{\Sigma}M)\hookrightarrow{TM}\oplus\mathfrak{so}({TM})=\mathfrak{so}[TM]$

with the projection $\mathfrak{so}[TM]\cong{\mathfrak{g}}\times
M\rightarrow{\mathfrak{g}}$.

###  4.5. The fundamental representation

Following §2.3, we now regard $A(f)$ as a subbundle of
$E={\mathfrak{g}}\times\Sigma$. As ${\mathfrak{g}}\times\Sigma$ is just the
vector bundle pullback of ${\mathfrak{g}}\times
M\cong\mathfrak{so}[TM]=TM\oplus\mathfrak{so}(TM)$, we may make the
identification

$E=T_{\Sigma}M\oplus\mathfrak{so}(T_{\Sigma}M).$

Then the inclusion of $A(f)=T\Sigma\oplus\mathfrak{so}(T_{\Sigma}M)$ into $E$
is the obvious one.

According to Proposition 4.3, the canonical flat connection on
${\mathfrak{g}}\times M\cong\mathfrak{so}[TM]$ is represented by the Cartan
connection $\nabla$ on $\mathfrak{so}[TM]$ defined by 4.2(1), and this
connection pulls back to a flat connection on
$E=T_{\Sigma}M\oplus\mathfrak{so}(T_{\Sigma}M)$ given by the same formula,
provided we again interpret $\nabla^{M}$ as the pullback of the Levi-Civita
connection to a connection on $T_{\Sigma}M$.

Under the identification ${\mathfrak{g}}\times M\cong\mathfrak{so}[TM]$ the
canonical algebraic bracket on sections of ${\mathfrak{g}}\times M$ coincides
with the torsion of $\nabla$, which is given explicitly in 4.2(2), 4.2(3)
above. From this we may determine the corresponding algebraic bracket on
sections of $E$:

(1) $\left\\{\begin{pmatrix}V_{1}\\\
\Phi_{1}\end{pmatrix},\begin{pmatrix}V_{2}\\\
\Phi_{2}\end{pmatrix}\right\\}=\begin{pmatrix}\Phi_{1}V_{2}-\Phi_{2}V_{1}\\\
[\Phi_{1},\Phi_{2}]+s(V_{1}^{\flat}\otimes V_{2}-V_{2}^{\flat}\otimes
V_{2})\end{pmatrix}\,.$

This bracket is necessarily $\nabla$-invariant. Since, furthermore, the Mauer-
Cartan equation 2.3(2) must hold, one now readily establishes the earlier
claim 4.4(1) regarding the Lie algebroid bracket on $A(f)$.

With the help of formula 4.2(1) for the connection $\nabla$ and formula (1)
for the algebraic bracket on $E$, the fundamental representation of
$A(f)=T\Sigma\oplus\mathfrak{so}(T_{\Sigma}M)$ on
$E=T_{\Sigma}M\oplus\mathfrak{so}(T_{\Sigma}M)$ is readily computed and given
by

(2) $\bar{\nabla}_{\begin{pmatrix}u\\\ \Phi\end{pmatrix}}\begin{pmatrix}V\\\
\Psi\end{pmatrix}=\begin{pmatrix}\nabla^{M}_{u}V+\Phi V\\\
\nabla^{M}_{u}\Psi+[\Phi,\Psi]\end{pmatrix}\enspace.$

###  4.6. Decomposing $T_{\Sigma}M$

Since $M$ and $\Sigma$ are oriented, $\Sigma$ has a well-defined unit normal
vector field, leading to identifications

(1) $T_{\Sigma}M\cong
T\Sigma\oplus({\mathbb{R}}\oplus\Sigma),\qquad\mathfrak{so}(T_{\Sigma}M)\cong\mathfrak{so}(T\Sigma)\oplus
T\Sigma.$

Elements of the bundles above will be written horizontally. Under the above
identifications the tautological representation of
$\mathfrak{so}(T_{\Sigma}M)$ on $T_{\Sigma}M$ becomes a representation of
$\mathfrak{so}(T\Sigma)\oplus T\Sigma$ on
$T\Sigma\oplus({\mathbb{R}}\times\Sigma)$, here denoted by
$\operatorname{ad}$, and given by

(2) $\operatorname{ad}_{\phi\oplus w}(v\oplus a)=(\phi
v+aw)\oplus-\langle\\!\langle w,v\rangle\\!\rangle,$

where $\langle\\!\langle\,\cdot\,,\,\cdot\,\rangle\\!\rangle$ denotes the
metric. The bracket on $\mathfrak{so}(T_{\Sigma}M)$ becomes a bracket on
$\mathfrak{so}(T\Sigma)\oplus T\Sigma$ given by

(3) $[\phi_{1}\oplus w_{1},\phi_{2}\oplus
w_{2}]=([\phi_{1},\phi_{2}]+w_{1}^{\flat}\otimes w_{2}-w_{2}^{\flat}\otimes
w_{1})\oplus(\phi_{1}w_{2}-\phi_{2}w_{1}).$

For the reader’s convenience we also record well-known expressions for the
pullback of the Levi-Connection $\nabla^{M}$ to connections on $T_{\Sigma}M$
and $\mathfrak{so}(T_{\Sigma}M)$, after making the identifications in (1):

(4) $\displaystyle\nabla^{M}_{u}(v\oplus a)$
$\displaystyle=(\nabla_{u}^{\Sigma}v-a\mathrm{I\\!I}(u)^{\sharp})\oplus(da(u)+\mathrm{I\\!I}(u,v)),$
(5) $\displaystyle\nabla^{M}_{u}(\phi\oplus w)$
$\displaystyle=(\nabla^{\Sigma}_{u}\phi+w^{\flat}\otimes\mathrm{I\\!I}(u)^{\sharp}-\mathrm{I\\!I}(u)\otimes
w)\oplus(\,\nabla^{\Sigma}_{u}w+\phi(\mathrm{I\\!I}(u)^{\sharp})\,).$

In these formulas $\nabla^{\Sigma}$ denotes the Levi-Cevita connection on
$\Sigma$ and $\mathrm{I\\!I}$ the second fundamental form. The map $V\mapsto
V^{\flat}\colon TM\rightarrow T^{*}\\!M$ is the canonical isomorphism
determined by the metric, and $\alpha\mapsto\alpha^{\sharp}$ is its inverse.

Having made the identifications in (1) we have the following updated model of
$E$:

(6) $E=\begin{matrix}T\Sigma\oplus({\mathbb{R}}\times M)\\\ \oplus\\\
\mathfrak{so}(T\Sigma)\oplus T\Sigma\end{matrix}\enspace.$

We are now writing bundle direct sums as we will be laying out corresponding
elements. The algebraic Lie bracket on $E$ is now given by the formula

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w_{1}\end{pmatrix},\begin{pmatrix}v_{2}\oplus a_{2}\\\ \phi_{2}\oplus
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}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys<EMAIL_ADDRESS>
where

1 $\displaystyle=\phi_{1}v_{2}-\phi_{2}v_{1}+a_{2}w_{1}-a_{1}w_{2},$ 2
$\displaystyle=\langle\\!\langle
w_{2},v_{1}\rangle\\!\rangle-\langle\\!\langle w_{1},v_{2}\rangle\\!\rangle,$
3 $\displaystyle=[\phi_{1},\phi_{2}]-w_{2}^{\flat}\otimes
w_{1}+w_{1}^{\flat}\otimes w_{2}-sv_{2}^{\flat}\otimes
v_{1}+sv_{1}^{\flat}\otimes v_{2},$ 4
$\displaystyle=\phi_{1}w_{2}-\phi_{2}w_{1}+sa_{1}v_{2}-sa_{2}v_{1}.$

Recall that $s=1,0$ or $-1$ according to whether $M=S^{n}$ ,
${\mathbb{R}}^{n}$ or ${\mathbb{H}}^{n}$ respectively. The canonical flat
connection $\nabla$ on $E$ is given by

(8) $\nabla_{u}\begin{pmatrix}v\oplus a\\\ \phi\oplus
w\end{pmatrix}=\begin{pmatrix}\text{{\tiny \leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\\ \text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{pmatrix}\enspace,$

where

1 $\displaystyle=\nabla^{\Sigma}_{u}v+\phi u-a\mathrm{I\\!I}(u)^{\sharp},$ 2
$\displaystyle=da(u)+\mathrm{I\\!I}(u,v)-\langle\\!\langle
u,w\rangle\\!\rangle,$ 3
$\displaystyle=\nabla^{\Sigma}_{u}\phi+w^{\flat}\otimes\mathrm{I\\!I}(u)^{\sharp}-\mathrm{I\\!I}(u)\otimes
w+s(v^{\flat}\otimes u-u^{\flat}\otimes v),$ 4
$\displaystyle=\nabla^{\Sigma}w+\phi(\mathrm{I\\!I}(u)^{\sharp})+sau.$

We may now also rewrite formula 4.5(2), expressing the fundamental
representation of

(9) $A(f)\cong\begin{matrix}T\Sigma\\\ \oplus\\\ \mathfrak{so}(T\Sigma)\oplus
T\Sigma\end{matrix}$

on $E$, as

$\bar{\nabla}_{\begin{pmatrix}u\\\ \phi\oplus
w\end{pmatrix}}\begin{pmatrix}v\oplus a\\\ \psi\oplus
x\end{pmatrix}=\begin{pmatrix}\text{{\tiny \leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\\ \text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{pmatrix},$

where

1 $\displaystyle=\nabla_{u}^{\Sigma}v+\phi v+a(w-\mathrm{I\\!I}(u)^{\sharp}),$
2 $\displaystyle=da(u)-\langle\\!\langle
w-\mathrm{I\\!I}(u)^{\sharp},v\rangle\\!\rangle,$ 3
$\displaystyle=\nabla^{\Sigma}_{u}\psi+[\phi,\psi]+(w-\mathrm{I\\!I}(u)^{\sharp})^{\flat}\otimes
x-x^{\flat}\otimes(w-\mathrm{I\\!I}(u)^{\sharp}),$ 4
$\displaystyle=\nabla_{u}^{\Sigma}x+\phi
x-\psi(\,w-\mathrm{I\\!I}(u)^{\sharp}\,).$

Under the identification in (9), formula 4.4(1) for the Lie bracket on $A(f)$
can be written

(10) $\begin{bmatrix}\begin{pmatrix}v_{1}\\\ \phi_{1}\oplus
w_{1}\end{pmatrix},\begin{pmatrix}v_{2}\\\ \phi_{2}\oplus
w_{2}\end{pmatrix}\end{bmatrix}=\begin{pmatrix}[v_{1},v_{2}]\\\ \text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
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} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
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}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
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}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{pmatrix},$

where

$\displaystyle\text{{\tiny \leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=$
$\displaystyle[\phi_{1},\phi_{2}]+\nabla^{\Sigma}_{v_{1}}\phi_{2}-\nabla^{\Sigma}_{v_{2}}\phi_{1}+w_{1}^{\flat}\otimes
w_{2}-w_{2}^{\flat}\otimes w_{1}$
$\displaystyle+w_{2}^{\flat}\otimes\mathrm{I\\!I}(v_{1})^{\sharp}-\mathrm{I\\!I}(v_{1})\otimes
w_{2}-w_{1}^{\flat}\otimes\mathrm{I\\!I}(v_{2})^{\sharp}+\mathrm{I\\!I}(v_{2})\otimes
w_{1}+s(v_{2}^{\flat}\otimes v_{1}-v_{1}^{\flat}\otimes v_{2}),$
$\displaystyle\text{{\tiny \leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}=$
$\displaystyle\nabla^{\Sigma}_{v_{1}}w_{2}+\phi_{1}w_{2}-\nabla^{\Sigma}_{v_{2}}w_{1}-\phi_{2}w_{1}-\phi_{1}(\mathrm{I\\!I}(v_{2})^{\sharp})+\phi_{2}(\mathrm{I\\!I}(v_{1})^{\sharp}).$

###  4.7. The Gauss-Codazzi equations

While the connection $\nabla$ on $E$ is flat (it corresponds to the canonical
flat connection on ${\mathfrak{g}}\times\Sigma$), this fact is hidden in the
present model of $E$. So we compute its curvature:

(1) $\operatorname{curv}\nabla(u_{1},u_{2})\begin{pmatrix}v\oplus a\\\
\phi\oplus w\end{pmatrix}=\begin{pmatrix}\text{{\tiny \leavevmode\hbox
to10.08pt{\vbox to10.08pt{\pgfpicture\makeatletter\hbox{\hskip
5.03984pt\lower-5.03984pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{1}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{2}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\\\ \text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{3}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\oplus\text{{\tiny
\leavevmode\hbox to10.08pt{\vbox
to10.08pt{\pgfpicture\makeatletter\hbox{\hskip 5.03984pt\lower-5.03984pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{
{{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{
}{{}{{{}}}{{}}{}{}{}{}{}{}{}{}{}{{}\pgfsys@moveto{4.83984pt}{0.0pt}\pgfsys@curveto{4.83984pt}{2.673pt}{2.673pt}{4.83984pt}{0.0pt}{4.83984pt}\pgfsys@curveto{-2.673pt}{4.83984pt}{-4.83984pt}{2.673pt}{-4.83984pt}{0.0pt}\pgfsys@curveto{-4.83984pt}{-2.673pt}{-2.673pt}{-4.83984pt}{0.0pt}{-4.83984pt}\pgfsys@curveto{2.673pt}{-4.83984pt}{4.83984pt}{-2.673pt}{4.83984pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{
} }{{{{}}\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-1.25pt}{-1.6111pt}\pgfsys@invoke{
}\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{4}}
}}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}
\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} }
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{{{}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}}\end{pmatrix}\enspace,$

where

1 $\displaystyle=\Xi(v)-a\Theta^{\sharp},$ 2 $\displaystyle=\Theta(v),$ 3
$\displaystyle=\operatorname{ad}_{\phi}\Xi+w^{\flat}\otimes\Theta^{\sharp}-\Theta\otimes
w,$ 4 $\displaystyle=\Xi(w)+\phi(\Theta^{\sharp})$

and

$\displaystyle\Xi$
$\displaystyle=\operatorname{curv}\nabla^{\Sigma}(u_{1},u_{2})+\mathrm{I\\!I}(u_{1})\otimes\mathrm{I\\!I}(u_{2})^{\sharp}-\mathrm{I\\!I}(u_{2})\otimes\mathrm{I\\!I}(u_{1})^{\sharp}+s(u_{1}^{\flat}\otimes
u_{2}-u_{2}^{\flat}\otimes u_{1}),$ $\displaystyle\Theta$
$\displaystyle=(\nabla_{u_{1}}^{\Sigma}\mathrm{I\\!I})u_{2}-(\nabla^{\Sigma}_{u_{2}}\mathrm{I\\!I})u_{1}.$

Here $(\operatorname{ad}_{\phi}\Xi)(u):=\phi(\Xi(u))-\Xi(\phi u)$. The
equations $\Xi=0$ and $\Theta=0$ are the well-known Gauss-Codazzi equations.
Putting $\operatorname{curv}\nabla=0$ we obtain the following well-known fact:

######  Proposition.

For any immersed submanifold $\Sigma\subset M$ the Gauss-Codazzi equations
hold:

(2)
$\begin{gathered}\operatorname{curv}\nabla^{\Sigma}(u_{1},u_{2})+\mathrm{I\\!I}(u_{1})\otimes\mathrm{I\\!I}(u_{2})^{\sharp}-\mathrm{I\\!I}(u_{2})\otimes\mathrm{I\\!I}(u_{1})^{\sharp}\hskip
60.00009pt\\\ \hfill+\enspace s(u_{1}^{\flat}\otimes
u_{2}-u_{2}^{\flat}\otimes u_{1})=0\\\
(\nabla_{u_{1}}^{\Sigma}\mathrm{I\\!I})u_{2}-(\nabla^{\Sigma}_{u_{2}}\mathrm{I\\!I})u_{1}=0.\end{gathered}$

###  4.8. The invariant filtration

We now turn attention to the first three terms

(1) $A(f)\supset A(f)^{2}\supset A(f)^{3}$

of the invariant filtration associated with the immersion
$f\colon\Sigma\rightarrow M$. Let $\mathfrak{so}[T\Sigma]_{\mathrm{I\\!I}}$
denote the isotropy of $\mathrm{I\\!I}$ under the representation of
$\mathfrak{so}[T\Sigma]$ on $S^{2}\,T^{*}\Sigma$ induced by the canonical
representation on $T\Sigma$, so that we have a filtration

(2)
$\mathfrak{so}[T^{+}\Sigma]\supset\mathfrak{so}[T\Sigma]\supset\mathfrak{so}[T\Sigma]_{\mathrm{I\\!I}}.$

Note that $\mathfrak{so}[T^{+}\Sigma]_{\mathrm{I\\!I}}$ is generally
intransitive.

######  Proposition.

Define a monomorphism ${i}\colon\mathfrak{so}[T^{+}\Sigma]\rightarrow E$ by

(3) $\begin{pmatrix}u\\\ \phi\end{pmatrix}\oplus
w\mapsto\begin{pmatrix}u\oplus 0\\\
\phi\oplus(\mathrm{I\\!I}(u)^{\sharp}+w)\end{pmatrix}.$

Then:

  * The monomorphism is a Lie algebroid isomorphism of $\mathfrak{so}[T^{+}\Sigma]$ onto $A(f)$ that maps the filtration (2) to the filtration (1).

Under the corresponding identification $A(f)\cong\mathfrak{so}[T^{+}\Sigma]$