# Einstein equations for a metric coupled to a trace-free symmetric tensor

Daniel J. F. Fox  Departamento de Matemática Aplicada a la Ingeniería
Industrial
Escuela Técnica Superior de Ingeniería y Diseño Industrial
Universidad Politécnica de Madrid
Ronda de Valencia 3
28012 Madrid España<EMAIL_ADDRESS>
###### Abstract.

There are described equations coupling a completely symmetric conformal
Killing or Codazzi tensor to the Einstein equations for a metric, in a manner
analogous to that used to obtain the Einstein-Maxwell equations by coupling a
two-form to the metric. Examples of solutions are constructed from mean
curvature zero immersions, affine spheres, isoparametric polynomials, and
regular graphs. There are deduced some constraints on the scalar curvature of
the metric occurring in a solution. Along the way, there are reviewed
Weitzenböck formulas, vanishing theorems, and related results for conformal
Killing and divergence free Codazzi tensors.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
  3. 3 Differential operators on trace-free symmetric tensors
  4. 4 Vanishing theorems for conformal Killing and divergence free Codazzi tensors
  5. 5 Coupling Einstein equations to symmetric tensors
  6. 6 Examples of solutions to the coupled equations
  7. 7 Refined Kato inequalities for trace-free symmetric tensors
  8. 8 Constraints on solutions

##  1\. Introduction

The paper considers consistent couplings of the Einstein equations for a
pseudo-Riemannian metric with a completely symmetric tensor
$\omega_{i_{1}\dots i_{k}}$. The couplings are defined for a class of tensors
that includes trace-free Codazzi tensors and conformal Killing tensors. For
example, for a trace-free symmetric tensor $\omega_{i_{1}\dots
i_{k}}=\omega_{(i_{1}\dots i_{k})}$ that is Codazzi, meaning
$D_{[i}\omega_{j]p_{1}\dots p_{k-1}}=0$, the resulting equations have the form

(1.1)
$\displaystyle\begin{split}\mathscr{R}_{ij}&-\tfrac{1}{2}\mathscr{R}_{h}h_{ij}+\tfrac{n-2}{2n}\kappa
h_{ij}=c\left(\operatorname{\rho}(\omega\owedge\omega)_{ij}-\tfrac{1}{2}|\omega|_{h}^{2}h_{ij}\right),\end{split}$

where $c$ is a constant, $\kappa$ is a function,
$\operatorname{\rho}(\omega\owedge\omega)_{ij}=\omega_{i}\,^{p_{1}\dots
p_{k-1}}\omega_{jp_{1}\dots p_{k-1}}$ is the Ricci trace of the Kulkarni-
Nomizu product $\omega\owedge\omega$, and $\mathscr{R}_{ij}$ and
$\mathscr{R}_{h}$ are the Ricci and scalar curvature of the Levi-Civita
connection $D$ of $h_{ij}$. What is important about the constant $c$ is its
sign, as its modulus can be absorbed into the tensor $\omega$; see Remark 5.8.
It follows from the differential Bianchi identity and the Codazzi condition on
$\omega_{i_{1}\dots i_{k}}$ that $\kappa$ must be constant.Note that (1.1) is
equivalent to

(1.2) $\displaystyle\mathscr{R}_{ij}-c\omega_{i}\,^{p_{1}\dots
p_{k-1}}\omega_{jp_{1}\dots p_{k-1}}=\kappa h_{ij}.$

In the form (1.1), the right-hand side can be viewed as a stress energy tensor
and $\tfrac{n-2}{2n}\kappa$ can be regarded as a cosmological constant.

Equations such as (1.1) can be seen as formal analogues of the Einstein-
Maxwell equations, with a completely symmetric tensor coupled to the metric in
place of a two-form. The equations (1.1) are a special case of the more
general equations (5.21) discussed in detail in Section 5, that allow for
coupling also with a trace-free conformal Killing tensor and a relaxation of
the Codazzi condition. (For expository simplicity the discussion here in the
introduction focuses on (1.1).)

One reason for considering conformal Killing and trace-free Codazzi tensors is
that on an oriented Riemann surface these are exactly the real parts of
holomorphic vector fields and holomorphic $k$-differentials [27].

The expression $(\omega\owedge\omega)_{ijkl}$ can be viewed as a curvature
term, and Lemma 5.10 shows that a metric $h$ and a tensor $\omega$ as in (1.1)
such that the modified curvature
$\mathscr{R}_{ijkl}-\tfrac{1}{4}(\omega\owedge\omega)_{ijkl}$ is projectively
flat, meaning it solves

(1.3)
$\displaystyle\mathscr{R}_{ijkl}-c(\omega\owedge\omega)_{ijkl}=-\tfrac{\kappa}{n(n-1)}(h\owedge
h)_{ijkl},$

for some $c\in\mathbb{R}$, yield a solution of (1.1). (Because when $n=3$ the
module of trace-free curvature tensors is empty, in the $n=3$ case the
equations (1.1) and (1.3) are equivalent, meaning that $h$ and $\omega$ solve
(1.3) for $c\in\mathbb{R}$ if and only if they solve (1.1).)

The relation of the equations (1.3) to the equations (1.1) is parallel to the
relation, to which it specializes when $\omega$ is identically zero, between
constant sectional curvature metrics and Einstein metrics.

Section 6 records examples of solutions of the equations (5.21) and (1.3). The
equations for a mean curvature zero nondegenerate immersion of a hypersurface
in a pseudo-Riemannian space form and the equations for an affine sphere are
special cases of the equations (1.3). In these cases the tensor $\omega$ is,
respectively, the second fundamental form or the cubic form of the immersion.
In both these contexts, a solution $(h,\omega)$ to the equations (1.1) can be
viewed as a more general geometric structure, not necessarily induced via an
immersion. These examples show that, at least for $k\leq 3$, solutions to
(5.22) abound, although it should be remembered that the basic existence
result for affine spheres, due to Cheng and Yau is highly nontrivial. (That
the general formalism recovers in the $k=2$ and $k=3$ cases these well-known
geometric settings serves also as a useful consistency check on the sometimes
involved preliminary computations in Sections 3 and 4.)

Lemma 6.8 shows that on any compact simple Lie group solutions to (1.1) can be
constructed from invariant polynomials on its Lie algebra. These solutions
show that (1.1) admits nontrivial solutions in arbitrarily high dimensions for
arbitrarily large $k$.

If $h$ is Euclidean, so flat, the curvature terms in (1.1) vanish, and there
remains a purely algebraic equation for the tensor $\omega$. When $k=3$, a
harmonic cubic polynomial solving the algebraic part of (1.1) can be viewed as
the structure tensor of a nonassociative algebra, and from this point of view
it can be seen that solutions abound, as the author has shown in [26, section
$8$] and [28, 29]. Here there are described two constructions of algebraic
solutions that work for larger $k$. First, Theorem 6.12 shows that all
isoparametric polynomials yield solutions. Second, Lemma 6.14 shows that such
a solution is associated with every $k$-regular graph.

General existence theorems are not studied here, but Section 8 describes a
priori constraints on solutions of the restricted system (1.3) when $h$ is
Riemannian. Such results are motivated by and generalize results about
immersed submanifolds that go back to Calabi [7], in the context of affine
spheres, Simons in the context of minimal immersions in spheres [53], and
Cheng-Yau [15, 63] in the context of hypersurfaces of various kinds, as well
as many others. The general pattern of such results is as follows. There is a
Weitzenböck identity for the Laplacian of the squared-norm of a tensor
satisfying some partial differential equation (here $\omega$). Depending on
the signs of some curvature terms there are two general classes of results.
One uses refined Kato inequalities and sharp algebraic bounds on quadratic
terms to obtain a differential inequality that, via a sort of argumentation
developed most prominently by Calabi and Cheng-Yau, yields an upper bounds on
$|\omega|^{2}$ that can be interpreted as an upper bound on the scalar
curvature of $h$ (working harder along the same lines one could obtain an
upper bound on the Ricci curvature of $h$). The second follows arguments from
Simons [53], and integrates the Weitzenböck formula to obtain integral bounds
on $|\omega|^{2}$. In both cases the results along these lines obtained here
are reported in Section 8.

Theorem 8.5 shows that if $h$ is a complete Riemannian metric on a manifold of
dimension $n>2$ which with a trace-free Codazzi tensor $\omega$ solves (1.3)
for $c>0$ and $\kappa\in\mathbb{R}$ then

  1. (1)

If $\kappa\geq 0$ then $\omega$ is identically zero, and $h$ is a metric of
constant sectional curvature.

  2. (2)

If $\kappa<0$ then the scalar curvature $\mathscr{R}_{h}=c|\omega|^{2}+\kappa$
of $h$ is nonpositive.

In the special case corresponding to the context of the cubic form of a
complete hyperbolic affine sphere, Calabi [7] showed the nonpositivity of the
Ricci curvature. It seems reasonable to expect that, perhaps with some
additional conditions, the nonpositivity of the scalar curvature in Theorem
8.5 can be improved to nonpositivity of the Ricci curvature. The remaining
technical issue is to extend to $k>3$ certain tensorial identities used in
Calabi’s argument for $k=3$.

The proof of Theorem 8.5 requires the Weitzenböck formulas and refined Kato
inequalities for trace-free Codazzi tensors and conformal Killing tensors
described in sections 3 and 7 and a result of Cheng-Yau on the growth of
solutions to a differential inequality of the form $\Delta u\geq
Bu^{1+\sigma}-Au$. These results should be understood as generalizations of
results for holomorphic tensors on surfaces and as a counterparts to classical
vanishing theorems for holomorphic symmetric tensors due to Kobayashi [35,
36].

Theorem 8.6 is the integral bound parallel to that of Theorem 8.5 for
solutions of (1.3). Its $k=2$ case recovers an integral estimate of the scalar
curvature of a compact mean curvature zero hypersurface in a round sphere due
to J. Simons [53], while its $k=3$ case recovers the analogous integral
estimate for the scalar curvature of a compact mean curvature zero Lagrangian
submanifold of a constant holomorphic sectional curvature Kähler manifold due
to B.-Y. Chen and K. Ogiue [13]. The corresponding result for $k>3$ obtained
here is not sharp, as are the results for $k\leq 3$, because certain
inequalities for norms of tensors used in intermediate steps that are sharp
for $k\leq 3$ can be improved when $k>3$. Even when $k\leq 3$, the method of
proof, uniform in $k$ and not supposing an immersion in an ambient space,
seems a conceptual improvement.

Section 2 describes background material. Section 3 describes Weitzenböck
formulas for trace-free Codazzi and conformal Killing tensors. Most of the
material recounted in sections 3 and 4 was presented in the author’s [26,
section $6$]. In any case, much of this material has been obtained before or
since by others in various forms; see in particular [56], [51], [33] and [31].
It is presented here to make the exposition self-contained, because notations
differ substantially between different authors, and because the cases of
trace-free Codazzi and conformal Killing tensors can be given a uniform
treatment and this seems clarifying.

The Weitzenböck identities are needed here for showing in full generality the
the equations (5.21) generalizing (1.1) that are considered here are well
formulated, and for understanding when their hypotheses are nontrivial. This
is described in Section 5. They are needed also to obtain the estimate leading
to Theorem 8.5. In the proof of Theorem 8.5 there are needed the refined Kato
inequalities for trace-free Codazzi and conformal Killing tensors described in
section 7. As is explained there, these can be deduced from general results in
[8] and [6] (see also [34]), but considerable work is required to translate
general representation theoretic statements into the concrete contexts here,
and it is simpler to give the direct proofs recorded here.

The Weitzenböck identities presented here were found by the author in [26],
and have been treated independently in [33] (there are notational differences
because of different curvature conventions and because here identities are
written for symmetric tensors that are assumed trace-free). These Weitzenböck
identities serve to deduce for conformal Killing tensors and trace-free
Codazzi tensors vanishing theorems that are of independent interest and which
are summarized as Corollary 4.5. This objective was incompletely realized in
[26], where there was obtained the partial result recalled here as Theorem
4.3. For conformal Killing tensors of rank $k=2$ this partial result coupled
with a theorem of Berger-Ebin [1] is enough to deduce the vanishing theorem
under the desired hypotheses of nonnegative or nonpositive sectional
curvature, but for $k>2$ the author then did not see some step necessary to
make the argument work and the desired vanishing theorems for trace-free
Codazzi and conformal Killing tensors were reported in [27, Corollary $3.1$]
only for surfaces. For trace-free Codazzi tensors the desired result for all
$k$ has recently been proved [51, Corollary $1$] along the lines similar to
those taken here. For conformal Killing tensors the desired result for all $k$
was obtained in [20] based on a different line of argument. In [33] an elegant
proof was given for conformal Killing tensors based on the Weitzenböck
formulas, and that proof adapts immediately to give the corresponding (for
nonnegative sectional curvature) vanishing theorem for trace-free Codazzi
tensors.

##  2\. Preliminaries

Throughout the paper $M$ denotes a connected, smooth manifold of dimension
$n$. The abstract index conventions in the sense of Penrose [46, chapter $2$]
are used; see also [61]. With these conventions indices are labels indicating
tensor type and symmetries and do not refer to any local frame. Let $h_{ij}$
be a pseudo-Riemannian metric with Levi-Civita connection $D$. Indices are
raised and lowered using $h_{ij}$ and the dual symmetric bivector $h^{ij}$
defined by $h^{ip}h_{pj}=\delta_{j}\,^{i}$. The inner product on a tensor
module used here is always that defined by complete contraction with $h_{ij}$,
$\langle\alpha,\beta\rangle=\alpha^{i_{1}\dots i_{k}}\beta_{i_{1}\dots
i_{k}}$, which generally differs from the metric induced by that on $TM$ by a
constant factor that depends on the symmetries of the tensors considered. The
curvature $\mathscr{R}_{ijk}\,^{l}$ of $D$ is defined by
$2D_{[i}D_{j]}\omega_{k}=-R_{ijk}\,^{p}\omega_{p}$ for
$\omega_{i}\in\Gamma(T^{\ast}M)$. The curvature tensor
$\mathscr{R}_{ijkl}=\mathscr{R}_{ijk}\,^{p}h_{pl}\in\Gamma(\mathscr{MC}(T^{\ast}M))$
is defined by lowering the last index. The Ricci and scalar curvatures of $h$
are defined by
$\mathscr{R}_{ij}=\operatorname{\rho}(\mathscr{R})_{ij}=R_{pij}\,^{p}$ and
$\mathscr{R}_{h}=h^{ij}\operatorname{\rho}(\mathscr{R})_{ij}$. The submodule
of $\otimes^{4}\mathbb{V}^{\ast}$ corresponding to the lexicographic filling
of the Young diagram with two rows of two boxes is

(2.1)
$\displaystyle\mathscr{MC}(\mathbb{V}^{\ast})=\\{\mathscr{Y}_{ijkl}\in\otimes^{4}\mathbb{V}^{\ast}:\mathscr{Y}_{[ij]kl}=\mathscr{Y}_{ijkl}=\mathscr{Y}_{ij[kl]},\mathscr{Y}_{[ijk]l}=0\\}.$

that comprises the tensors $\mathscr{Y}_{ijkl}$ having _metric curvature
tensor type_. If $\mathscr{R}_{ijk}\,^{l}$ is the curvature tensor of the
Levi-Civita connection of a pseudo-Riemannian metric $h_{ij}$, the tensor
$\mathscr{R}_{ijkl}=\mathscr{R}_{ijk}\,^{p}h_{pl}$ takes values in the vector
bundle $\mathscr{MC}(T^{\ast}M)$.

The space $\mathsf{Pol}(\mathbb{V})$ of polynomials on the vector space
$\mathbb{V}$ comprises those functions on $\mathbb{V}$ that are polynomials
with respect to any choice of coordinates $x^{1},\dots,x^{n}$ such that
$dx^{1},\dots,dx^{n}$ is a parallel frame with respect to the affine structure
on $\mathbb{V}$ determined by the lines in $\mathbb{V}$. The centroaffine
structure on $\mathbb{V}$ induces a graded algebra structure,
$\mathsf{Pol}(\mathbb{V})=\oplus_{k\geq 0}\mathsf{Pol}^{k}(\mathbb{V})$, where
$\mathsf{Pol}^{k}(\mathbb{V})$ denotes the subspace of polynomials which are
homogeneous of degree $k$ with respect to dilations centered on the origin.
The graded symmetric algebra $S(\mathbb{V}^{\ast})=\oplus_{k\geq
0}S^{k}(\mathbb{V}^{\ast})$ of finite linear combinations of completely
symmetric covariant tensors on $\mathbb{V}$ is canonically isomorphic to
$\mathsf{Pol}(\mathbb{V})$ via the linear map sending $\omega\in
S^{k}(\mathbb{V}^{\ast})\to P^{\omega}(x)=\omega_{i_{1}\dots
i_{k}}x^{i_{1}}\dots x^{i_{k}}\in\mathsf{Pol}^{k}(\mathbb{V})$. Polarization
yields the inverse linear map sending
$P\in\mathsf{Pol}^{k}(\mathbb{V})\to\omega^{P}_{i_{1}\dots
i_{k}}=\tfrac{1}{k!}D_{i_{1}}\dots D_{i_{k}}P\in S^{k}(\mathbb{V}^{\ast})$.
The product $\alpha\odot\beta$ of $\alpha\in S^{k}(\mathbb{V}^{\ast})$ and
$\beta\in S^{l}(\mathbb{V}^{\ast})$ such that
$P^{\alpha}P^{\beta}=P^{\alpha\odot\beta}$ is given by the symmetrized tensor
product $(\alpha\odot\beta)_{i_{1}\dots i_{k+l}}=\alpha_{(i_{1}\dots
i_{k}}\beta_{i_{k+1}\dots i_{k+l})}$. For a constant nondegenerate symmetric
tensor $h_{ij}$ on $\mathbb{V}$, or, what is the same, a pseudo-Riemannian
metric on $\mathbb{V}$ parallel with respect to the standard flat affine
connection $D$, define

(2.2) $\displaystyle\operatorname{\mathsf{tr}}(\omega)_{i_{1}\dots
i_{k-2}}=\omega_{i_{1}\dots i_{k-2}p}\,^{p},$
$\displaystyle\mathsf{h}(\omega)=h\odot\omega=h_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k+2})},$

for $\omega\in S^{k}(\mathbb{V}^{\ast})$. By convention,
$\operatorname{\mathsf{tr}}(\omega)=0$ when $k=1$. These operators are
adjoints, meaning that
$\langle\alpha,\mathsf{h}(\beta)\rangle=\langle\operatorname{\mathsf{tr}}(\alpha),\beta\rangle$
for $\alpha\in S^{k}(\mathbb{V})$ and $\beta\in S^{k-2}(\mathbb{V})$. In
particular, the orthogonal complement in $S^{k}(\mathbb{V}^{\ast})$ of the
image of $\mathsf{h}:S^{k-2}(\mathbb{V}^{\ast})\to S^{k}(\mathbb{V}^{\ast})$
is the space
$S^{k}_{0}(\mathbb{V}^{\ast})=S^{k}(\mathbb{V}^{\ast})\cap\ker\operatorname{\mathsf{tr}}$
of trace-free elements of $S^{k}(\mathbb{V}^{\ast})$.

Define a symmetric bilinear map $\owedge:S^{k}(\mathbb{V}^{\ast})\times
S^{k}(\mathbb{V}^{\ast})\to\mathscr{MC}(\mathbb{V}^{\ast})$ by

(2.3) $\displaystyle(\alpha\owedge\beta)_{ijkl}=\alpha_{k[i}\,^{p_{1}\dots
p_{k-2}}\beta_{j]lp_{1}\dots p_{k-2}}-\alpha_{l[i}\,^{p_{1}\dots
p_{k-2}}\beta_{j]kp_{1}\dots p_{k-2}}.$

It is immediate from the definition that $(\alpha\owedge\beta)_{[ijk]l}=0$, so
that $(\alpha\owedge\beta)_{ijkl}\in\mathscr{MC}(\mathbb{V}^{\ast})$. When
$k=2$ the map $\owedge$ is half what is usually called the Kulkarni-Nomizu
product. There hold

(2.4)
$\displaystyle\begin{split}\operatorname{\rho}(\alpha\owedge\beta)_{ij}&=\alpha_{(i}\,^{p_{1}\dots
p_{k-1}}\beta_{j)p_{1}\dots p_{k-1}}-\tfrac{1}{2}\alpha_{ij}\,^{p_{1}\dots
p_{k-2}}\operatorname{\mathsf{tr}}(\beta)_{p_{1}\dots
p_{k-2}}-\tfrac{1}{2}\beta_{ij}\,^{p_{1}\dots
p_{k-2}}\operatorname{\mathsf{tr}}(\alpha)_{p_{1}\dots p_{k-2}},\\\
\operatorname{s}(\alpha\owedge\beta)&=\langle\alpha,\beta\rangle-\langle\operatorname{\mathsf{tr}}\alpha,\operatorname{\mathsf{tr}}\beta\rangle.\end{split}$

For $\alpha\in S^{2}(\mathbb{V}^{\ast})$ there hold
$\operatorname{\rho}(\alpha\owedge
h)_{ij}=\tfrac{2-n}{2}\left(\alpha_{ij}+\tfrac{1}{n-2}\operatorname{\mathsf{tr}}(\alpha)h_{ij}\right)$
and $\operatorname{s}(\alpha\owedge h)=(1-n)\operatorname{\mathsf{tr}}\alpha$.
Taking $\alpha_{ij}=h_{ij}$, yields that $(h\owedge h)_{ijkl}=2h_{k[i}h_{j]l}$
satisfies $\operatorname{\rho}(h\owedge h)_{ij}=(1-n)h_{ij}$ and
$\operatorname{s}(h\owedge h)=-n(n-1)$. It follow thats the trace-free part
$\operatorname{\mathsf{tf}}(\mathscr{Y})_{ijkl}$ of
$\mathscr{Y}_{ijkl}\in\mathscr{MC}(\mathbb{V}^{\ast})$ is given by

(2.5)
$\displaystyle\begin{split}\operatorname{\mathsf{tf}}(\mathscr{Y})_{ijkl}&=\mathscr{Y}_{ijkl}+\tfrac{2}{n-2}(\operatorname{\rho}(\mathscr{Y})\owedge
h)_{ijkl}-\tfrac{1}{(n-2)(n-1)}\operatorname{s}(\mathscr{Y})(h\owedge
h)_{ijkl}\\\
&=\mathscr{Y}_{ijkl}+\tfrac{2}{n-2}(\operatorname{\mathsf{tf}}(\operatorname{\rho}(\mathscr{Y}))\owedge
h)_{ijkl}+\tfrac{1}{n(n-1)}\operatorname{s}(\mathscr{Y})(h\owedge
h)_{ijkl}.\end{split}$

Suppose that $h$ is positive definite. An immediate consequence of (2.5) is

(2.6)
$\displaystyle\begin{split}|\mathscr{Y}|^{2}_{h}&=|\operatorname{\mathsf{tf}}\mathscr{Y}|^{2}_{h}+\tfrac{4}{n-2}|\operatorname{\mathsf{tf}}\operatorname{\rho}(\mathscr{Y})|^{2}_{h}+\tfrac{2}{n(n-1)}(\operatorname{s}(\mathscr{Y}))^{2}.\end{split}$

Any $\mathscr{Y}_{ijkl}\in\mathscr{MC}(\mathbb{V}^{\ast})$ determines a self-
adjoint endomorphism of $S^{2}\mathbb{V}^{\ast}$ defined by $a_{ij}\in
S^{2}\mathbb{V}^{\ast}\to\mathscr{Y}_{ipjq}a^{pq}\in S^{2}\mathbb{V}^{\ast}$.
In general this endomorphism does not preserve the subspace
$S^{2}_{0}\mathbb{V}^{\ast}$. However, the modified endomorphism
$a_{ij}\to\wideparen{\mathscr{Y}}(a)_{ij}=a^{pq}(\mathscr{Y}_{ipjq}+\operatorname{\rho}(\mathscr{Y})_{p(i}h_{j)q})$
restricts to a self-adjoint endomorphism of $S^{2}_{0}\mathbb{V}^{\ast}$. This
is the $k=2$ special case of the following construction that goes back to A.
Lichnerowicz in [39, section $10$].

######  Lemma 2.1.

For $\mathscr{Y}_{ijkl}\in\mathscr{MC}(\mathbb{V}^{\ast})$, the linear
operator $\wideparen{\mathscr{Y}}:S^{k}(\mathbb{V}^{\ast})\to
S^{k}(\mathbb{V}^{\ast})$ defined by

(2.7) $\displaystyle\begin{split}\wideparen{\mathscr{Y}}(\omega)_{i_{1}\dots
i_{k}}&=\operatorname{\rho}(\mathscr{Y})_{p(i_{1}}\omega_{i_{2}\dots
i_{k})}\,^{p}+(1-k)\mathscr{Y}_{p(i_{1}i_{2}}\,^{q}\omega_{i_{3}\dots
i_{k})q}\,^{p}.\end{split}$

has the following properties:

  1. (1)

The operators $\mathsf{h}:S^{k}(\mathbb{V}^{\ast})\to
S^{k+2}(\mathbb{V}^{\ast})$ and
$\operatorname{\mathsf{tr}}:S^{k}(\mathbb{V}^{\ast})\to
S^{k-2}(\mathbb{V}^{\ast})$ defined in (2.2) commute with
$\wideparen{\mathscr{Y}}$ in the sense that there hold

(2.8)
$\displaystyle(k+2)\wideparen{\mathscr{Y}}(\mathsf{h}(\omega))=k\mathsf{h}(\wideparen{\mathscr{Y}}(\omega)),$
$\displaystyle(k-2)\wideparen{\mathscr{Y}}(\operatorname{\mathsf{tr}}\omega)=k\operatorname{\mathsf{tr}}\wideparen{\mathscr{Y}}(\omega),$

for $\omega\in S^{k}\mathbb{V}^{\ast}$. In particular
$\wideparen{\mathscr{Y}}(h^{\odot k})=0$ for all $k\geq 1$.

  2. (2)

$\wideparen{\mathscr{Y}}(\operatorname{\mathsf{tf}}\omega)=\operatorname{\mathsf{tf}}\wideparen{\mathscr{Y}}(\omega)$
for all $\omega\in S^{k}(\mathbb{V}^{\ast})$. Hence $\wideparen{\mathscr{Y}}$
restricts to an endomorphism of $S^{k}_{0}\mathbb{V}^{\ast}$.

###### Proof.

It is claimed that

(2.9)
$\displaystyle\begin{split}\operatorname{\rho}(\mathscr{Y})_{p(i_{1}}\mathsf{h}(\omega)_{i_{2}\dots
i_{k+2})}\,^{p}&=\tfrac{k}{k+2}h_{(i_{1}i_{2}}\operatorname{\rho}(\mathscr{Y})^{p}\,_{i_{3}}\omega_{i_{4}\dots
i_{k+2})p}+\tfrac{2}{k+2}\operatorname{\rho}(\mathscr{Y})_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k+2})},\\\
\mathscr{Y}^{p}\,_{(i_{1}i_{2}}\,{}^{q}\mathsf{h}(\omega)_{i_{3}\dots
i_{k+2})pq}&=\tfrac{k(k-1)}{(k+2)(k+1)}h_{(i_{1}i_{2}}\mathscr{Y}^{p}\,_{i_{3}i_{4}}\,{}^{q}\omega_{i_{5}\dots
i_{k+2})pq}+\tfrac{2}{(k+2)(k+1)}\operatorname{\rho}(\mathscr{Y})_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k+2})}.\end{split}$

Combining (2.9) using (2.7) yields the first identity of (2.8). The validity
of (2.9) is shown as follows. Write

(2.10)
$\displaystyle\begin{split}\tbinom{k+2}{2}\mathsf{h}(\omega)_{i_{1}\dots
i_{k+1}p}&=h_{pi_{k+1}}\omega_{i_{3}\dots
i_{k}}+kh_{p(i_{1}}\omega_{i_{2}\dots i_{k})i_{k+1}}\\\
&+kh_{i_{k+1}(i_{1}}\omega_{i_{2}\dots
i_{k})p}+\tbinom{k}{2}h_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k})i_{k+1}p}.\end{split}$

Contracting (2.10) with $\operatorname{\rho}(\mathscr{Y})_{i_{k+2}}\,^{p}$
yields

(2.11)
$\displaystyle\begin{split}\tbinom{k+2}{2}\operatorname{\rho}(\mathscr{Y})_{i_{k+2}}\,^{p}\mathsf{h}(\omega)_{i_{1}\dots
i_{k+1}p}&=\operatorname{\rho}(\mathscr{Y})_{i_{k+1}i_{k+2}}\omega_{i_{3}\dots
i_{k}}+k\operatorname{\rho}(\mathscr{Y})_{i_{k+2}(i_{1}}\omega_{i_{2}\dots
i_{k})i_{k+1}}\\\
&+k\operatorname{\rho}(\mathscr{Y})_{i_{k+2}}\,^{p}h_{i_{k+1}(i_{1}}\omega_{i_{2}\dots
i_{k})p}+\tbinom{k}{2}\operatorname{\rho}(\mathscr{Y})_{i_{k+2}}\,^{p}h_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k})i_{k+1}p}.\end{split}$

Symmetrizing (2.11) over the uncontracted indices yields the first identity in
(2.9). Relabeling $i_{k+1}$ as $q$ in (2.10) and contracting the result with
$\mathscr{Y}^{p}\,_{i_{k+1}i_{k+2}}\,{}^{q}$ yields

(2.12)
$\displaystyle\begin{split}\tbinom{k+2}{2}&\mathscr{Y}^{p}\,_{i_{k+1}i_{k+2}}\,{}^{q}\mathsf{h}(\omega)_{i_{1}\dots
i_{k}pq}\\\
&=\mathscr{Y}^{p}\,_{i_{k+1}i_{k+2}}\,{}^{q}\left(h_{pq}\omega_{i_{3}\dots
i_{k}}+kh_{p(i_{1}}\omega_{i_{2}\dots i_{k})q}+kh_{q(i_{1}}\omega_{i_{2}\dots
i_{k})p}+\tbinom{k}{2}h_{(i_{1}i_{2}}\omega_{i_{3}\dots i_{k})pq}\right)\\\
&=\operatorname{\rho}(\mathscr{Y})_{i_{k+1}i_{k+2}}\omega_{i_{1}\dots
i_{k}}-k\mathscr{Y}_{i_{k+2}}\,^{q}\,{}_{i_{k+1}(i_{1}}\omega_{i_{2}\dots
i_{k})q}\\\
&\quad-k\mathscr{Y}_{i_{k+1}}\,^{q}\,{}_{i_{k+2}(i_{1}}\omega_{i_{2}\dots
i_{k})q}+\tbinom{k}{2}\mathscr{Y}^{p}\,_{i_{k+1}i_{k+2}}\,{}^{q}h_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k})pq}\end{split}$

Symmetrizing (2.12) over the uncontracted indices yields the second identity
in (2.9). Similarly,

(2.13)
$\displaystyle\begin{split}k&(\operatorname{\mathsf{tr}}\wideparen{\mathscr{Y}}(\omega))_{i_{1}\dots
i_{k-2}}=kh^{i_{k-1}i_{k-2}}\left(\operatorname{\rho}(\mathscr{Y})_{p(i_{1}}\omega_{i_{1}\dots
i_{k})}\,^{p}+(1-k)\mathscr{Y}_{p(i_{1}i_{2}}\,^{q}\omega_{i_{3}\dots
i_{k})q}\,^{p}\right)\\\
&=\left(2\operatorname{\rho}(\mathscr{Y})^{pq}\omega_{i_{1}\dots
i_{k-2}pq}+(k-2)\operatorname{\rho}(\mathscr{Y})_{p(i_{1}}(\operatorname{\mathsf{tr}}\omega)_{i_{2}\dots
i_{k-2})}\,^{p}\right)\\\
&\quad-2\left(\operatorname{\rho}(\mathscr{Y})_{p}\,^{q}\omega_{i_{1}\dots
i_{k-2}q}\,^{p}+\tbinom{k-2}{2}\mathscr{Y}_{p(i_{1}i_{2}}\,^{q}(\operatorname{\mathsf{tr}}\omega)_{i_{3}\dots
i_{k-2})q}\,^{p}\right)\\\
&=(k-2)\left(\operatorname{\rho}(\mathscr{Y})_{p(i_{1}}(\operatorname{\mathsf{tr}}\omega)_{i_{2}\dots
i_{k-2})}\,^{p}+(3-k)\mathscr{Y}_{p(i_{1}i_{2}}\,^{q}(\operatorname{\mathsf{tr}}\omega)_{i_{3}\dots
i_{k-2})q}\,^{p}\right)=(k-2)\wideparen{\mathscr{Y}}(\operatorname{\mathsf{tr}}\omega)_{i_{1}\dots
i_{k-2}}.\end{split}$

That $\wideparen{\mathscr{Y}}(h^{\odot k})=0$ follows from
$\wideparen{\mathscr{Y}}(h)=0$ and (2.8) by induction.

By (2.8),
$k\wideparen{\mathscr{Y}}(\mathsf{h}^{i}(\omega))=(k-2i)\mathsf{h}^{i}\wideparen{\mathscr{Y}}(\omega)$
and
$k\operatorname{\mathsf{tr}}^{i}\wideparen{\mathscr{Y}}(\omega)=(k-2i)\wideparen{\mathscr{Y}}(\operatorname{\mathsf{tr}}^{i}\omega)$
for all $\omega\in S^{k}(\mathbb{V}^{\ast})$. Consequently,
$\wideparen{\mathscr{Y}}(\mathsf{h}^{i}\operatorname{\mathsf{tr}}^{i}\omega)=\mathsf{h}^{i}\operatorname{\mathsf{tr}}^{i}\wideparen{\mathscr{Y}}(\omega)$.
Since $\omega-\operatorname{\mathsf{tf}}\omega$ is in the image of
$\mathsf{h}$, there results
$\wideparen{\mathscr{Y}}(\operatorname{\mathsf{tf}}\omega)=\operatorname{\mathsf{tf}}\wideparen{\mathscr{Y}}(\omega)$
for all $\omega\in S^{k}(\mathbb{V}^{\ast})$. That $\wideparen{\mathscr{Y}}$
maps trace-free symmetric tensors to trace-free symmetric tensors follows from
this observation or from the second identity of (2.8). A more conceptual proof
of this claim goes as follows. Define
$Y(\omega)=k\wideparen{\mathscr{Y}}(\omega)$ for $\omega\in
S^{k}(\mathbb{V}^{\ast})$. Then (2.8) means that $Y$ commutes with the
$\mathfrak{sl}(2,\mathbb{R})$ triple determined by the operators $E$, $F$, and
$H$, so must preserve the decomposition of symmetric tensors into their
primitive parts. ∎

##  3\. Differential operators on trace-free symmetric tensors

This section studies some operators acting on trace-free symmetric tensors.
Let $E$ and $F$ be bundles of tensors on $M$. A metric $h$ on $M$ determines a
pairing
$\left(\alpha,\beta\right)=\int_{M}\langle\alpha,\beta\rangle\,d\operatorname{vol}_{h}$
of sections $\alpha,\beta\in\Gamma(E)$, at least one of which is compactly
supported. Write $\lVert\omega\rVert^{2}=(\omega,\omega)$.

Because the fibers of $TM$ and $T^{\ast}M$ carry canonically dual flat
centroaffine structures, constructions applicable to dual vector spaces
$\mathbb{V}$ and $\mathbb{V}^{\ast}$ apply fiberwise to $TM$ and $T^{\ast}M$
without change. By definition $S(TM)=\oplus_{k\geq 0}\Gamma(S^{k}(TM))$
(respectively $S_{0}(TM)=\oplus_{k\geq 0}\Gamma(S^{k}_{0}(TM)))$ is the graded
vector bundle comprising finite linear combinations of (trace-free) completely
symmetric tensors. It becomes a graded algebra when equipped with the
fiberwise multiplication $\odot$. By definition $\mathsf{Pol}(T^{\ast}M)$ is
the graded subalgebra of $C^{\infty}(T^{\ast}M)$ comprising functions
polynomial in the fibers of $T^{\ast}M$ and of globally bounded degree.
Regarding $\mathsf{Pol}(T^{\ast}M)$ as a subspace of $C^{\infty}(T^{\ast}M)$
it acquires from the tautological Poisson structure on $T^{\ast}M$ a Poisson
structure defined, for $X\in\Gamma(S^{k}(TM))$ and $Y\in\Gamma(S^{l}(TM))$ and
any torsion-free affine connection $\nabla$, by

(3.1) $\displaystyle\\{X,Y\\}^{i_{1}\dots i_{k+l-1}}=kX^{p(i_{1}\dots
i_{k-1}}\nabla_{p}Y^{i_{k}\dots i_{k+l-1})}-lY^{p(i_{1}\dots
i_{l-1}}\nabla_{p}X^{i_{l}\dots i_{k+l-1})}.$

Given a metric $h_{ij}$, the connection in (3.1) may be taken to be its Levi-
Civita connection $D$, and $\\{h,X\\}=2D^{(i_{1}}X^{i_{2}\dots i_{k+1})}$ for
any $X\in\Gamma(S^{k}(TM))$ (in the bracket $\\{h,X\\}$ the notation $h$
refers to the dual bivector $h^{ij}$). Using $h$, $S(TM)$ and $S(T^{\ast}M)$
are identified by index raising and lowering, and for $X\in\Gamma(S^{k}(TM))$
and $\omega\in\Gamma(S^{k}(T^{\ast}M))$ one defines $X^{\flat}_{i_{1}\dots
i_{k}}=X^{j_{1}\dots j_{k}}h_{i_{1}j_{1}}\dots
h_{i_{k}j_{k}}\in\Gamma(S^{k}(T^{\ast}M))$ and defines
$\omega^{\sharp}\in\Gamma(S^{k}(TM))$ dually. Since index raising and lowering
induce symmetric algebra isomorphisms, there results on $S(T^{\ast}M)$ the
Poisson bracket $\\{\alpha,\beta\\}$ defined by
$\\{\alpha,\beta\\}=\\{\alpha^{\sharp},\beta^{\sharp}\\}^{\flat}$.

The divergence operator $\operatorname{\mathsf{div}}:\Gamma(T^{\ast}M\otimes
E)\to\Gamma(E)$ defined by

(3.2) $\displaystyle\operatorname{\mathsf{div}}(\omega)_{i_{1}\dots
i_{k}}=D^{p}\omega_{pi_{1}\dots i_{k}},$
$\displaystyle\omega\in\Gamma(T^{\ast}M\otimes E),$

is the formal adjoint of the negative of the covariant derivative $-D$.

######  Lemma 3.1.

Let $h$ be a pseudo-Riemannian metric on $M$. The operator
$\mathscr{H}:\Gamma(S^{k}(T^{\ast}M))\to\Gamma(S^{k+1}(T^{\ast}M))$ defined by
$\mathscr{H}(\alpha)=\tfrac{k+1}{2}\\{h,\alpha\\}$ satisfies
$[e^{\operatorname{\mathsf{tr}}},\mathscr{H}]=2\operatorname{\mathsf{div}}e^{\operatorname{\mathsf{tr}}}$.

###### Proof.

The identity
$[e^{\operatorname{\mathsf{tr}}},\mathscr{H}]=2\operatorname{\mathsf{div}}e^{\operatorname{\mathsf{tr}}}$
follows from the validity of

(3.3)
$\displaystyle\begin{split}\tfrac{k+1}{2}\operatorname{\mathsf{tr}}^{i}\\{h,\alpha\\}=\tfrac{k+1-2i}{2}\\{h,\operatorname{\mathsf{tr}}^{i}\alpha\\}+2i\operatorname{\mathsf{div}}(\operatorname{\mathsf{tr}}^{i-1}\alpha),\end{split}$

for all $\alpha\in\Gamma(S^{k}(T^{\ast}M))$ and $i\geq 1$, which is proved by
induction on $i$. The case $i=1$ is a direct computation:

(3.4)
$\displaystyle\begin{split}\tfrac{k+1}{2}&(\operatorname{\mathsf{tr}}\\{h,\alpha\\})_{i_{1}\dots
i_{k-1}}=(k+1)h^{pq}D_{(i_{1}}\alpha_{i_{2}\dots i_{k-1}pq)}\\\
&=2D^{p}\alpha_{i_{1}\dots
i_{k-1}p}+(k-1)D_{(i_{1}}(\operatorname{\mathsf{tr}}\alpha)_{i_{2}\dots
i_{k-1})}=2\operatorname{\mathsf{div}}(\alpha)_{i_{1}\dots
i_{k-1}}+\tfrac{k-1}{2}\\{h,\operatorname{\mathsf{tr}}\alpha\\}_{i_{1}\dots
i_{k-1}}.\end{split}$

Note that $\operatorname{\mathsf{tr}}$ and $\operatorname{\mathsf{div}}$
commute. If there holds (3.3), then, using (3.3) and (3.4),

(3.5)
$\displaystyle\begin{split}\tfrac{k+1}{2}&\operatorname{\mathsf{tr}}^{i+1}\\{h,\alpha\\}=\operatorname{\mathsf{tr}}\left(\tfrac{k+1}{2}\operatorname{\mathsf{tr}}^{i}\\{h,\alpha\\}\right)=\tfrac{k+1-2i)}{2}\operatorname{\mathsf{tr}}\\{h,\operatorname{\mathsf{tr}}^{i}\alpha\\}+2i\operatorname{\mathsf{div}}(\operatorname{\mathsf{tr}}^{i}\alpha)\\\
&=\tfrac{k-1-2i}{2}\\{h,\operatorname{\mathsf{tr}}^{i+1}\alpha\\}+2\operatorname{\mathsf{div}}(\operatorname{\mathsf{tr}}^{i}\alpha)+2i\operatorname{\mathsf{div}}(\operatorname{\mathsf{tr}}^{i}\alpha)=\tfrac{k+1-2(i+1)}{2}\\{h,\operatorname{\mathsf{tr}}^{i+1}\alpha\\}+2(i+1)\operatorname{\mathsf{div}}(\operatorname{\mathsf{tr}}^{i}\alpha),\end{split}$

and this proves (3.3). ∎

Let
$\operatorname{\mathsf{tf}}:\Gamma(S^{k}(T^{\ast}M))\to\Gamma(S^{k}_{0}(T^{\ast}M))$
be the $h$-orthogonal projection onto the completely trace-free part and
define $\mathscr{L}:\Gamma(S^{k}(T^{\ast}M))\to\Gamma(S^{k+1}(T^{\ast}M))$ by

(3.6)
$\displaystyle\mathscr{L}(\omega)=\tfrac{1}{2}\operatorname{\mathsf{tf}}\\{h,\omega\\},$

which is the trace-free part of the completely symmetrized covariant
derivative of $\omega$. Explicitly, for
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$,
$\mathscr{L}(\omega)=\tfrac{1}{2}\\{h,\omega\\}-\tfrac{k}{n+2(k-1)}h\odot\operatorname{\mathsf{div}}(\omega)$,
or

(3.7) $\displaystyle\begin{split}\mathscr{L}(\omega)_{i_{1}\dots
i_{k+1}}&=D_{(i_{1}}\omega_{i_{2}\dots
i_{k+1})}-\tfrac{k}{n+2(k-1)}h_{(i_{1}i_{2}}\operatorname{\mathsf{div}}(\omega)_{i_{3}\dots
i_{k+1})}.\end{split}$

By definition, $\mathscr{L}$ is the formal adjoint of the composition
$-\operatorname{\mathsf{div}}\circ\operatorname{\mathsf{tf}}$, meaning
$\left(\mathscr{L}(\alpha),\beta\right)=-\left(\alpha,\operatorname{\mathsf{div}}\operatorname{\mathsf{tf}}(\beta)\right)$
for $\alpha\in\Gamma(S^{k}(T^{\ast}M))$ and
$\beta\in\Gamma(S^{k+1}(T^{\ast}M))$, at least one of which has compact
support. If $X\in\Gamma(TM)$, then $\mathscr{L}(X^{\flat})$ is the Lie
derivative of the conformal structure $[h]$ along $X$, for

(3.8) $\displaystyle
2\mathscr{L}(X^{\flat})_{ij}=2\operatorname{\mathsf{tf}}(DX^{\flat})=2D_{(i}X_{j)}-\tfrac{2}{n}D_{p}X^{p}h_{ij}=\operatorname{\mathsf{tf}}(\mathfrak{L}_{X}h)_{ij}=(\mathfrak{L}_{X}[h])_{ij}.$

The identity (3.8) motivates using the notation, $\mathscr{L}$, resembling
that for the Lie derivative.

Let
$\mathscr{C}^{k+1}(\mathbb{V}^{\ast})\subset\otimes^{k+1}\mathbb{V}^{\ast}$ be
the the space of trace-free $(k+1)$-tensors $\phi_{iji_{1}\dots i_{k-1}}$
having the symmetries determined by the Young projector given by
symmetrization over the rows followed by anti-symmetrization over the columns
of the Young diagram corresponding to the partition $(n1)$, so satisfying
$\phi_{iji_{1}\dots i_{k-1}}=\phi_{[ij]i_{1}\dots i_{k-1}}=\phi_{ij(i_{1}\dots
i_{k-1})}$ and $\phi_{[iji_{s}]i_{1}\dots\hat{i}_{s}\dots i_{k-1}}=0$ for any
$1\leq s\leq k-1$. Define a differential operator
$\mathscr{K}:\Gamma(S^{k}(T^{\ast}M))\to\Gamma(\mathscr{C}^{k+1}(T^{\ast}M))$
to be the trace-free part of $D_{[i}\omega_{j]i_{1}\dots i_{k-1}}$. If $k>1$
and $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$,

(3.9) $\displaystyle\begin{split}\mathscr{K}&(\omega)_{iji_{1}\dots
i_{k-1}}=D_{[i}\omega_{j]i_{1}\dots
i_{k-1}}-\tfrac{1}{n+k-3}\sum_{s=1}^{k-1}h_{i_{s}[i}\operatorname{\mathsf{div}}(\omega)_{j]i_{1}\dots\hat{i}_{s}\dots
i_{k-1}}\\\ &=D_{[i}\omega_{j]i_{1}\dots
i_{k-1}}-\tfrac{k-1}{2(n+k-3)}\left(h_{i(i_{1}}\operatorname{\mathsf{div}}(\omega)_{i_{2}\dots
i_{k-1})j}-h_{j(i_{1}}\operatorname{\mathsf{div}}(\omega)_{i_{2}\dots
i_{k-1})i}\right),\qquad\text{if}\,\,k>1,\\\ \end{split}$

which is the completely trace-free part of $D_{[i}\omega_{j]i_{1}\dots
i_{k-1}}$. For $\gamma\in\Gamma(T^{\ast}M)=\Gamma(S^{1}_{0}(T^{\ast}M))$,
$\mathscr{K}(\gamma)_{ij}=D_{[i}\gamma_{j]}=\tfrac{1}{2}d\gamma_{ij}$. In the
$k=1$ case recall that the convention here is
$S^{1}_{0}(T^{\ast}M)=T^{\ast}M$. Checking the equality of the two different
expressions for the trace part of (3.9) is straightforward. By definition,
$\mathscr{K}(\omega)_{iji_{1}\dots
i_{k-1}}=\mathscr{K}(\omega)_{[ij]i_{1}\dots i_{k-1}}$,
$\mathscr{K}(\omega)_{iji_{1}\dots i_{k-1}}=\mathscr{K}(\omega)_{ij(i_{1}\dots
i_{k-1})}$, and there vanishes the antisymmetrization of
$\mathscr{K}(\omega)_{iji_{1}\dots i_{k-1}}$ over $ij$ and any $i_{s}$. From
the definition of $\mathscr{K}$, it follows that its formal adjoint
$\mathscr{K}^{\ast}:\Gamma(\mathscr{C}^{k+1}(T^{\ast}M))\to\Gamma(S^{k}_{0}(T^{\ast}M))$
satisfies

(3.10) $\displaystyle\mathscr{K}^{\ast}(\phi)_{i_{1}\dots
i_{k}}=-D^{p}\phi_{p(i_{1}\dots i_{k})},$
$\displaystyle\mathscr{K}^{\ast}\mathscr{K}(\omega)_{i_{1}\dots
i_{k}}=-D^{p}\mathscr{K}(\omega)_{p(i_{1}\dots i_{k})},$

for $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$, and $\phi_{iji_{1}\dots
i_{k-1}}\in\Gamma(\mathscr{C}^{k+1}(T^{\ast}M))$. If
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ then $D\omega$ has a pure trace part,
and parts in the submodules of $\Gamma(S^{k+1}_{0}(T^{\ast}M))$ with
symmetries corresponding to the partitions $(n+1)$ and $(n1)$. Lemma 3.2
describes explicitly the decomposition of $D\omega$ into these parts. Some
definitions are needed for its statement. The linear map
$\mathsf{i}_{h}:\Gamma(S^{k-1}_{0}(T^{\ast}M))\to\Gamma(T^{\ast}M\otimes
S^{k}_{0}(T^{\ast}M))$ defined by

(3.11) $\displaystyle\mathsf{i}_{h}(\omega)_{ii_{1}\dots
i_{k}}=\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}h_{i(i_{1}}\omega_{i_{2}\dots
i_{k})}+\tfrac{k(1-k)}{(n+k-3)(n+2(k-1))}h_{(i_{1}i_{2}}\omega_{i_{3}\dots
i_{k})i},$

is characterized by the properties that its image is contained in
$\Gamma(T^{\ast}M\otimes S^{k}_{0}(T^{\ast}M))$ and that the nontrivial traces
of $\mathsf{i}_{h}(\omega)$ equal $\omega$. In particular, it is injective.
For $f\in C^{\infty}(M)=\Gamma(S^{0}_{0}(T^{\ast}M))$,
$\mathsf{i}_{h}(f)_{ij}=\tfrac{1}{n}fh_{ij}$ while for
$\gamma_{i}\in\Gamma(T^{\ast}M)=\Gamma(S^{1}_{0}(T^{\ast}M))$,
$\mathsf{i}_{h}(\gamma)_{ijk}=\tfrac{2}{(n-1)(n+2)}\left(nh_{i(j}\gamma_{k)}-\gamma_{i}h_{jk}\right)$.

For $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ define
$\mathscr{T}(\omega)_{ii_{1}\dots
i_{k}}=\tfrac{2k}{k+1}\mathscr{K}(\omega)_{i(i_{1}\dots i_{k})}$, which is
completely trace-free and satisfies $\mathscr{T}(\omega)_{i(i_{1}\dots
i_{k})}=\mathscr{T}(\omega)_{ii_{1}\dots i_{k}}$ and
$\mathscr{T}(\omega)_{(i_{1}\dots i_{k+1})}=0$. Using
$\mathscr{K}(\omega)_{[iji_{1}]\dots i_{k-1}}=0$ it can be checked that
$\mathscr{T}(\omega)_{[ij]i_{1}\dots
i_{k-1}}=\mathscr{K}(\omega)_{iji_{1}\dots i_{k-1}}$. The identity (3.13) of
Lemma 3.2 is a special case of [5, Lemma $2.24$].

######  Lemma 3.2.

For $\omega_{i_{1}\dots i_{k}}\in\Gamma(S^{k}_{0}(T^{\ast}M))$ there hold

(3.12)
$\displaystyle\begin{split}D\omega&=\mathscr{L}(\omega)+\mathscr{T}(\omega)+\mathsf{i}_{h}(\operatorname{\mathsf{div}}(\omega)),\end{split}$
(3.13)
$\displaystyle\begin{split}|D\omega|^{2}&=|\mathscr{L}(\omega)|^{2}+|\mathscr{T}(\omega)|^{2}+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}|\operatorname{\mathsf{div}}(\omega)|^{2}\\\
&=|\mathscr{L}(\omega)|^{2}+\tfrac{2k}{k+1}|\mathscr{K}(\omega)|^{2}+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}|\operatorname{\mathsf{div}}(\omega)|^{2}.\end{split}$

###### Proof.

Substituting the definitions of $\mathscr{L}(\omega)$ and
$\mathscr{K}(\omega)$ into

(3.14) $\displaystyle D_{i}\omega_{i_{1}\dots i_{k}}=D_{(i}\omega_{i_{1}\dots
i_{k})}+\tfrac{2}{k+1}\sum_{s=1}^{k}D_{[i}\omega_{i_{s}]i_{1}\dots\hat{i}_{s}\dots
i_{k}}.$

and simplifying the trace terms yields (3.12). Alternatively, it is
straightforward to check that the right sides of (3.14) and (3.12) are the
same modulo pure trace terms. On the other hand from the properties
characterizing $\mathsf{i}_{h}$ it follows that the traces of the right sides
of (3.14) and (3.12) are the same. This verifies (3.12). Contracting (3.12)
with $D^{i}\omega^{i_{1}\dots i_{k}}$ gives (3.13) ∎

It is immediate from (3.13) that $\ker
D\cap\Gamma(S^{k}_{0}(T^{\ast}M))=\ker\mathscr{L}\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$.

######  Lemma 3.3.

Let $(M,h)$ be an $n$-dimensional Riemannian manifold.

  1. (1)

The differential operator $\mathscr{L}:S^{k}_{0}(T^{\ast}M)\to
S^{k+1}_{0}(T^{\ast}M)$ has injective symbol, so

(3.15) $\displaystyle
S^{k+1}_{0}(T^{\ast}M)=\mathscr{L}(S^{k}_{0}(T^{\ast}M))\oplus(\ker\operatorname{\mathsf{div}}\cap
S^{k+1}_{0}(T^{\ast}M)),$

and $\operatorname{\mathsf{div}}\mathscr{L}:S^{k}_{0}(T^{\ast}M)\to
S^{k}_{0}(T^{\ast}M)$ is an elliptic operator. If $M$ is compact, then
$\operatorname{\mathsf{div}}\mathscr{L}$ is nonpositive and
$\ker\operatorname{\mathsf{div}}\mathscr{L}=\ker\mathscr{L}$.

  2. (2)

For any $c\geq\tfrac{n+2(k-2)}{2(n+k-3)}$, the operator
$\diamond_{c}=(\mathscr{K},\sqrt{c}\operatorname{\mathsf{div}}):\Gamma(S^{k}_{0}(T^{\ast}M))\to\Gamma(\mathscr{C}^{k+1}(T^{\ast}M)\oplus
S^{k-1}_{0}(T^{\ast}M))$ has injective symbol, so

(3.16)
$\displaystyle\diamond_{c}^{\ast}\diamond_{c}=-\mathscr{K}^{\ast}\mathscr{K}+c\mathscr{L}\operatorname{\mathsf{div}}$

is an elliptic operator. If $M$ is compact, then $\diamond_{c}$ is nonpositive
and $\ker\diamond_{c}=\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$.

  3. (3)

For any $c\geq\tfrac{(k+1)(n+2(k-1))}{2k(n+k-3)}$, the operator
$\square_{c}=(\mathscr{K},\sqrt{c}\mathscr{L}):\Gamma(S^{k}_{0}(T^{\ast}M))\to\Gamma(\mathscr{C}^{k+1}(T^{\ast}M)\oplus
S^{k+1}_{0}(T^{\ast}M))$ has injective symbol, so

(3.17)
$\displaystyle\square_{c}^{\ast}\square_{c}=-\mathscr{K}^{\ast}\mathscr{K}+c\operatorname{\mathsf{div}}\mathscr{L}$

is an elliptic operator. If $M$ is compact, then $\square_{c}$ is nonpositive
and $\ker\square_{c}=\ker\mathscr{K}\cap\ker\mathscr{L}$.

###### Proof.

Write $\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)$,
$\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)$, and
$\mathsf{\sigma}_{\operatorname{\mathsf{div}}}(Z)(\phi)$ for the symbols of
$\mathscr{L}$, $\mathscr{K}$, and $\operatorname{\mathsf{div}}$ applied to the
vector $Z^{i}$ and $\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))$. Write
$(i(Z)\phi)_{i_{1}\dots i_{k-1}}=Z^{p}\phi_{pi_{1}\dots i_{k-1}}$.
Straightforward computations show

(3.18)
$\displaystyle\begin{split}|\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)|^{2}&=\tfrac{1}{k+1}|Z|^{2}|\phi|^{2}+\tfrac{k(n+2(k-2))}{(k+1)(n+2(k-1))}|i(Z)\phi|^{2},\end{split}$
(3.19)
$\displaystyle\begin{split}|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)|^{2}&=\tfrac{1}{2}\left(|Z|^{2}|\phi|^{2}-|i(Z)\phi|^{2}\right)-\tfrac{k-1}{2(n+k-3)}|i(Z)\phi|^{2}=\tfrac{1}{2}\left(|\phi|^{2}-\tfrac{n+2(k-2)}{n+k-3}|i(Z)\phi|^{2}\right).\end{split}$

When $k=1$ and $n=2$ the coefficient of the pure trace terms in (3.19) should
be understood in a limiting sense. By (3.18), if
$\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)=0$ for nonzero $Z$, then $\phi=0$, so
$\mathscr{L}$ has injective symbol. The remaining claims follow from standard
elliptic operator theory as in [1, section $4$]. If $M$ is compact, then
$\left(\operatorname{\mathsf{div}}\mathscr{L}\omega,\omega\right)=-\lVert\mathscr{L}(\omega)\rVert^{2}-c\lVert\operatorname{\mathsf{div}}\omega\rVert^{2}\leq
0$ and $\operatorname{\mathsf{div}}\mathscr{L}\omega=0$ if and only if
$\mathscr{L}(\omega)=0$.

Combining
$|\mathsf{\sigma}_{\operatorname{\mathsf{div}}}(Z)(\phi)|^{2}=|\iota(Z)\phi|^{2}$
with (3.19) yields

(3.20)
$\displaystyle|\mathsf{\sigma}_{\diamond_{c}}(Z)(\phi)|^{2}=\tfrac{1}{2}|Z|^{2}|\phi|^{2}+\left(c-\tfrac{1}{2}\tfrac{n+2(k-2)}{n+k-3}\right)|i(Z)\phi|^{2},$

from which the injectivity of $\mathsf{\sigma}_{\diamond_{c}}(Z)$ is apparent.
The ellipticity of $\diamond_{c}$ follows from standard elliptic operator
theory as in [1, section $6$]. If $M$ is compact, then
$\left(\diamond_{c}\omega,\omega\right)=-\lVert\mathscr{K}(\omega)\rVert^{2}-c\lVert\operatorname{\mathsf{div}}\omega\rVert^{2}\leq
0$ and $\diamond_{c}\omega=0$ if and only if $\mathscr{K}(\omega)=0$ and
$\operatorname{\mathsf{div}}\omega=0$.

Combining (3.18) and (3.19) shows

(3.21)
$\displaystyle|\mathsf{\sigma}_{\diamond_{c}}(Z)(\phi)|^{2}=\tfrac{k+1+2c}{2(k+1)}|Z|^{2}|\phi|^{2}+\left(\tfrac{ck(n+2(k-2)}{(k+1)(n+2(k-1))}-\tfrac{1}{2}\tfrac{n+2(k-2)}{n+k-3}\right)|i(Z)\phi|^{2},$

from which the injectivity of $\mathsf{\sigma}_{\square_{c}}(Z)$ is apparent.
The ellipticity of $\square_{c}$ follows from standard elliptic operator
theory. If $M$ is compact, then
$\left(\square_{c}\omega,\omega\right)=-\lVert\mathscr{K}(\omega)\rVert^{2}-c\lVert\mathscr{L}\omega\rVert^{2}\leq
0$ and $\square_{c}\omega=0$ if and only if $\mathscr{K}(\omega)=0$ and
$\mathscr{L}(\omega)=0$. ∎

Let $\mathscr{R}_{ijk}\,^{l}$ be the curvature tensor of the Levi-Civita
connection $D$ of the metric $h$. For $\omega\in\Gamma(S^{k}(T^{\ast}M))$
there holds $2D_{[i}D_{j]}\omega_{i_{1}\dots
i_{k}}=-k\mathscr{R}_{ij(i_{1}}\,^{p}\omega_{i_{2}\dots i_{k})p}$. Tracing
this in $i$ and $i_{k}$ yields

(3.22) $\displaystyle D^{p}D_{j}\omega_{i_{1}\dots
i_{k-1}p}-D_{j}D^{p}\omega_{i_{1}\dots i_{k-1}p}=R_{j}\,^{p}\omega_{i_{1}\dots
i_{k-1}p}+(1-k)R^{p}\,_{j(i_{1}}\,{}^{q}\omega_{i_{2}\dots i_{k-1})pq}.$

Symmetrizing over the free indices yields

(3.23) $\displaystyle D^{p}D_{(i_{1}}\omega_{i_{2}\dots
i_{k})p}-D_{(i_{1}}D^{p}\omega_{i_{2}\dots
i_{k})p}=\wideparen{\mathscr{R}}(\omega)_{i_{1}\dots i_{k}}.$

Thus $\wideparen{\mathscr{R}}(\omega)$ measures the failure of the
commutativity of the symmetrized covariant derivative and the divergence
operator. For this reason $\wideparen{\mathscr{R}}(\omega)$ occurs in
Weitzenböck type formulas.

######  Lemma 3.4.

Let $(M,h)$ be a Riemannian manifold of dimension $n\geq 2$. For
$\alpha\in\mathbb{R}$ define a formally self-adjoint second order elliptic
differential operator
$\square_{\alpha}:\Gamma(S^{k}_{0}(T^{\ast}M))\to\Gamma(S^{k}_{0}(T^{\ast}M))$
by
$\square_{\alpha}\omega=\Delta_{h}\omega+\alpha\wideparen{\mathscr{R}}(\omega)$.

  1. (1)

If $\alpha=-1$, then

(3.24)
$\displaystyle\square_{-1}=\Delta_{h}\omega-\wideparen{\mathscr{R}}(\omega)$
$\displaystyle=\tfrac{n+2(k-2)}{n+k-3}\mathscr{L}\operatorname{\mathsf{div}}(\omega)-2\mathscr{K}^{\ast}\mathscr{K}(\omega),$

is an elliptic operator on $\Gamma(S^{k}_{0}(T^{\ast}M))$. If $M$ is compact,
then $\square_{-1}$ is nonpositive and
$\ker\square_{-1}\cap\Gamma(S^{k}_{0}(T^{\ast}M))=\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$.

  2. (2)

If $\alpha=\tau(k)=\tfrac{k}{n+k-2}$, then

(3.25)
$\displaystyle\begin{split}\square_{\tau(k)}\omega=\Delta_{h}\omega+\tfrac{k}{n+k-2}\wideparen{\mathscr{R}}(\omega)&=\tfrac{n+2(k-1)}{n+k-2}\operatorname{\mathsf{div}}\mathscr{L}(\omega)-\tfrac{2k(n+k-3)}{(k+1)(n+k-2)}\mathscr{K}^{\ast}\mathscr{K}(\omega),\end{split}$

is an elliptic operator on $\Gamma(S^{k}_{0}(T^{\ast}M))$. If $M$ is compact,
then $\square_{\tau(k)}$ is nonpositive and
$\ker\square_{\tau(k)}\cap\Gamma(S^{k}_{0}(T^{\ast}M))=\ker\mathscr{K}\cap\ker\mathscr{L}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$.

  3. (3)

If $-1<\alpha<\tau(k)=\tfrac{k}{n+k-2}$ then $\square_{\alpha}$ is an elliptic
operator on $\Gamma(S^{k}_{0}(T^{\ast}M))$. If $M$ is compact, then
$\square_{\alpha}$ is nonpositive and
$\ker\square_{\alpha}\cap\Gamma(S^{k}_{0}(T^{\ast}M))=\ker
D\cap\Gamma(S^{k}_{0}(T^{\ast}M))$.

###### Proof.

For $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$, straightforward computations
using the Ricci identity and (3.7) show

(3.26)
$\displaystyle\begin{split}\mathscr{L}\operatorname{\mathsf{div}}(\omega)_{i_{1}\dots
i_{k}}&=D_{(i_{1}}D^{p}\omega_{i_{2}\dots
i_{k})p}+\tfrac{1-k}{(n+2(k-2))}h_{(i_{1}i_{2}}D^{p}D^{q}\omega_{i_{3}\dots
i_{k})pq},\end{split}$ (3.27)
$\displaystyle\begin{split}\Delta_{h}\omega+k\wideparen{\mathscr{R}}(\omega)&=(k+1)\operatorname{\mathsf{div}}\mathscr{L}(\omega)-\tfrac{k(n+2(k-2))}{n+2(k-1)}\mathscr{L}\operatorname{\mathsf{div}}(\omega).\end{split}$

Contracting (3.12) with $D^{i}$ and using (3.26) and (3.10) gives

(3.28)
$\displaystyle\begin{split}\Delta_{h}\omega&=\operatorname{\mathsf{div}}\mathscr{L}(\omega)+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}\mathscr{L}\operatorname{\mathsf{div}}(\omega)-\tfrac{2k}{k+1}\mathscr{K}^{\ast}\mathscr{K}(\omega).\end{split}$

Solving (3.27) for $\Delta_{h}\omega$ and substituting the result into (3.28)
yields

(3.29)
$\displaystyle\begin{split}&\wideparen{\mathscr{R}}(\omega)=\operatorname{\mathsf{div}}\mathscr{L}(\omega)-\tfrac{(n+k-2)(n+2(k-2))}{(n+k-3)(n+2(k-1))}\mathscr{L}\operatorname{\mathsf{div}}(\omega)+\tfrac{2}{k+1}\mathscr{K}^{\ast}\mathscr{K}(\omega).\end{split}$

Equation (3.27) and (3.29) are the analogues of the corresponding identities
for operators on antisymmetric forms, for example [50, Equations $(2.8)$ and
$(2.9)$]. Rewriting (3.28) in two different ways using (3.27) gives (3.24) and
(3.25). The ellipticity of $\square_{\alpha}$ in these cases, and its
nonpositivity when $M$ is compact, follow from Lemma 3.3. Being convex
combinations of the elliptic operators $\square_{-1}$ and $\square_{\tau(k)}$,
the operators $\square_{\alpha}$ for $-1<\alpha<\tau(k)$ are elliptic. If $M$
is compact, the same argument shows that $\square_{\alpha}$ is nonpositive. By
(3.24),

(3.30)
$\displaystyle\square_{\alpha}\omega=(1+\alpha)\operatorname{\mathsf{div}}\mathscr{L}(\omega)+\tfrac{(n+2(k-2))(k-\alpha(n+k-2))}{(n+k-3)(n+2(k-1))}\mathscr{L}\operatorname{\mathsf{div}}(\omega)+\tfrac{2(\alpha-k)}{k+1}\mathscr{K}^{\ast}\mathscr{K}(\omega),$

from which follows
$\ker\square_{\alpha}\cap\Gamma(S^{k}_{0}(T^{\ast}M))\supset\ker
D\cap\Gamma(S^{k}_{0}(T^{\ast}M))$. If $M$ is compact and
$\omega\in\ker\square\cap\Gamma(S^{k}_{0}(T^{\ast}M))$, integrating the right
side of (3.30) gives

(3.31) $\displaystyle
0=(1+\alpha)\lVert\mathscr{L}(\omega)\rVert^{2}+\tfrac{(n+2(k-2))(k-\alpha(n+k-2))}{(n+k-3)(n+2(k-1))}\lVert\operatorname{\mathsf{div}}(\omega)\rVert^{2}+\tfrac{2(k-\alpha)}{k+1}\lVert\mathscr{K}(\omega)\rVert^{2},$

and together with (3.12) this implies
$\ker\square_{\alpha}\cap\Gamma(S^{k}_{0}(T^{\ast}M))\subset\ker
D\cap\Gamma(S^{k}_{0}(T^{\ast}M))$. ∎

######  Remark 3.5.

Let $M$ be compact. Define a functional $\mathcal{C}_{\alpha}$ with arguments
a Riemannian metric $h_{ij}$ and a tensor
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ by
$\mathcal{C}_{\alpha}(h,\omega)=-\left(\omega,\square_{\alpha}\omega\right)$.
For fixed $h$ the first variation of $\mathcal{C}_{\alpha}(h,\omega)$ in
$\omega$ yields the equation $\square_{\alpha}\omega=0$. Lemma 3.4 implies
that for $-1\leq\alpha\leq\tfrac{k}{n+k-2}$ the functional
$\mathcal{C}_{\alpha}(h,\omega)$ is nonnegative.

######  Remark 3.6.

The _Lichnerowicz Laplacian_ $\diamond_{L}$ is the formally self-adjoint
operator which acts on an arbitrary rank $k$ covariant tensor
$\omega_{i_{1}\dots i_{k}}$ by
$-\Delta_{h}\omega+k\wideparen{\mathscr{R}}(\omega)$ (see [39, page $27$] for
the definition of $\wideparen{\mathscr{R}}(\omega)$ for general tensors
$\omega$). The linearization of the Ricci curvature of the metric $h$ at the
symmetric two-tensor $a_{ij}$ is
$\tfrac{1}{2}\diamond_{L}a_{ij}+D_{(i}D^{p}a_{j)p}$. On differential forms the
Lichnerowicz Laplacian restricts to the usual Hodge Laplacian. The
Lichnerowicz operator restricts to $-\square_{-k}$ on
$\Gamma(S^{k}_{0}(T^{\ast}M))$.

######  Remark 3.7.

The special case of $\square_{-1}=\Delta_{h}-\wideparen{\mathscr{R}}$ acting
on sections of $S^{2}_{0}(T^{\ast}M)$ was studied by J. Simons in [53], and
this case of Lemma 3.4 is given in [1, section $6$.c].

######  Remark 3.8.

As is shown in [1], an infinitesimal deformation of an Einstein metric $h$ on
a compact manifold is identified with an
$\omega\in\Gamma(S^{2}_{0}(T^{\ast}M))\cap\ker\operatorname{\mathsf{div}}$
solving
$\Delta\omega=2\wideparen{\mathscr{R}}(\omega)-\tfrac{2\mathscr{R}}{n}\omega$.
As is summarized in [2, section $12$.H] (the notations there are different
than those here), using this equation in conjunction with the positivity
conditions given by integrating (3.27) and (3.24) gives a proof of the
criterion of N. Koiso [37, Theorem $3.3$], for the rigidity of an Einstein
metric, in particular showing that an Einstein metric of negative sectional
curvature is rigid provided $n\geq 3$.

Because the operator $\wideparen{\mathscr{Y}}$ associated with
$\mathscr{Y}\in\mathscr{MC}(\mathbb{V}^{\ast})$ is self-adjoint it determines
a quadratic form defined by
$\mathscr{Q}_{\mathscr{Y}}(\omega)=\langle\omega,\wideparen{\mathscr{Y}}(\omega)\rangle$
for $\omega\in E$ on any $\wideparen{\mathscr{Y}}$-invariant subspace
$E\subset\otimes^{k}\mathbb{V}^{\ast}$.

######  Corollary 3.9.

Let $(M,h)$ be a Riemannian manifold of dimension $n\geq 2$. For
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ there hold

(3.32) $\displaystyle\tfrac{1}{2}\Delta_{h}|\omega|^{2}$
$\displaystyle=|D\omega|^{2}+(k+1)\langle\omega,\operatorname{\mathsf{div}}\mathscr{L}(\omega)\rangle-\tfrac{k(n+2(k-2))}{n+2(k-1)}\langle\omega,\mathscr{L}\operatorname{\mathsf{div}}(\omega)\rangle-k\mathscr{Q}_{\mathscr{R}}(\omega).$
(3.33)
$\displaystyle\begin{split}\tfrac{1}{2}\Delta_{h}|\omega|^{2}&=|D\omega|^{2}+\tfrac{n+2(k-1)}{n+k-2}\langle\omega,\operatorname{\mathsf{div}}\mathscr{L}(\omega)\rangle-\tfrac{2k(n+k-3)}{(k+1)(n+k-2)}\langle\omega,\mathscr{K}^{\ast}\mathscr{K}(\omega)\rangle-\tfrac{k}{n+k-2}\mathscr{Q}_{\mathscr{R}}(\omega).\end{split}$
(3.34) $\displaystyle\tfrac{1}{2}\Delta_{h}|\omega|^{2}$
$\displaystyle=|D\omega|^{2}+\tfrac{n+2(k-2)}{n+k-3}\langle\omega,\mathscr{L}\operatorname{\mathsf{div}}(\omega)\rangle-2\langle\omega,\mathscr{K}^{\ast}\mathscr{K}(\omega)\rangle+\mathscr{Q}_{\mathscr{R}}(\omega).$

###### Proof.

Contracting (3.27), (3.25), and (3.24) with $\omega$ yields (3.32)-(3.34).
(Any two of (3.32)-(3.34) imply the third.) ∎

######  Remark 3.10.

If
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\operatorname{\mathsf{div}}\cap\ker\mathscr{K}$,
(3.34) and

(3.35)
$\displaystyle\begin{split}\mathscr{Q}_{\mathscr{R}}(\omega)&=\mathscr{Q}_{\mathscr{W}}(\omega)+\tfrac{n+2(k-2)}{n-2}\langle\operatorname{\rho}(\omega\owedge\omega),\operatorname{\rho}(\mathscr{R})\rangle+\tfrac{1-k}{(n-1)(n-2)}\mathscr{R}_{h}|\omega|_{h}^{2}.\end{split}$

together yield

(3.36) $\displaystyle\tfrac{1}{2}\Delta_{h}|\omega|^{2}$
$\displaystyle=|D\omega|^{2}+\mathscr{Q}_{\mathscr{W}}(\omega)+\tfrac{n+2(k-2)}{n-2}\langle\operatorname{\rho}(\omega\owedge\omega),\operatorname{\rho}(\mathscr{R})\rangle+\tfrac{1-k}{(n-1)(n-2)}\mathscr{R}_{h}|\omega|_{h}^{2},$

where $\mathscr{W}_{ijkl}\in\mathscr{MC}(\mathbb{V}^{\ast})$ is the conformal
Weyl tensor. This recovers [40, Corollary $4.2$].

##  4\. Vanishing theorems for conformal Killing and divergence free Codazzi
tensors

This section defines conformal Killing and trace-free Codazzi tensors and
proves vanishing theorems for them. The vanishing theorems for conformal
Killing tensors and trace and divergence free Codazzi tensors are analogous to
the somewhat stronger vanishing theorems for symmetric tensors on Kähler
manifolds obtained by S. Kobayashi in [35, 36].

Because $\mathscr{L}$ and $\mathscr{K}$ are constructed by taking trace-free
parts they behave well with respect to conformal changes of the metric. The
Levi-Civita connections $\tilde{D}$ and $D$ of conformally related pseudo-
Riemannian metrics $\tilde{h}_{ij}=fh_{ij}$ are related by
$\tilde{D}-D=2\sigma_{(i}\delta_{j)}\,^{k}-h_{ij}h^{kp}\sigma_{p}$ with
$2\sigma_{i}=d\log f_{i}$ and $\sigma^{i}=h^{ip}\sigma_{p}$. Write
$\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)$ for the symbol of $\mathscr{L}$
applied to the vector $Z^{i}$ and $\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))$, and
similarly for $\mathscr{K}$ and $\operatorname{\mathsf{div}}$. For
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$, $0<f\in C^{\infty}(M)$, and
$\alpha\in\mathbb{R}$, there hold

(4.1) $\displaystyle\begin{aligned}
&\mathscr{L}_{\tilde{h}}(f^{\alpha}\omega)=f^{\alpha}\left(\mathscr{L}_{h}(\omega)+2(\alpha-k)\mathsf{\sigma}_{\mathscr{L}}(\sigma^{\sharp})(\omega)\right),&\\\
&\mathscr{K}_{\tilde{h}}(f^{\alpha}\omega)=f^{\alpha}\left(\mathscr{K}_{h}(\omega)+(2\alpha+1-k)\mathsf{\sigma}_{\mathscr{K}}(\sigma^{\sharp})(\omega)\right),\\\
&f\operatorname{\mathsf{div}}_{\tilde{h}}(f^{\alpha}\omega)=f^{\alpha}\left(\operatorname{\mathsf{div}}_{h}(\omega)+(n-2+2\alpha)\iota(\sigma^{\sharp})\omega\right),\end{aligned}$

so that $\mathscr{L}$, $\mathscr{K}$, and $\operatorname{\mathsf{div}}$ are
conformally invariant in the sense that for $0<f\in C^{\infty}(M)$ there hold

(4.2)
$\displaystyle\mathscr{L}_{\tilde{h}}(f^{k}\omega)=f^{k}\mathscr{L}_{h}(\omega),$
$\displaystyle\mathscr{K}_{\tilde{h}}(f^{(k-1)/2}\omega)=f^{(k-1)/2}\mathscr{K}_{h}(\omega),$
$\displaystyle
f\operatorname{\mathsf{div}}_{\tilde{h}}(f^{1-n/2}\omega)=f^{1-n/2}\operatorname{\mathsf{div}}_{h}(\omega).$

Define $\mathscr{L}^{\sharp}:\Gamma(S^{k}_{0}(TM))\to\Gamma(S^{k+1}_{0}(TM))$
by $\mathscr{L}^{\sharp}(X)=\mathscr{L}(X^{\flat})^{\sharp}$. Then (4.2)
implies the invariance
$f\mathscr{L}^{\sharp}_{\tilde{h}}(X)=\mathscr{L}^{\sharp}_{h}(X)$, so that
while $\mathfrak{L}^{\sharp}$ depends on $h$, the subspace
$\ker\mathscr{L}^{\sharp}\cap\Gamma(S^{k}_{0}(TM))$ does not. A _conformal
Killing tensor_ of rank $k$ is an element of
$\ker\mathscr{L}^{\sharp}\cap\Gamma(S^{k}_{0}(TM))$.

A _conformal Codazzi tensor_ is an element of
$\ker\mathscr{K}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$. A divergence-free element
of $\ker\mathscr{K}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$ is a _trace-free Codazzi
tensor_. These have been studied previously in [56, 51].

######  Remark 4.1.

Except in ranks one and two, conformal Killing tensors are not as well studied
as their antisymmetric counterparts, the conformal Killing forms, for which
[50] is a good reference. Probably their most natural occurrence is as the
symbols of symmetries of the Laplacian; see [23, 30, 38, 43, 57]. For further
background on Killing and conformal Killing tensors see also [32, 42, 47, 48,
58, 62].

The _Cartan product_ $\alpha\circledcirc\beta\in S^{k+l}_{0}\mathbb{V}^{\ast}$
of $\alpha\in S^{k}_{0}\mathbb{V}^{\ast}$ and $\beta\in
S^{l}_{0}\mathbb{V}^{\ast}$ defined by
$\alpha\circledcirc\beta=\operatorname{\mathsf{tf}}(\alpha\odot\beta)$ makes
$S_{0}(\mathbb{V}^{\ast})=\oplus_{k\geq 0}S^{k}_{0}\mathbb{V}^{\ast}$ into a
graded associative algebra (see [22, Supplement] or [24] for background). This
claim can be justified by showing that the ideal in
$(S(\mathbb{V}^{\ast}),\odot)$ generated by $h$ equals the kernel
$\ker\operatorname{\mathsf{tf}}$ of the graded linear map
$\operatorname{\mathsf{tf}}:S(\mathbb{V}^{\ast})\to S_{0}(\mathbb{V}^{\ast})$
sending $\alpha\in S^{k}(\mathbb{V})$ to its trace-free part. That the
symmetrized covariant derivative is a derivation of the graded algebra of
symmetric Killing tensors is [57, Lemma $1.3$]. Lemma 4.2 is the corresponding
statement, that the operator $\mathscr{L}$ acting on conformal Killing tensors
is a derivation with respect to the Cartan product.

######  Lemma 4.2.

On a pseudo-Riemannian manifold $(M,h)$, the operator $\mathscr{L}$ is a
derivation with respect to the Cartan product on $\Gamma(S_{0}(T^{\ast}M))$.
Consequently, on a conformal manifold $(M,[h])$, the subspace
$\mathsf{CK}(TM,[h])=\oplus_{k\geq
0}\ker\mathscr{L}^{\sharp}\cap\Gamma(S^{k}_{0}(TM))\subset\Gamma(S_{0}(TM))$
comprising finite linear combinations of conformal Killing tensors is a
subalgebra with respect to the Cartan product.

###### Proof.

For $\alpha\in\Gamma(S^{k}_{0}(T^{\ast}M))$ and
$\beta\in\Gamma(S^{l}_{0}(T^{\ast}M))$,
$\alpha\odot\beta=\operatorname{\mathsf{tf}}(\alpha\odot\beta)+h\odot\gamma$
for $\gamma\in\Gamma(S^{k+l-2}(T^{\ast}M))$, so

(4.3)
$\displaystyle\begin{split}\\{h,\operatorname{\mathsf{tf}}(\alpha\odot\beta)\\}=\\{h,\alpha\odot\beta\\}+\\{h,h\odot\gamma\\}=\\{h,\alpha\odot\beta\\}+h\odot\\{h,\gamma\\}.\end{split}$

Hence, because
$\operatorname{\mathsf{tf}}:(S(T^{\ast}M),\odot)\to(S_{0}(T^{\ast}M),\circledcirc)$
is a graded linear homomorphism,

(4.4)
$\displaystyle\begin{split}2\mathscr{L}(\alpha\circledcirc\beta)&=\operatorname{\mathsf{tf}}\\{h,\alpha\circledcirc\beta\\}=\operatorname{\mathsf{tf}}\\{h,\operatorname{\mathsf{tf}}(\alpha\odot\beta)\\}=\operatorname{\mathsf{tf}}\\{h,\alpha\odot\beta\\}=\operatorname{\mathsf{tf}}\left(\\{h,\alpha\\}\odot\beta+\alpha\odot\\{h,\beta\\}\right)\\\
&=\operatorname{\mathsf{tf}}(\\{h,\alpha\\})\circledcirc\beta+\alpha\circledcirc\operatorname{\mathsf{tf}}(\\{h,\beta\\})=2\left(\mathscr{L}(\alpha)\circledcirc\beta+\alpha\circledcirc\mathscr{L}(\beta)\right).\end{split}$

The identity (4.4) shows $\mathsf{CK}(T^{\ast}M,h)=\oplus_{k\geq
0}\ker\mathscr{L}\cap\Gamma(S^{k}_{0}(T^{\ast}M))\subset\Gamma(S_{0}(T^{\ast}M))$
is a subalgebra under Cartan product. While $\mathsf{CK}(T^{\ast}M,h)$ depends
on the choice of $h\in[h]$, it is linearly isomorphic to
$\mathsf{CK}(T^{\ast}M,e^{f}h)$ by the graded linear map sending
$\omega_{i_{1}\dots i_{k}}$ to $f^{k}\omega_{i_{1}\dots i_{k}}$, and both are
are identified with $\mathsf{CK}(TM,[h])$ via index raising, so
$\mathscr{L}^{\sharp}$ is a derivation of $S_{0}(TM)$ with kernel
$\mathsf{CK}(TM,[h])$. ∎

It is convenient to say that $\mathscr{Q}_{\mathscr{R}}$ is positive or
negative (semi-)definite on $S^{k}(T^{\ast}M)$ or $S^{k}_{0}(T^{\ast}M)$ if it
is positive (semi-)definite or negative (semi-)definite as a quadratic form on
$\Gamma(S^{k}(T^{\ast}M))$ or $\Gamma(S^{k}_{0}(T^{\ast}M))$. Since, by (2.8),
$\mathscr{Q}_{\mathscr{R}}(h^{\odot k})=0$ for any $k\geq 1$,
$\mathscr{Q}_{\mathscr{R}}$ is not definite on $\Gamma(S^{2k}(T^{\ast}M))$ for
any $k\geq 1$.

######  Theorem 4.3 ([26]).

Let $(M,h)$ be a compact Riemannian manifold of dimension $n>2$.

  1. (1)

If $\mathscr{Q}_{\mathscr{R}}$ is nonnegative on $S^{k}_{0}(T^{\ast}M)$ then
any
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$
is parallel. If moreover $\mathscr{Q}_{\mathscr{R}}$ is at some point of $M$
strictly positive on $S^{k}_{0}(T^{\ast}M)$ then
$\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}=\\{0\\}$.

  2. (2)

If $\mathscr{Q}_{\mathscr{R}}$ is nonpositive on $S^{k}_{0}(T^{\ast}M)$ then
any rank $k$ conformal Killing tensor is parallel, and if, moreover,
$\mathscr{Q}_{\mathscr{R}}$ is at some point of $M$ strictly negative on
$S^{k}_{0}(T^{\ast}M)$, then any rank $k$ conformal Killing tensor is
identically zero.

###### Proof.

For a compactly supported $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$, integrating
any of (3.32)-(3.34) by parts against the Riemannian volume
$\operatorname{vol}_{h}$ and simplifying the result using (3.13) yields

(4.5)
$\displaystyle\begin{split}\tfrac{2}{k+1}\lVert|\mathscr{K}(\omega)\rVert^{2}&+\tfrac{(n+k-2)(n+2(k-2))}{(n+k-3)(n+2(k-1))}\lVert\operatorname{\mathsf{div}}(\omega)\rVert^{2}-\lVert\mathscr{L}(\omega)\rVert^{2}=\int_{M}\mathscr{Q}_{\mathscr{R}}(\omega)\,d\operatorname{vol}_{h}.\end{split}$

The identity (4.5) generalizes the usual integrated Bochner identities for
harmonic one-forms and conformal Killing vector fields. If
$\omega\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$ and
$\mathscr{Q}_{\mathscr{R}}\geq 0$ then (4.5) shows that
$\omega\in\ker\mathscr{L}$ and from (3.13) it follows that $D\omega=0$. If
moreover $\mathscr{Q}_{\mathscr{R}}$ is somewhere positive then (3.34) shows
$\omega\equiv 0$. If $\omega\in\ker\mathscr{L}$ and
$\mathscr{Q}_{\mathscr{R}}\leq 0$ then (4.5) shows that
$\omega\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$ and from (3.13)
it follows that $D\omega=0$. If, moreover, $\mathscr{Q}_{\mathscr{R}}$ is
somewhere negative then (3.33) shows $\omega\equiv 0$. ∎

######  Remark 4.4.

A result very similar to (1) of Theorem 4.3 was obtained in [56, Theorem $2$].
The difference is that the operator $\wideparen{\mathscr{R}}$ and
corresponding quadratic form $\mathscr{Q}_{\mathscr{R}}$ considered in [56]
are slightly different (they differ by a term involving the Ricci curvature).
The claim here is very slightly more general, but, when the curvature is
assumed to have a definite sign, since this sign is inherited by the Ricci
tensor, the ambit of application of the claims is the same.

######  Corollary 4.5.

Let $h$ be a Riemannian metric on a compact manifold $M$ of dimension $n>2$.

  1. (1)

(Stepanov, [56, Theorem $2$]; see also [51, Corollary $1$]) If $h$ has
nonnegative sectional curvature, then
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$
is parallel. If, moreover, the sectional curvature is strictly positive at
some point of $M$ then
$\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}=\\{0\\}$.

  2. (2)

(Dairbekov-Sharafutdinov, [20, Theorem $1.6$]; see also [33, Proposition
$6.6$]) If $h$ has nonpositive sectional curvature, then a rank $k$ conformal
Killing tensor is parallel, and if, moreover, the sectional curvature is
strictly negative at some point of $M$, then a rank $k$ conformal Killing
tensor is identically zero.

###### Proof.

Both claims follow from Theorem 4.3 once it is known that a sign condition on
the sectional curvature of $h$ implies the same sign condition for
$\mathscr{Q}_{\mathscr{R}}$ on $\Gamma(S^{k}_{0}(T^{\ast}M))$. When $k=2$ this
follows from the proof of [1, Proposition $6.1$], but that argument (which is
direct) does not extend straightforwardly to the case $k>2$. For conformal
Killing tensors, claim (2) is [20, Theorem $1.6$]. A proof of the result of
Dairbekov-Sharafutdinov for all $k$ based on Weitzenböck formulas as in
Theorem 4.3 was given as [33, Proposition $6.6$]; they show via an elegant
integration in the fibers argument that the condition (2) of Theorem 4.3
follows from the nonpositivity of the sectional curvature. Their argument
works equally well assuming nonnegativity of the sectional curvature, and
combined with Theorem 4.3, this yields corollary 4.5. Alternatively, in the
case of trace-free Codazzi tensors, the proof of [51, Corollary $1$] shows how
to deduce the required nonnegativity for $k>2$ from the Berger-Ebin argument.
∎

######  Remark 4.6.

The $n=2$ case of Corollary 4.5 was proved in [27, section $3$]. The $k=2$
case of (1) of Corollary 4.5 is stated as [40, Theorem $5.1$], where this
statement is generalized to higher rank traceless Codazzi tensors supposing
the background metric is conformally flat.

##  5\. Coupling Einstein equations to symmetric tensors

This section describes Einstein-like equations coupling a metric to a trace-
free symmetric $k$-tensor.

Given $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ define one-forms by

(5.1)
$\displaystyle\begin{split}&\mathscr{A}_{h}(\omega)_{i}=\omega^{p_{1}\dots
p_{k}}\mathscr{K}_{h}(\omega)_{ip_{1}\dots
p_{k}},\qquad\mathscr{B}_{h}(\omega)_{i}=\omega^{p_{1}\dots
p_{k}}\mathscr{L}_{h}(\omega)_{ip_{1}\dots p_{k}},\\\
&\mathscr{D}_{h}(\omega)_{i}=\omega_{i}\,^{p_{1}\dots
p_{k-1}}\operatorname{\mathsf{div}}_{h}(\omega)_{p_{1}\dots
p_{k-1}}.\end{split}$

By (4.2), with $\sigma_{i}=\tfrac{1}{2}d\log f_{i}$,

(5.2)
$\displaystyle\begin{split}f\mathscr{A}_{fh}(f^{\alpha}\omega)&=f^{2\alpha+1-k}\left(\mathscr{A}_{h}(\omega)+\tfrac{2\alpha+1-k}{2}\left(|\omega|^{2}_{h}\sigma_{i}-\tfrac{n+2(k-2)}{n+k-3}\iota(\sigma^{\sharp})\operatorname{\rho}(\omega\owedge\omega)\right)\right),\\\
f\mathscr{B}_{fh}(f^{\alpha}\omega)&=f^{2\alpha+1-k}\left(\mathscr{B}_{h}(\omega)+\tfrac{2(\alpha-k)k}{k+1}\left(|\omega|^{2}_{h}\sigma_{i}+\tfrac{n+2(k-2)}{n+2(k-1)}\iota(\sigma^{\sharp})\operatorname{\rho}(\omega\owedge\omega)\right)\right),\\\
f\mathscr{D}_{fh}(f^{\alpha}\omega)&=f^{2\alpha+1-k}\left(\mathscr{D}_{h}(\omega)+(n-2+2\alpha)\iota(\sigma^{\sharp})\operatorname{\rho}(\omega\owedge\omega)\right).\end{split}$

######  Lemma 5.1.

Let $M$ be a manifold of dimension $n\geq 2$ and let $h$ be a pseudo-
Riemannian metric. For $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ there hold

(5.3)
$\displaystyle\begin{split}\operatorname{\mathsf{div}}(\operatorname{\rho}(\omega\owedge\omega))_{i}&=-\tfrac{2}{k+1}\mathscr{A}(\omega)_{i}+\mathscr{B}(\omega)_{i}+\left(1+\tfrac{n-2}{(n+k-3)(n+2(k-1))}\right)\mathscr{D}(\omega)_{i}\\\
&=-\tfrac{2}{k+1}\mathscr{A}(\omega)_{i}+\mathscr{B}(\omega)_{i}+\tfrac{(n+k-2)(n+2(k-2))+2(n-2)}{(n+k-3)(n+2(k-1))}\mathscr{D}(\omega)_{i},\end{split}$
(5.4) $\displaystyle\tfrac{1}{2}\operatorname{\mathsf{div}}(|\omega|^{2}h)$
$\displaystyle=\tfrac{1}{2}D_{i}|\omega|^{2}=\tfrac{2k}{k+1}\mathscr{A}(\omega)_{i}+\mathscr{B}(\omega)_{i}+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}\mathscr{D}(\omega)_{i},$
(5.5)
$\displaystyle\tfrac{1}{2}D_{i}|\omega|^{2}-\operatorname{\mathsf{div}}(\operatorname{\rho}(\omega\owedge\omega))_{i}$
$\displaystyle=2\mathscr{A}(\omega)_{i}-\tfrac{n-2}{n+k-3}\mathscr{D}(\omega)_{i},$
(5.6)
$\displaystyle\tfrac{1}{2k}D_{i}|\omega|^{2}+\operatorname{\mathsf{div}}(\operatorname{\rho}(\omega\owedge\omega))_{i}$
$\displaystyle=\tfrac{k+1}{k}\mathscr{B}(\omega)_{i}+\tfrac{n+2k}{n+2(k-1)}\mathscr{D}(\omega)_{i},$

where when $k=1$ (5.3)-(5.6) have sense as written if
$\operatorname{\rho}(\omega\owedge\omega)_{ij}$ is interpreted as
$\omega_{i}\omega_{j}$.

###### Proof.

Suppose $k\geq 2$ and let $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$. Contracting
$\omega_{i_{1}\dots i_{k}}$ with $D_{i}\omega_{i_{1}\dots i_{k}}$ and using
(3.12) yields

(5.7)
$\displaystyle\begin{split}\tfrac{1}{2}D_{i}|\omega|^{2}&=\omega^{i_{1}\dots
i_{k}}D_{i}\omega_{i_{1}\dots
i_{k}}=\tfrac{2k}{k+1}\mathscr{A}(\omega)_{i}+\mathscr{B}(\omega)_{i}+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}\mathscr{D}(\omega)_{i},\end{split}$

which is (5.3). Contracting $\omega_{i_{1}\dots i_{k}}$ with
$D_{i_{1}}\omega_{i_{2}\dots i_{k}i}$ and using (3.12) yields

(5.8) $\displaystyle\begin{split}\omega^{i_{1}\dots
i_{k}}D_{i_{1}}\omega_{i_{2}\dots
i_{k}i}&=-\tfrac{2}{k+1}\mathscr{A}(\omega)_{i}+\mathscr{B}(\omega)_{i}+\tfrac{2(1-k)}{(n+k-3)(n+2(k-1))}\mathscr{D}(\omega)_{i}.\end{split}$

Differentiating
$\operatorname{\rho}(\omega\owedge\omega)_{ij}=\omega_{ia_{1}\dots
a_{k-1}}\omega_{j}\,^{a_{1}\dots a_{k-1}}$ yields

(5.9) $\displaystyle\begin{split}w^{i_{1}\dots
i_{k}}D_{i_{1}}\omega_{i_{2}\dots
i_{k}i}&=\operatorname{\mathsf{div}}(\operatorname{\rho}(\omega\owedge\omega))_{i}-\mathscr{D}(\omega)_{i}.\end{split}$

Substitututing (5.9) in (5.8) yields (5.4). Taking linear combinations of
(5.3) and (5.4) in different ways yields (5.5) and (5.6). The $k=1$ cases of
(5.5) and (5.6) follow from

(5.10)
$\displaystyle\begin{split}\operatorname{\mathsf{div}}(\gamma\otimes\gamma-\tfrac{1}{2}|\gamma|^{2}_{h}h)_{i}&=\operatorname{\mathsf{div}}(\gamma)\gamma_{i}-2\gamma^{p}D_{[i}\gamma_{p]}=\operatorname{\mathsf{div}}(\gamma)\gamma_{i}-2\gamma^{p}\mathscr{K}(\gamma)_{ip},\\\
\operatorname{\mathsf{div}}(\gamma\otimes\gamma+\tfrac{1}{2}|\gamma|^{2}_{h}h)_{i}&=\operatorname{\mathsf{div}}(\gamma)\gamma_{i}+2\gamma^{p}D_{(i}\gamma_{p)}=\tfrac{n+2}{2}\operatorname{\mathsf{div}}(\gamma)\gamma_{i}+2\gamma^{p}\mathscr{L}(\gamma)_{ip},\end{split}$

valid for any $\gamma\in\Gamma(T^{\ast}M)$. ∎

######  Remark 5.2.

Among the possible linear combinations of (5.3) and (5.4), the combinations
(5.6) and (5.5) are distinguished by the absence of $\mathscr{A}(\omega)$ or
$\mathscr{B}(\omega)$. This is relevant because the conformal scaling of
$\mathscr{A}(\omega)$, $\mathscr{B}(\omega)$, and $\mathscr{D}(\omega)$ is
different. The first two rescale in a way that depends on $k$, while the last
rescales in a way that depends only on dimension. This means that given a
conformal structure $[h]$ it makes sense to impose that either
$\mathscr{A}(\omega)$ or $\mathscr{B}(\omega)$ vanish, but does not make sense
to require that both vanish. The vanishing of $\mathscr{D}(\omega)$ can then
be treated as condition selecting within the conformal class.

Recall the convention $S^{1}_{0}\mathbb{V}^{\ast}=\mathbb{V}^{\ast}$. Define
linear operators
$\mathscr{T}^{\pm}:\Gamma(S^{k}_{0}(T^{\ast}M))\to\Gamma(S^{2}T^{\ast}M)$ by

(5.11) $\displaystyle\begin{aligned}
\mathscr{T}^{+}(\omega)_{ij}&=\begin{cases}(\omega\otimes\omega)_{ij}-\tfrac{1}{2}|\omega|^{2}_{h}h_{ij}&k=1,\\\
\operatorname{\rho}(\omega\owedge\omega)_{ij}-\tfrac{1}{2}|\omega|^{2}_{h}h_{ij}&k\geq
2,\end{cases}&&\mathscr{T}^{-}(\omega)_{ij}&=\begin{cases}(\omega\otimes\omega)_{ij}+\tfrac{1}{2}|\omega|^{2}_{h}h_{ij}&k=1,\\\
\operatorname{\rho}(\omega\owedge\omega)_{ij}+\tfrac{1}{2k}|\omega|^{2}_{h}h_{ij}&k\geq
2.\end{cases}\end{aligned}$

Note the asymmetry in the definitions; there is a $k$ in the definition of
$\mathscr{T}^{-}$ that is not present in the definition of $\mathscr{T}^{+}$.
The $\pm$ in the notation is motivated by thinking of sections of
$S^{k}_{0}TM$ as sections of $S^{-k}_{0}T^{\ast}M$ (although this has no
sense), so corresponding to negative integers. By definition,

(5.12)
$\displaystyle\operatorname{\mathsf{tr}}\mathscr{T}^{+}(\omega)=\tfrac{2-n}{2}|\omega|^{2}_{h},$
$\displaystyle\operatorname{\mathsf{tr}}\mathscr{T}^{-}(\omega)=\tfrac{n+2k}{2k}|\omega|^{2}_{h}.$

The operators $\mathscr{T}^{\pm}$ depend on $h$. When it is necessary to
indicate this dependence there is written $\mathscr{T}^{\pm}_{h}$. From the
identities, for $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$,

(5.13) $\displaystyle\omega\owedge_{fh}\omega=f^{2-k}\omega\owedge\omega,$
$\displaystyle\operatorname{\rho}_{fh}(\omega\owedge_{fh}\omega)=f^{1-k}\operatorname{\rho}_{h}(\omega\owedge\omega),$
$\displaystyle|\omega|_{fh}^{2}=f^{-k}|\omega|^{2}_{h},$

for $0<f\in C^{\infty}(M)$, it follows that the operators $\mathscr{T}^{\pm}$
are conformally invariant in the sense that

(5.14)
$\displaystyle\mathscr{T}^{\pm}_{fh}(f^{\alpha}\omega)_{ij}=f^{2\alpha+1-k}\mathscr{T}^{\pm}_{h}(\omega)_{ij},$
$\displaystyle\omega\in\Gamma(S^{k}_{0}(T^{\ast}M)).$

######  Corollary 5.3.

Let $(M,h)$ be a pseudo-Riemannian manifold of dimension $n\geq 2$. For
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$,

(5.15)
$\displaystyle\operatorname{\mathsf{div}}(\mathscr{T}^{+}(\omega))_{i}=-2\mathscr{A}(\omega)_{i}+\tfrac{n-2}{n+k-3}\mathscr{D}(\omega)_{i},$
$\displaystyle\operatorname{\mathsf{div}}(\mathscr{T}^{-}(\omega))_{i}=\tfrac{k+1}{k}\mathscr{B}(\omega)_{i}+\tfrac{n+2k}{n+2(k-1)}\mathscr{D}(\omega)_{i}.$

###### Proof.

This is immediate from Lemma 5.1 and (5.11). ∎

######  Lemma 5.4.

Let $(M,h)$ be a pseudo-Riemannian manifold $M$ of dimension $n\geq 3$.
Consider sequences of tensors
$\\{\omega(k)\in\Gamma(S^{k}_{0}(T^{\ast}M)):k\geq 1\\}$ and
$\\{\gamma(k)\in\Gamma(S^{k}_{0}(T^{\ast}M)):k\geq 1\\}$ such that

  * •

$\omega^{(k)}\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{D}\cap\ker\mathscr{A}$,

  * •

$\gamma^{(k)}\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{D}\cap\ker\mathscr{B}$,
and

  * •

the tensors $\mathscr{T}^{\text{cod}}_{ij}=\sum_{k\geq
1}a_{k}\mathscr{T}^{+}(\omega^{(k)})_{ij}$ and
$\mathscr{T}^{\text{kill}}_{ij}=\sum_{k\geq
1}b_{k}\mathscr{T}^{-}(\omega^{(k)})_{ij}$ converge pointwise to smooth
sections of $S^{2}(T^{\ast}M)$ for some sequences $\\{a_{k}\in\mathbb{R}:k\geq
1\\}$ and $\\{b_{k}\in\mathbb{R}:k\geq 1\\}$.

Then the equations

(5.16)
$\displaystyle\mathscr{R}_{ij}-\tfrac{1}{2}\mathscr{R}_{h}h_{ij}+\tfrac{n-2}{2n}\kappa
h_{ij}=\mathscr{T}^{\text{cod}}_{ij}+\mathscr{T}^{\text{kill}}_{ij}$

are consistent and

(5.17) $\displaystyle\kappa=\mathscr{R}_{h}-\sum_{k\geq
1}a_{k}|\omega^{(k)}|_{h}^{2}+\sum_{k\geq
1}\tfrac{n+2}{k(n-2)}b_{k}|\gamma^{(k)}|^{2}_{h}$

is a constant. If $n>2$ the equations (5.16) hold for some
$\kappa\in\mathbb{R}$ if and only if

(5.18)
$\displaystyle\mathscr{R}_{ij}-a_{1}\omega^{(1)}_{i}\omega^{(1)}_{j}-\sum_{k\geq
2}a_{k}\operatorname{\rho}(\omega^{(k)}\owedge\omega^{(k)})_{ij}-b_{1}\gamma^{(1)}_{i}\gamma^{(1)}_{j}-\sum_{k\geq
2}b_{k}\operatorname{\rho}(\gamma^{(k)}\owedge\gamma^{(k)})_{ij}=\tfrac{\kappa}{n}h_{ij},$

in which case $\kappa$ has the form (5.17).

######  Remark 5.5.

Note that the equations (5.16) make sense for $h$ of any signature, although
for indefinite signature $h$ the expressions $|\omega^{(k)}|^{2}_{h}$ need not
be positive.

###### Proof of Lemma 5.4.

By Corollary 5.3, $\mathscr{T}^{\text{cod}}_{ij}$ and
$\mathscr{T}^{\text{kill}}_{ij}$ are divergence free. Taking the divergence of
both sides of (5.16) shows that $\kappa$ must be constant if (5.16) is to
admit solutions. The form (5.17) for $\kappa$ follows by tracing both sides of
(5.16). Evidently (5.16) implies (5.18). If there holds (5.18), taking the
divergence of (5.18) and using the traced differential Bianchi identity
$2D^{p}\mathscr{R}_{ip}=D_{i}\mathscr{R}_{h}$ and Corollary 5.3, shows

(5.19) $\displaystyle 0=\tfrac{n-2}{n}\left(\mathscr{R}_{h}-\sum_{k\geq
1}a_{k}|\omega^{(k)}|_{h}^{2}+\sum_{k\geq
1}\tfrac{n+2}{k(n-2)}b_{k}|\gamma^{(k)}|^{2}_{h}\right),$

so that, since $n>2$, $\kappa$ as in (5.17) is constant. With (5.18) this
implies (5.16). ∎

######  Corollary 5.6.

If $h$ is a Riemannian metric on a compact manifold $M$ of dimension $n\geq
2$, and
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker(\Delta_{h}-\wideparen{\mathscr{R}})$,
then $\omega$ satisfies the hypotheses of Lemma 5.4.

###### Proof.

Since $M$ is compact, and $\Delta_{h}-\wideparen{\mathscr{R}}=\square_{-1}$,
Lemma 3.4 shows that
$\ker(\Delta_{h}-\wideparen{\mathscr{R}})\cap\Gamma(S^{k}_{0}(T^{\ast}M))=\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}\cap\Gamma(S^{k}_{0}(T^{\ast}M))$.
∎

Let $\mathscr{T}_{ij}\in\Gamma(S^{2}T^{\ast}M)$. A pseudo-Riemannian metric
$g_{ij}$ solves the _Einstein equations with energy momentum tensor
$\mathscr{T}_{ij}$ and cosmological constant $\Lambda$_ if

(5.20)
$\displaystyle\mathscr{R}_{ij}-\tfrac{1}{2}\mathscr{R}_{g}g_{ij}=8\pi\mathscr{T}_{ij}-\Lambda
g_{ij}.$

In this case, it follows from the traced differential Bianchi identity that
$\mathscr{T}_{ij}$ is divergence free.

######  Corollary 5.7.

On an $n$-dimensional manifold $M$ a pair $(h,\omega)$ comprising a pseudo-
Riemannian metric $h_{ij}$ and a tensor
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ solves the system

(5.21)
$\displaystyle\begin{split}0&=-2(n+k-3)\mathscr{A}(\omega)+(n-2)\mathscr{D}(\omega),\\\
\mathscr{R}_{ij}&-\tfrac{1}{2}\mathscr{R}_{h}h_{ij}+\tfrac{n-2}{2n}\kappa
h_{ij}=c\left(\operatorname{\rho}(\omega\owedge\omega)_{ij}-\tfrac{1}{2}|\omega|_{h}^{2}h_{ij}\right)=c\mathscr{T}^{+}(\omega)_{ij},\end{split}$

for some real constants $\kappa$ and $c$ if and only if $h$ solves the
Einstein equations with energy momentum tensor
$\tfrac{c}{8\pi}\mathscr{T}^{+}(\omega)_{ij}$ and cosmological constant
$\Lambda=\tfrac{n-2}{2n}\kappa=\tfrac{n-2}{2n}\left(\mathscr{R}_{h}-c|\omega|^{2}_{h}\right)$.

The system (5.21) has been written so as to make readily apparent its formal
resemblance to the Einstein-Maxwell system for a metric and a two-form.

######  Remark 5.8.

In (5.21), the absolute value of $c$ can always be absorbed into $\omega$, but
its sign cannot, and the qualitative properties of the solutions of the
resulting equations depend on this sign. If $c$ has the wrong sign, the
solutions of (5.21) may not admit any possible physical interpretation.
Physical considerations focus attention on solutions of the Einstein equations
for which the energy momentum tensor satisfies some energy condition, such as
the _weak energy condition_ that $x^{i}x^{j}\mathscr{T}_{ij}\geq 0$ for all
timelike vector fields $x^{i}$, where $x^{i}$ is _timelike_ if
$|x|^{2}_{h}<0$. (Such a condition is vacuous if $h_{ij}$ is Riemannian.)
Changing the sign of $c$ destroys such a condition. However, as examples
coming from the study of submanifolds show, the equations (5.21) have
mathematical interest even without such an energy condition (see Example 6.1).

######  Example 5.9.

In the case $k=1$, for a solution $(h,\omega)$ of (5.21), the one-form
$\omega$ is $h$-harmonic. If, moreover, $\omega=d\phi$ for some $\phi\in
C^{\infty}(M)$, then $\phi$ solves the wave equation $\Delta_{h}\phi=0$, and
in Lorentzian signature can be interpreted as a massless scalar field, so the
pair $(h,d\phi)$ satisfies the Einstein scalar field equations. In the
Riemannian case, were $M$ compact, then $\phi$ would be harmonic, so constant,
and $\omega$ would vanish identically, but if $\omega$ is not required to be
exact, there can still be interesting solutions in Riemannian signature.

Thinking of $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$ as a
$S^{k-1}_{0}(T^{\ast}M)$-valued one-form, $(\omega\owedge\omega)_{ijkl}$ can
be viewed as a curvature term. Lemma 5.10 shows that a metric and
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\operatorname{\mathsf{div}}\cap\ker\mathscr{K}$
such that the modified curvature
$\mathscr{R}_{ijkl}-\tfrac{1}{4}(\omega\owedge\omega)_{ijkl}$ is projectively
flat, meaning it is a multiple of $(h\owedge h)_{ijkl}$, yield a solution of
(5.21).

######  Lemma 5.10.

Let $h_{ij}$ be a pseudo-Riemannian metric on a manifold $M$ of dimension
$n\geq 3$. Suppose
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\operatorname{\mathsf{div}}$
satisfies $\omega^{a_{1}\dots a_{k}}\mathscr{K}(\omega)_{ia_{1}\dots a_{k}}=0$
and that there is $\kappa\in C^{\infty}(M)$ such that the curvature
$\mathscr{R}_{ijkl}$ of $h_{ij}$ satisfies

(5.22)
$\displaystyle\mathscr{R}_{ijkl}-c(\omega\owedge\omega)_{ijkl}=-\tfrac{\kappa}{n(n-1)}(h\owedge
h)_{ijkl},$

for some $c\in\mathbb{R}$. Then $h$ and $\omega$ solve the equations (5.21).
In particular, $\kappa$ is a constant.

If $n=3$, then $h$ and $\omega$ solve (5.22) for $c\in\mathbb{R}$ if and only
if they solve (5.21).

###### Proof.

By (2.4), tracing (5.22) yields
$\mathscr{R}_{ij}=\tfrac{\kappa}{n}h_{ij}+c\operatorname{\rho}(\omega\owedge\omega)_{ij}$.
Tracing this yields $\mathscr{R}_{h}=\kappa+c|\omega|^{2}_{h}$, and
calculating the trace-free part of $\mathscr{R}_{ij}$ yields the last equation
of (5.21). From Lemma 5.4 it follows that $\kappa$ is a constant. If
$(h,\omega)$ solves (5.21), then, by (2.5) and (5.21),

(5.23)
$\displaystyle\begin{split}&\mathscr{R}_{ijkl}-c(\omega\owedge\omega)_{ijkl}\\\
&=\mathscr{W}_{ijkl}-c\operatorname{\mathsf{tf}}(\omega\owedge\omega)_{ijkl}-\tfrac{2}{n-2}\operatorname{\mathsf{tf}}\left(\operatorname{\rho}(\mathscr{R})-c\operatorname{\rho}(\omega\owedge\omega)\right)\owedge
h-\tfrac{1}{n(n-1)}\left(\mathscr{R}_{h}-c|\omega|^{2}_{h}\right)h\owedge h\\\
&=\mathscr{W}_{ijkl}-c\operatorname{\mathsf{tf}}(\omega\owedge\omega)_{ijkl}-\tfrac{\kappa}{n(n-1)}h\owedge
h,\end{split}$

where the conformal Weyl tensor $\mathscr{W}_{ijkl}$ of $h$ is the trace-free
part of $\mathscr{R}_{ijkl}$. If $n=3$, then there vanish $\mathscr{W}_{ijkl}$
and $\operatorname{\mathsf{tf}}(\omega\owedge\omega)_{ijkl}$, and (5.23) shows
that $h$ and $\omega$ solve (5.22). ∎

The qualitative properties of solutions of (5.22) depend strongly on the signs
of the parameters $c$ and $\kappa$.

##  6\. Examples of solutions to the coupled equations

This section records examples of solutions of the equations (5.21).

######  Example 6.1.

Lemma 6.2 yields solutions of (5.21) with $c$ having either sign.

######  Lemma 6.2.

Let $(N,g)$ be an $(n+1)$-dimensional pseudo-Riemannian space form with scalar
curvature $\mathscr{R}_{g}$ and let $i:M\to N$ be an immersion of an
$n$-dimensional hypersurface $M$ such that $h_{ij}=i^{\ast}(g)_{ij}$ is
nondegenerate. Let $\Pi_{ij}$ be the second fundamental form of the immersion
$i$ defined with respect to a unimodular transverse vector field $Z^{I}$
orthogonal to $i(M)$ and satisfying $\epsilon=|Z|^{2}_{g}\in\\{\pm 1\\}$. If
$g$ has constant curvature
$\hat{R}_{IJK}\,^{L}=-\tfrac{2\mathscr{R}_{g}}{n(n+1)}g_{K[I}\delta_{J]}\,^{L}$,
and $i(M)$ has mean curvature zero, then $(h,\Pi)$ solves (5.22) with
$c=-\epsilon$ and $\kappa=\tfrac{n-1}{n+1}\mathscr{R}_{g}$.

###### Proof.

Let $\Pi_{ij}$ and $S_{i}\,^{j}$ be the second fundamental form and shape
operator with respect to the normal field $Z^{I}$. Then $\Pi_{ij}=\epsilon
S_{i}\,^{p}h_{pj}$. Let $D$ be the Levi-Civita connection of $h_{ij}$. The
Gauss-Codazzi equations imply that $D_{[i}\Pi_{j]k}=0$ and
$\mathscr{R}_{ijkl}=\mathscr{R}_{ijk}\,^{p}h_{pl}=-\epsilon(\Pi\owedge\Pi)_{ijkl}-\tfrac{\mathscr{R}_{g}}{n(n+1)}(h\owedge
h)_{ijkl}$. If the immersion has mean curvature zero, then $\Pi_{p}\,^{p}=0$,
so $D^{p}\Pi_{ip}=D_{i}\Pi_{p}\,^{p}=0$ and
$\Pi_{ij}\in\Gamma(S^{2}_{0}(T^{\ast}M))\cap\ker\operatorname{\mathsf{div}}\cap\ker\mathscr{K}$.
∎

######  Example 6.3.

The notions described next are special cases of more general ones introduced
in [26]. Let $h_{ij}$ be a pseudo-Riemannian metric and let $\nabla$ be a
torsion-free affine connection satisfying $\nabla_{[i}h_{j]k}=0$ and
$\nabla_{i}\det h=0$. The torsion-free _conjugate_ connection $\bar{\nabla}$
defined by $\bar{\nabla}=\nabla+h^{kp}\nabla_{i}h_{jp}$ satisfies
$\bar{\nabla}_{i}h_{jk}=-\nabla_{i}h_{jk}$, so also
$\bar{\nabla}_{[i}h_{j]k}=0$ and $\bar{\nabla}_{i}\det h=0$ too. The conjugate
connection of $\bar{\nabla}$ is $\nabla$, so conjugacy is an involution on the
space of torsion-free affine connections satisfying $\nabla_{[i}h_{j]k}=0$ and
$\nabla_{i}\det h=0$. Such a connection has _self-conjugate_ curvature if its
curvature tensor equals the curvature tensor of its conjugate connection.

A cooriented $n$-dimensional hypersurface $M$ in $(n+1)$-dimensional flat
affine space equipped with the standard parallel volume form is
_nondegenerate_ if its second fundamental form is everywhere nondegenerate.
This condition does not depend on the choice of a vector field transverse to
the hypersurface. With the given coorientation, the second fundamental form
determines on $M$ a conformal structure $[h]$. A choice of transverse vector
field determines an induced torsion-free affine connection, $\nabla$, and a
pseudo-Riemannian metric, $h$, representing $[h]$ (the second fundamental form
with respect to the given transversal). The equiaffine normal is the
transverse vector field determined by the requirements that it be consistent
with the given coorientation and satisfy $\nabla_{[i}h_{j]k}=0$ and
$\nabla_{i}\det h=0$. The last condition is equivalent to the requirement that
the volume density determined by $h$ equals that determined by interior
multiplying the transversal in the ambient volume form. The connection,
$\nabla$, and metric, $h$, associated with the equiaffine normal are called
the _equiaffine connection_ and _equiaffine (or Blaschke) metric_.

The conormal map associating with $p\in M$ the annihilator of the tangent
space $T_{p}M$ takes values in the projectivization of the vector space
$\mathbb{V}$ dual to the ambient affine space. Because $M$ is nondegenerate
this conormal map is an immersion and the connection $\bar{\nabla}$ conjugate
to $\nabla$ represents the flat projective structure on the hypersurface
obtained by pulling back that on the projectivization
$\mathbb{P}(\mathbb{V})$. In particular $\bar{\nabla}$ is projectively flat
(one says that $\nabla$ is conjugate projectively flat).

Let $\tfrac{1}{2}L_{ij}\,^{k}$ be the difference tensor of the Levi-Civita
connection, $D$, of the equiaffine metric and the equiaffine connection,
$\nabla$, so that $D=\nabla+\tfrac{1}{2}L_{ij}\,^{k}$, and write
$L_{ijk}=L_{ij}\,^{p}h_{pk}$. Then $\nabla_{i}h_{jk}=L_{i(jk)}$, and from
$\nabla_{[i}h_{j]k}=0$ it follows that $L_{ijk}=L_{(ijk)}$, while from
$\nabla_{i}\det h=0$ it follows that $L_{ip}\,^{p}=0$, so that
$L_{ijk}\in\Gamma(S^{3}_{0}(T^{\ast}M))$. The tensor $L_{ijk}$ is known as the
_Fubini-Pick form_ ; the constant factor $1/2$ is conventional.

The _equiaffine mean curvature_ , $\mathscr{H}$, of $M$ is the average of the
eigenvalues of the shape operator associated with the equiaffine normal. The
hypersurface $M$ is an _affine sphere_ if the lines spanned by its equiaffine
normals are all parallel or all meet in a point (called the _center_ of the
affine sphere). Equivalently, the equiaffine shape operator is a multiple of
the identity endomorphism (in this case the multiple equals the equiaffine
mean curvature and is constant). The curvature of the equiaffine metric $h$ of
an affine sphere satisfies $\mathscr{R}_{ijkl}-\tfrac{1}{4}(L\owedge
L)_{ijkl}=-2\mathscr{H}(h\owedge h)_{ijkl}$, so $(h,L)$ is a pair satisfying
the hypotheses of Lemma 5.10. For background on affine spheres see [7, 41]
(these treat only the case of convex affine spheres, but the local
computations are the same in any signature).

A convex affine sphere $M$ cooriented to the concave side and having
equiaffine mean curvature $\mathscr{H}<0$ is said to be _hyperbolic_. By a
theorem due to S. Y. Cheng and S.-T. Yau [16, 17, 41], the interior of a
pointed proper open convex cone is foliated by hyperbolic affine spheres
having center at its vertex, asymptotic to the cone, and for which the
equiaffine metric is complete. The Cheng-Yau theorem on the existence of
hyperbolic affine spheres thus guarantees many solutions to (5.16) when $k=3$,
moreover for which $h$ is a complete Riemannian metric.

Theorem 6.4 shows that on a compact manifold a solution of a particular case
of (5.21) is equivalent to the existence of a torsion-free affine connection
that is Einstein-like in the sense that its Ricci curvature is a constant
multiple of the metric. Example 6.3 shows that these conditions are satisfied
by the equiaffine metric and affine connection induced on a convex
nondegenerate hypersurface in flat affine space (in this case the Codazzi
tensor is twice the usual cubic form). Since in this case the affine
connection has projectively flat conjugate and there hold the stronger
equations (5.22), this shows that (5.21) is a strict relaxation of (5.22) in
the sense that a solution of (5.21) need not come from a solution of (5.22).

######  Theorem 6.4.

Let $M$ be a manifold of dimension $n>2$, and let $h$ be a Riemannian metric
on $M$ having Levi-Civita connection $D$ and curvature
$\mathscr{R}_{ijk}\,^{l}$.

  1. (1)

Suppose $\omega\in\Gamma(S^{3}_{0}(T^{\ast}M))$ is Codazzi, meaning
$D_{[i}\omega_{j]kl}=0$, and $(h,\omega)$ is a solution of (5.21) for
$\kappa\in\mathbb{R}$ and $c=1$. The torsion-free affine connection
$\nabla^{\pm}=D\pm\omega_{ij}\,^{k}$ has the following properties:

    1. (a)

$\nabla^{\pm}_{[i}h_{j]k}=0$ and $\nabla_{i}\det h=0$.

    2. (b)

The curvature $R^{\pm}_{ijk}\,{}^{l}$ satisfies
$R^{+}_{ijkl}=R^{+}_{ijk}\,{}^{p}h_{pl}=\mathscr{R}_{ijkl}+2\omega_{pl[i}\omega_{j]k}\,^{p}=R^{-}_{ijk}\,{}^{p}h_{pl}=R^{-}_{ijkl}$.
In particular, $R^{+}_{ijkl}=R^{-}_{ijkl}$ has the symmetries of a metric
curvature tensor.

    3. (c)

$R^{\pm}_{ij}=R^{\pm}_{pij}\,{}^{p}=\mathscr{R}_{ij}-\omega_{ip}\,^{q}\omega_{jq}\,^{p}=\tfrac{\kappa}{n}h_{ij}$
and $R^{\pm}=h^{ij}R^{\pm}_{ij}=\mathscr{R}_{h}-|\omega|^{2}=\kappa$ is
constant.

  2. (2)

Suppose $\nabla$ is a torsion-free affine connection satisfying
$\nabla_{[i}h_{j]k}=0$ and $\nabla_{i}\det h=0$ having self-conjugate
curvature $R_{ijk}\,^{l}$ satisfying
$R_{ij}=R_{pij}\,^{p}=\tfrac{\kappa}{n}h_{ij}$ for some constant $\kappa$.
Then the difference tensor $\omega_{ij}\,^{k}=\nabla-D$ satisfies
$\omega_{ijk}=\omega_{ij}\,^{p}h_{pk}\in\Gamma(S^{3}_{0}(T^{\ast}M))$,
$D_{[i}\omega_{j]kl}=0$, and $\kappa=\mathscr{R}_{h}-|\omega|^{2}$ is
constant, and the pair $(h,\omega)$ solves (5.21) with constants $\kappa$ and
$c=1$.

###### Proof.

First suppose the conditions of (1). Claims (1a)-(1c) follow from
straightforward computations as follows. By the definition of $\nabla^{\pm}$
and the symmetry of $\omega_{ijk}$, $\nabla^{\pm}h_{ij}=\mp 2\omega_{ijk}$.
Antisymmetrizing yields $\nabla^{\pm}_{[i}h_{j]k}=0$. Because $\omega_{ijk}$
is trace-free, $\nabla^{\pm}_{i}\det h=h^{pq}\nabla^{pm}_{i}h_{pq}=\mp
2\omega_{ip}\,^{p}=0$. This proves (1a). The curvature of $\nabla^{\pm}$
satisfies

(6.1) $\displaystyle R^{\pm}_{ijk}\,{}^{l}=\mathscr{R}_{ijk}\,^{l}\pm
2D_{[i}\omega_{j]k}\,^{l}+2\omega_{p[i}\,^{l}\omega_{j]k}\,^{p}=\mathscr{R}_{ijk}\,^{l}+2\omega_{p[i}\,^{l}\omega_{j]k}\,^{p},$

the last equality because $D_{[i}\omega_{j]kl}=0$. This shows (1b) and (1c)
follows by taking traces. The constancy of $\kappa$ was assumed, and is
equivalent to the constancy of $R^{\pm}$, for, Now suppose $\nabla$ is as in
(2). Define $\omega_{ijk}=\omega_{ij}\,^{p}h_{pk}$ by
$\nabla-D=\omega_{ij}\,^{k}$. Because $\nabla$ and $D$ are torsion-free,
$\omega_{[ij]k}=0$. Then $\nabla_{i}h_{jk}=-2\omega_{i(jk)}$, so
$0=\nabla_{[i}h_{j]k}=-\omega_{[ij]k}-\omega_{k[ij]}=-\omega_{k[ij]}$, showing
that $\omega_{ijk}=\omega_{(ijk)}$ and $\nabla_{i}h_{jk}=-2\omega_{ijk}$.
Similarly, $0=\nabla_{i}\det h=h^{pq}\nabla_{i}h_{pq}=-\omega_{ip}\,^{p}$, so
$\omega_{ijk}\in\Gamma(S^{3}_{0}(T^{\ast}M))$. The connection $\bar{\nabla}$
conjugate to $\nabla$ is defined by
$\bar{\nabla}=\nabla+h^{kp}\nabla_{i}h_{jp}=D+\omega_{ij}\,^{k}-2\omega_{ij}\,^{k}=D-\omega_{ij}\,^{k}$,
so $\nabla$ and $\bar{\nabla}$ have the forms
$\nabla^{\pm}=D\pm\omega_{ij}\,^{k}$. The curvature $R^{\pm}_{ijk}\,{}^{l}$ of
$\nabla^{\pm}$ satisfies

(6.2) $\displaystyle
R^{\pm}_{ijkl}=R^{\pm}_{ijk}\,{}^{p}h_{pl}=\mathscr{R}_{ijkl}\pm
2D_{[i}\omega_{j]kl}+2\omega_{pl[i}\omega_{j]k}\,^{p},$

so that $\nabla=\nabla^{+}$ has self-conjugate curvature if and only if
$D_{[i}\omega_{j]kl}=0$. In this case $\kappa
h_{ij}=R_{ij}=R_{pij}\,^{p}=\mathscr{R}_{ij}-\omega_{ip}\,^{q}\omega_{jq}\,^{p}$
and $n\kappa=h^{ij}R_{ij}=\mathscr{R}_{h}-|\omega|^{2}$. Using the
differential Bianchi identity, $D_{[i}\omega_{j]kl}=0$, and
$\omega_{ip}\,^{p}=0$ there results

(6.3)
$\displaystyle\begin{split}nD_{i}\kappa&=D_{i}(\mathscr{R}-|\omega|^{2})=2D^{p}\mathscr{R}_{ip}-2\omega^{abc}D_{i}\omega_{abc}=2D^{p}\mathscr{R}_{ip}-2\omega^{abc}D_{a}\omega_{bci}\\\
&=2D^{p}\mathscr{R}_{ip}-2D^{a}(\omega^{abc}\omega_{bci})+2\omega_{bci}D^{a}\omega_{a}\,^{bc}=2D^{p}(\mathscr{R}_{ip}-\omega_{ia}\,^{b}\omega_{pb}\,^{a})=2D^{p}R^{\pm}_{ip}=2D_{i}\kappa.\end{split}$

Since $n>2$ this implies $\kappa=\mathscr{R}_{h}-|\omega|^{2}$ is constant. ∎

######  Example 6.5.

Let $(N,\Omega)$ be a $2n$-dimensional symplectic manifold and let $i:M\to N$
be a Lagrangian immersion. Tensors on $M$ are labeled with lowercase Latin
indices and tensors on $N$ are labeled with uppercase Latin indices. A
_(para/pseudo)-Kähler structure_ is a triple $(G,A,\Omega)$ comprising a
(pseudo-)Riemannian metric $G_{IJ}$, an integrable (para)complex structure,
and a symplectic form $\Omega_{IJ}$, which are _compatible_ in the sense that

(6.4) $\displaystyle A_{I}\,^{P}A_{P}\,^{J}=\epsilon\delta_{I}\,^{J},$
$\displaystyle\Omega_{IJ}=A_{I}\,^{P}G_{PJ},$ $\displaystyle
A_{I}\,^{P}\Omega_{PJ}=\epsilon G_{IJ},$
$\displaystyle\Omega_{IP}G^{JP}=A_{I}\,^{J},$

where, when $\epsilon=1$ the qualifier _para_ applies, and when $\epsilon=-1$
the qualifier _pseudo_ is used if $G_{IJ}$ is not Riemannian signature.

A (para/pseudo)-Kähler manifold $(N,G,A)$ has _constant (para)-holomorphic
sectional curvature $4c$_ if its Levi-Civita connection $\widehat{\nabla}$ has
curvature $\hat{R}_{IJK}\,^{L}$ of the form

(6.5)
$\displaystyle\hat{R}_{IJK}\,^{L}=2c\left(\delta_{[I}\,^{L}G_{J]K}-\epsilon
A_{[I}\,^{L}\Omega_{J]K}+\epsilon\Omega_{IJ}A_{K}\,^{L}\right).$

If $\dim N=2n$, then the Ricci and scalar curvatures are
$\hat{R}_{IJ}=2c(n+1)G_{IJ}$ and $\hat{R}=4cn(n+1)$.

An immersion $i:M\to N$ into a (para/pseudo)-Kähler manifold $(N,\Omega,A,G)$
is _nondegenerate_ if the induced tensor $h=i^{\ast}(G)$ is nondegenerate. Let
$(N,G,A,\Omega)$ be a $2n$-dimensional (para/pseudo)-Kähler manifold with
canonical (Levi-Civita) connection $\widehat{\nabla}$ and let $i:M\to N$ be a
nondegenerate Lagrangian immersion with second fundamental form $\Pi(X,Y)$
equal to the projection of $\widehat{\nabla}_{X}Ti(Y)$ onto the normal bundle
of $M$. Define
$\Pi_{ijk}=\Pi_{(ijk)}=\Pi^{\widehat{\nabla}}_{ij}\,{}^{Q}\Omega_{Qk}$ on $M$,
so $\Pi(X,Y,Z)=\Omega(\widehat{\nabla}_{X}Ti(Y),Ti(Z))$. The tensor $\Pi$ is
symmetric because $\widehat{\nabla}$ is torsion-free and the immersion is
Lagrangian. Let $D$ be the Levi-Civita connection of the metric
$h_{ij}=i^{\ast}(G)_{ij}$ on $M$. Tensors defined on $M$ are raised and
lowered using $h_{ij}$ and $h^{ij}$. Let $\hat{R}_{IJK}\,^{L}$ be the
curvature of $\widehat{\nabla}$ and let $\mathscr{R}_{ijk}\,^{l}$ be the
curvature of $D$. Define $\Pi_{ij}\,^{k}=h^{kp}\Pi_{ijp}=\Pi_{ij}\,^{k}$. Note
that $\Pi_{ij}\,^{k}$ is not the second fundamental form as such, as its upper
index is twisted by the (para)-complex structure. Let
$P\in\Gamma(\operatorname{End}(i^{\ast}(TN)))$ be projection onto $Ti(TM)$
along $ATi(TM)$. It is claimed that, for $X,Y\in\Gamma(TM)$ there hold

(6.6) $\displaystyle\begin{aligned} &Ti(D_{X}Y)=P\widehat{\nabla}_{X}Ti(Y),\\\
&\widehat{\nabla}_{X}Ti(Y)=Ti(D_{X}Y)+\epsilon
ATi(\Pi^{\sharp}(X,Y)),&&\widehat{\nabla}_{X}ATi(Y)=ATi(D_{X}Y)+Ti(\Pi^{\sharp}(X,Y)).\end{aligned}$

Because $i$ is nondegenerate $Ti(TM)$ and $ATi(TM)$ are transverse, so $P$ is
well-defined. By definition of the second fundamental form there is induced on
$M$ a torsion-free connection $D$ such that
$\widehat{\nabla}_{X}Ti(Y)=Ti(D_{X}Y)+\Pi(X,Y)$ where $\Pi(X,Y)$ is the
projection of $\widehat{\nabla}_{X}Ti(Y)$ onto $ATi(TM)$ along $Ti(TM)$. In
particular, $Ti(D_{X}Y)=P\widehat{\nabla}_{X}Ti(Y)$ by definition of $D$. Then

(6.7)
$\displaystyle\begin{split}(D_{X}h)(Y,X)&=XG(Ti(Y),Ti(Z))-G(Ti(D_{X}Y),Ti(Z))-G(Ti(Y),Ti(D_{X}Z))\\\
&=(\widehat{\nabla}_{X}G)(Ti(Y),Ti(Z))+G(\pi(X,Y),Ti(Z))+G(Ti(Y),\pi(X,Z))=0,\end{split}$

shows that $D$ is the Levi-Civita connection of $h$. By the definition of
$\Pi^{\sharp}$ there holds

(6.8)
$\displaystyle\begin{split}\Omega(\pi(X,Y),Ti(Z))&=\Omega(\widehat{\nabla}_{X}Ti(Y),Ti(Z))=\Pi(X,Y,Z)=h(\Pi^{\sharp}(X,Y),Z)\\\
&=G(Ti(\Pi^{\sharp}(X,Y)),Ti(Z))=\epsilon\Omega(ATi(\Pi^{\sharp}(X,Y)),Ti(Z)),\end{split}$

for all $X,Y,Z\in\Gamma(TM)$, and by the nondegeneracy of $\Omega$ this shows
$\pi(X,Y)=\epsilon ATi(\Pi^{\sharp}(X,Y))$, so that
$\widehat{\nabla}_{X}Ti(Y)=Ti(D_{X}Y)+\epsilon ATi(\Pi^{\sharp}(X,Y))$. The
identity $\widehat{\nabla}_{X}ATi(Y)=ATi(D_{X}Y)+Ti(\Pi^{\sharp}(X,Y))$
follows because $A$ is $\widehat{\nabla}$-parallel.

The Lagrangian immersion has _mean curvature zero_ if $\Pi_{ip}\,^{p}=0$.

######  Lemma 6.6.

Let $i:M\to N$ be a nondegenerate mean curvature zero Lagrangian immersion of
the $n$-dimensional manifold $M$ in the $2n$-dimensional (para/pseudo-)Kähler
manifold $(N,G,\Omega,A)$ with constant (para)-holomorphic curvature
$4\hat{c}$. Let $h_{ij}=i^{\ast}(G)_{ij}$ be the induced metric. Then
$(h,\Pi)$ solves (5.22) with $c=\epsilon$ and $\kappa=\hat{c}n(n-1)$.

###### Proof.

From (6.6) it follows that there are tensors $\mathscr{T}_{ijk}\,^{l}$ and
$\mathscr{N}_{ijk}\,^{l}$ on $M$ having the algebraic symmetries of a
curvature tensor of metric type and such that

(6.9) $\displaystyle\hat{R}(X,Y)Ti(Z)=Ti(\mathscr{T}(X,Y)Z)+\epsilon
ATi(\mathscr{N}(X,Y)Z)$

for all $X,Y,Z\in\Gamma(TM)$. Straightforward computations using (6.6) show
that

(6.10)
$\displaystyle\mathscr{T}_{ijk}\,^{l}=\mathscr{R}_{ijk}\,^{l}-2\epsilon(\Pi\owedge\Pi)_{ijkl},$
$\displaystyle\mathscr{N}_{ijk}\,^{l}=2D_{[i}\Pi_{j]k}\,^{l}=2\mathscr{K}(\Pi).$

Let $\mathscr{T}_{ij}=\mathscr{T}_{pij}\,^{p}$,
$\mathscr{N}_{ij}=\mathscr{N}_{pij}\,^{p}$,
$\mathscr{T}=\mathscr{T}_{p}\,^{p}$, and $\mathscr{N}_{p}\,^{p}=0$. Tracing
(6.10) yields

(6.11)
$\displaystyle\mathscr{T}_{ij}=\mathscr{R}_{ij}-\epsilon\operatorname{\rho}(L\owedge
L)_{ij},$ $\displaystyle\mathscr{T}=\mathscr{R}-\epsilon|L|^{2}_{h},$
$\displaystyle\mathscr{N}_{ij}=\operatorname{\mathsf{div}}(\Pi)_{ij}.$

Let $\hat{R}_{IJKL}=\hat{R}_{IJK}\,^{P}G_{PL}$. Suppressing notation
indicating the differential of $i$, it follows from (6.9) that
$\mathscr{T}_{ijkl}=\hat{R}_{ijkl}$ and
$\mathscr{N}_{ijkl}=A_{i}\,^{P}\hat{R}_{Pjkl}$. If $G$ has constant
(para)-holomorphic sectional curvature, it follows from (6.5) that
$\hat{R}_{IJKL}=2\hat{c}(G_{L[I}G_{J]K}+\epsilon\Omega_{L[I}\Omega_{J]K}+\epsilon\Omega_{IJ}\Omega_{KL})$
and
$A_{I}\,^{P}\hat{R}_{PJKL}=2\hat{c}(\Omega_{L[I}G_{J]K}+G_{K[I}\Omega_{J]L}+2G_{IJ}\Omega_{KL})$.
Pulling these back to $M$ via $i$ yields
$\mathscr{T}_{ijkl}=\hat{R}_{ijkl}=2ch_{l[i}h_{j]k}$ and
$\mathscr{N}_{ijkl}=A_{i}\,^{P}\hat{R}_{Pjkl}=0$. The remaining identities in

(6.12) $\displaystyle\mathscr{N}_{ijk}\,^{l}=0,$
$\displaystyle\mathscr{T}_{ijk}\,^{l}=2\hat{c}\delta_{[i}\,^{l}h_{j]k},$
$\displaystyle\mathscr{T}_{ij}=\hat{c}(n-1)h_{ij},$
$\displaystyle\mathscr{T}=cn(n-1).$

are obtained by taking traces. Substituting (6.12) in (6.10), and (6.11) shows
that $\Pi_{ijk}\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$ and with
the curvature $\mathscr{R}_{ijk}\,^{l}$ of the Levi-Civita connection $D$ of
$h_{ij}$ satisfies

(6.13) $\displaystyle\mathscr{R}_{ijkl}$ $\displaystyle=-\hat{c}(h\owedge
h)_{ijkl}+\epsilon(\Pi\owedge\Pi)_{ijkl},$

and so also
$\mathscr{R}_{ij}=\hat{c}(n-1)h_{ij}+\epsilon\operatorname{\rho}(\Pi\owedge\Pi)_{ij}$
and $\mathscr{R}=\hat{c}n(n-1)+\epsilon|\Pi|^{2}_{h}$. ∎

There seems to be no general existence result already known for the case
$k>3$.

######  Example 6.7.

Let $G$ be a connected compact simple Lie group of dimension greater than $3$
with Lie algebra $\mathfrak{g}$, let $h_{ij}=-B_{ij}$ be the bi-invariant
metric determined by the negative of the Killing form of $\mathfrak{g}$, and
note that $h_{ij}$ is Einstein with Ricci curvature
$\mathscr{R}_{ij}=\tfrac{1}{4}h_{ij}$.

Let $\omega_{i_{1}\dots i_{k}}\in S^{k}(\mathfrak{g}^{\ast})$ be the complete
polarization of a homogeneous degree $k$ polynomial $P$ invariant under the
adjoint action of $G$ on $\mathfrak{g}$. More precisely, suppose that $P$ is
one of a set of homogeneous generators of the ring of invariant polynomials on
$\mathfrak{g}$ and that $k\geq 3$. If $\mathfrak{g}$ has rank $l$ then the
ring of invariant polynomials on $\mathfrak{g}$ is generated by $l$
algebraically independent homogeneous elements (see e.g. [3, section
VIII.$8$]). The degrees $2=u_{1}<\dots<u_{l}$ of the homogeneous generators
are given in terms of the exponents $m_{1}<\dots<m_{l}$ of the Weyl group $W$
of $\mathfrak{g}$ by $u_{i}=m_{i}+1$, and satisfy $u_{1}\cdot\dots\cdot
u_{l}=|W|$ and $2(u_{1}+\dots+u_{l})=\dim\mathfrak{g}+l$. Since $G$ acts on
$\mathfrak{g}$ orthogonally, the $h$-Laplacian is invariant under the
$G$-action, so $\Delta_{h}P$ is again an invariant polynomial. Let $E$ be the
invariant polynomial corresponding to $h$. The harmonic part $Q$ of $P$ is
obtained by subtracting from $P$ a linear combination of terms of the form
$E^{s}\Delta_{h}^{s}P$ ($s>0$). Since each of these terms is $G$-invariant, so
is $Q$. Since the homogeneous generators of the ring of invariant polynomials
are algebraically independent, it cannot be that $P$ is a linear combination
of powers of $E$, so $Q$ is not null. It follows that it may be supposed from
the beginning that $P$ is $\Delta_{h}$-harmonic, or, equivalently, that
$\omega_{i_{1}\dots i_{k}}$ is trace-free. In particular, this shows that on
$\mathfrak{g}$ there exists a $\Delta_{h}$-harmonic homogeneous $G$-invariant
polynomial of degree at least $3$.

######  Lemma 6.8.

Let $G$ be a connected compact simple Lie group of dimension greater than $3$
with Lie algebra $\mathfrak{g}$, let $h_{ij}=-B_{ij}$ be the bi-invariant
metric on $G$ determined by the negative of the Killing form $B_{ij}$ of
$\mathfrak{g}$, and let $D$ be its Levi-Civita connection. Suppose $k\geq 3$
and let $\omega_{i_{1}\dots i_{k}}\in S^{k}(\mathfrak{g}^{\ast})$ be the
complete polarization of a $\Delta_{h}$-harmonic homogeneous $G$-invariant
polyonomial $P$ of degree $k$. The pair $(h,\omega)$ solves the equations
(5.16) on $G$.

###### Proof.

The invariance of $P$ means that $0=kc_{i(i_{1}}\,^{p}\omega_{i_{2}\dots
i_{k})p}=-2D_{i}\omega_{i_{1}\dots i_{k}}$, so that $\omega$ is parallel, and
so in the kernel of $\square_{-1}$ by (3.29). Let
$\sigma_{ij}=\omega_{ii_{1}\dots i_{k-1}}\omega_{j}\,^{i_{1}\dots i_{k-1}}$.
Then

(6.14)
$\displaystyle\begin{split}c_{ij}\,^{p}\sigma_{pk}&=-c_{ji}\,^{p}\omega_{pi_{1}\dots
i_{k-1}}\omega_{k}\,^{i_{1}\dots
i_{k-1}}=(k-1)c_{j(i_{1}}\,^{p}\omega_{i_{2}\dots
i_{k-1})pi}\omega_{k}\,^{i_{1}\dots i_{k-1}}\\\
&=(k-1)c_{j(p}\,^{i_{1}}\omega_{i_{2}\dots
i_{k-1})ki_{1}}\omega_{i}\,^{pi_{2}\dots
i_{k-1}}=-c_{jk}\,^{i_{1}}\omega_{i_{1}pi_{2}\dots
i_{k-1}}\omega_{i}\,^{pi_{2}\dots
i_{k-1}}=-c_{jk}\,^{p}\sigma_{ip},\end{split}$

showing that $\sigma_{ij}$ is an invariant bilinear form. By the simplicity of
$\mathfrak{g}$ there is a constant $c$ such that $\sigma_{ij}=ch_{ij}$, and
tracing this equality shows that $c=|\omega|^{2}_{h}/\dim\mathfrak{g}$. It
follows that $(h,\omega)$ solves the equations (5.16). ∎

######  Example 6.9.

When the metric $h_{ij}$ is flat, the equations (5.16) admit purely algebraic
solutions. In this example $h_{ij}$ is a flat Riemannian metric on an
$n$-dimensional vector space $\mathbb{V}$, although many of the claims make
sense in other signatures. Let $x^{i}$ denote the radial Euler vector field.
Let $E(x)=|x|^{2}_{h}$. Let $\Delta=\Delta_{h}$. If $F(x)$ is a function on
$\mathbb{V}$, let $F_{i_{1}\dots i_{k}}=D_{i_{1}}\dots D_{i_{k}}F$. Sometimes
it is convenient to write $D^{(k)}F$ for the $k$-fold covariant derivative of
$F$ (for example $D^{(2)}F$ is the Hessian of $F$). If $F$ is a polynomial
homogeneous of degree $g$ then $x^{i_{1}}\dots x^{i_{j}}F_{i_{1}\dots
i_{j}}=g(g-1)\dots(g-j+1)F$. In this case $F_{i_{1}\dots i_{g}}$ is a constant
tensor, so parallel.

######  Lemma 6.10.

Let $F$ be an $h$-harmonic polynomial homogeneous of degree $g\geq 2$ on the
$n$-dimensional Euclidean vector space $(\mathbb{V},h)$. Let
$\omega_{i_{1}\dots i_{g}}=F_{i_{1}\dots i_{g}}$. The pair $(h,\omega)$ solves
the equations (5.16) (with $\gamma=0$) if and only if there is
$c\in\mathbb{R}$ such that $P$ solves any one of the following equivalent
equations:

(6.15) $\displaystyle 0$ $\displaystyle=F_{ip_{1}\dots
p_{g-1}}F_{j}\,^{p_{1}\dots p_{g-1}}-ch_{ij},$ (6.16) $\displaystyle 0$
$\displaystyle=D_{i}D_{j}\left(|D^{(g-1)}F|^{2}-cE\right),$ (6.17)
$\displaystyle 0$ $\displaystyle=|D^{(g-1)}F|^{2}-cE.$

In this case $c=|D^{(g)}F|^{2}$.

###### Proof.

The equivalence of (6.15) and (6.16) follows from the vanishing of
$F_{i_{1}\dots i_{g+1}}$. Because $|D^{(g-1)}F|^{2}-cE$ is a homogeneous
quadratic polynomial, its Hessian vanishes if and only if it vanishes
identically, so the equations (6.15) and (6.16) are equivalent to (6.17). ∎

######  Example 6.11.

A hypersurface in a Riemannian space form is _isoparametric_ if its principal
curvatures are constant. The question of classifying isoparametric
hypersurfaces was posed and partially solved by E. Cartan in [11, 12, 10]. See
[19], [52], and [59] and for background. In [45, 44] it is shown that for an
isoparametric hypersurface in a constant curvature $(n-1)$-dimensional sphere
$\mathbb{S}^{n-1}=\\{x\in\mathbb{V}:E(x)=1\\}$:

  * •

the number $g$ of distinct principal curvatures satisfies $g\in\\{1,2,3,6\\}$;

  * •

if the distinct principal curvatures are ordered
$\lambda_{1}>\lambda_{2}>\dots>\lambda_{g}$, the multiplicities, $m_{i}$, of
the $\lambda_{i}$ satisfy $m_{i}=m_{i+2}$ (indices modulo $6$), so that there
are at most two distinct multiplicities $m_{1}$ and $m_{2}$ (moreover, if
$g<4$ then $m_{1}=m_{2}$ always); and

  * •

every such hypersurface arises as a level set of the restriction to the sphere
of a polynomial $P:\mathbb{V}\to\mathbb{R}$ homogeneous of degree $g$ and
satisfying the equations

(6.18) $\displaystyle|dP|^{2}=g^{2}E^{g-1},$ $\displaystyle\Delta
P=\tfrac{m_{2}-m_{1}}{2}g^{2}E^{\tfrac{g}{2}-1}.$

A polynomial $P$ solving (6.18) is called a _Cartan-Münzner polynomial_.
Examples of solutions for which the resulting hypersurfaces are not
extrinsically homogeneous are known when $g=4$; see for example [25].

######  Theorem 6.12.

Let $P$ be a Cartan-Münzner polynomial homogeneous of degree $g\geq 2$ and
having multiplicities $m_{1}$ and $m_{2}$ on the $n$-dimensional Euclidean
vector space $(\mathbb{V},h)$. Then the trace-free part $\omega_{i_{1}\dots
i_{g}}$ of $P_{i_{1}\dots i_{g}}$ solves (6.15), so the pair $(h,\omega)$
solves the equations (5.16) (with $\gamma=0$).

###### Proof.

It suffices to show that $P$ solves an equation of the form (6.17). The
following identity is needed. For any polynomial $F$ homogeneous of degree $g$
there holds

(6.19)
$\displaystyle\begin{split}\Delta(E^{i}F)&=2i(n+2(g+i-1))E^{i-1}F+E^{i}\Delta
F.\end{split}$

In particular, the special case $f=1$ yields $\Delta
E^{i}=2i(n+2(i-1))E^{i-1}$.

In the case $m_{1}=m_{2}$, the polynomial $P$ is harmonic and so
$\omega_{i_{1}\dots i_{g}}=P_{i_{1}\dots i_{g}}$. In this case applying
$\Delta^{g-2}$ to the first equation of (6.18) and simplifying the result
using (6.19) yields

(6.20) $\displaystyle
2^{g-2}g(g!)(n+2(g-2))\dots(n+2)E=\Delta^{g-2}(g^{2}E^{g-1})=\Delta^{g-2}|dP|^{2}=2^{g-2}|D^{(g-1)}P|^{2}.$

Differentiating this yields

(6.21) $\displaystyle
D_{i}D_{j}|D^{(g-1)}P|^{2}=2g(g!)(n+2(g-2))\dots(n+2)h_{ij},$

which suffices to show that $P$ solves (6.17). The argument in the general
case is similar, but more involved. Since $m_{1}=m_{2}$, if $g<4$, it can be
supposed that $g\in\\{4,6\\}$. In particular, $g$ is even. Let
$P=\sum_{i=0}^{g/2}E^{i}Q^{(g-2i)}$ be the Lefschetz decomposition of $P$ into
its harmonic components. Here $Q^{(g-2i)}$ is a harmonic polynomial
homogeneous of degree $g-2i$, and the decomposition is uniquely determined.
Applying $\Delta$ to both sides and using (6.19) yields

(6.22) $\displaystyle\tfrac{m_{2}-m_{1}}{2}g^{2}E^{\tfrac{g}{2}-1}=\Delta
P=\sum_{i=1}^{g/2}2i(n+2(g-i-1))E^{i-1}Q^{(g-2i)}.$

By the uniqueness of the Lefschetz decomposition, this implies $Q^{(g-2i)}=0$
if $0<i<g/2$. Hence

(6.23) $\displaystyle P=Q+\tfrac{(m_{2}-m_{1})g}{2(n+g-2)}E^{g/2}$

where $Q$ is a harmonic polynomial homogeneous of degree $g$. Note that the
desired tensor $\omega_{i_{1}\dots i_{g}}$ equals $Q_{i_{1}\dots,i_{g}}$.
Calculating the differential of (6.23) using the homogeneity of $Q$ yields

(6.24) $\displaystyle
g^{2}E^{g-1}=|dP|^{2}=|dQ|^{2}+\tfrac{(m_{2}-m_{1})g^{3}}{(n+g-2)}E^{g/2-1}Q+\tfrac{(m_{2}-m_{1})^{2}g^{2}}{(n+g-2)^{2}}E^{g-1},$

so that

(6.25) $\displaystyle
0=|dQ|^{2}+g^{3}\tfrac{(m_{2}-m_{1})}{(n+g-2)}E^{g/2-1}Q+g^{2}\left(\tfrac{(m_{2}-m_{1})^{2}}{(n+g-2)^{2}}-1\right)E^{g-1}.$

Applying $\Delta^{g-2}$ to both sides of (6.25) and simplifying using (6.19)
yields

(6.26) $\displaystyle
0=2^{g-2}\left(|D^{(g-1)}Q|^{2}+\left(\tfrac{(m_{2}-m_{1})^{2}}{(n+g-2)^{2}}-1\right)g(g!)(n+2(g-2))(n+2(g-3))\dots(n+2)E\right).$

Hence

(6.27) $\displaystyle Q_{ip_{1}\dots p_{g-1}}Q_{j}\,^{p_{1}\dots
p_{g-1}}=\left(1-\tfrac{(m_{2}-m_{1})^{2}}{(n+g-2)^{2}}\right)g(g!)(n+2(g-2))(n+2(g-3))\dots(n+2)h_{ij}.$

Because $\omega_{i_{1}\dots i_{g}}$ is parallel, this suffices to prove the
claim. ∎

######  Example 6.13.

Let $E=\\{e_{1},\dots,e_{n}\\}$ be an enumeration of the edge set of a finite
$k$-regular graph with vertex set $V$. The partial Steiner system
$\mathcal{B}$ determined by the incidence of edges in the given graph is the
collection of $k$-element subsets (blocks) of $\bar{n}=\\{1,\dots,n\\}$ such
that $I=\\{i_{1},\dots,i_{k}\\}\in\mathcal{B}$ if and only if the edges
$e_{i_{1}},\dots,e_{i_{k}}$ are incident at some vertex in $V$. Let
$\mathbb{V}$ be the $n$-dimensional real vector space generated by $E$ and
equip $\mathbb{V}$ with the flat Riemannian metric $h$ with respect to which
$E$ is an ordered orthonormal basis. Let $x_{1},\dots,x_{n}$ be the
coordinates of $x\in\mathbb{V}$ with respect to the ordered basis $E$. The
quadratic form $Q(x)$ determined by $h$ is $Q(x)=\sum_{i=1}^{n}x_{i}^{2}$.
Associate with $I\in\mathcal{B}$ the monomial $x_{I}=x_{i_{1}}\dots
x_{i_{k}}$. Let $\epsilon\in\\{\pm 1\\}^{\mathcal{B}}$, so that
$\epsilon_{I}\in\\{\pm 1\\}$ for each $I\in\mathcal{B}$.

######  Lemma 6.14.

Let $\mathcal{B}$ be the partial Steiner system determined by the incidence of
edges in a finite $k$-regular graph. For the homogeneity $k$ polynomial
$P(x)=\sum_{I\in\mathcal{B}}\epsilon_{I}x_{I}$ associated with a $k$-regular
graph and a choice of signs $\epsilon\in\\{\pm 1\\}^{\mathcal{B}}$, the pair
$(h,D^{(k)}P)$ solves (5.16).

###### Proof.

The component $\tfrac{\partial^{k-1}P}{\partial x_{i_{1}}\dots\partial
x_{i_{k-1}}}$ is nonzero if and only if $\\{i_{1},\dots,i_{k-1}\\}$ is
contained in some block $I=\\{i_{1},\dots,i_{k}\\}$ in $\mathcal{B}$ (because
any $k-1$ edges meet at most one vertex there is at most one such block). In
this case $\tfrac{\partial^{k-1}P}{\partial x_{i_{1}}\dots\partial
x_{i_{k-1}}}=\epsilon_{I}x_{i_{k}}$. Since there are $(k-1)!$ orderings of the
distinct indices $i_{1},\dots,i_{k-1}$ and since the index of a given edge
appears in exactly two blocks there results $|D^{(k-1)}P|^{2}_{h}=2(k-1)!Q$.
Because no variable $x_{i}$ appears in any monomial of $P$ with a power higher
than one, $P$ is $h$-harmonic. By Lemma 6.10 this shows that the pair
$(h,D^{(k)}P)$ solves (5.16). ∎

##  7\. Refined Kato inequalities for trace-free symmetric tensors

The inequality (7.1) of Lemma 7.1 generalizes the estimate for the second
fundamental form of a minimal hypersurface proved in [49]. With a $1$ in place
of $\tfrac{n+k-2}{n+2(k-1)}$, it follows from the Cauchy-Schwarz inequality,
and is known as a _Kato inequality_. The estimates (7.1)-(7.3) are _refined
Kato inequalities_ in the sense of [8] and [6], and can be deduced from the
results in either of those papers. In particular the results of [8, Section
$6$] include Lemma 7.1, and the discussion at the very end of [8, Section $6$]
gives the explicit constants of (7.1)-(7.3) for the cases $k=1,2$. (The $k=2$
case of (7.1) had earlier been stated in [4, section $4$].) To keep the
exposition self-contained, here there is given a direct proof of (7.1)-(7.3)
following the general procedure described in the introduction of [6], and not
utilizing general representation theoretic machinery.

######  Lemma 7.1.

Let $M$ be an $n$-dimensional manifold with a Riemannian metric $h$ having
Levi-Civita connection $D$, and let $\phi_{i_{1}\dots
i_{k}}\in\Gamma(S^{k}_{0}(T^{\ast}M))$, where $k\geq 1$. If
$\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$
(that is $D_{[i}\phi_{j]i_{1}\dots i_{k-1}}=0$) then where $|\phi|^{2}\neq 0$
there holds

(7.1) $\displaystyle|d|\phi||^{2}\leq\tfrac{n+k-2}{n+2(k-1)}|D\phi|^{2}.$

If
$\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{L}\cap\ker\operatorname{\mathsf{div}}$
(that is $D_{(i_{1}}\phi_{i_{2}\dots i_{k+1})}=0$) then where $|\phi|^{2}\neq
0$ there holds

(7.2) $\displaystyle|d|\phi||^{2}\leq\tfrac{k}{k+1}|D\phi|^{2}.$

If $\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{L}\cap\ker\mathscr{K}$
(that is $D\phi$ is pure trace) then where $|\phi|^{2}\neq 0$ there holds

(7.3) $\displaystyle|d|\phi||^{2}\leq\tfrac{k}{n+2(k-1)}|D\phi|^{2}.$

###### Proof.

Write $\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)$ for the symbol of $\mathscr{L}$
applied to the vector $Z^{i}$ and $\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))$, and
similarly for $\mathscr{K}$ and $\operatorname{\mathsf{div}}$. Write
$(i(Z)\phi)_{i_{1}\dots i_{k-1}}=Z^{p}\phi_{pi_{1}\dots i_{k-1}}$ and suppose
$Z^{i}$ has unit norm. When $k=1$ and $n=2$ the coefficient of the pure trace
terms in (3.19) should be understood in a limiting sense. The nonnegativity of
$|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)|^{2}$ in (3.19) yields

(7.4) $\displaystyle\tfrac{n+2(k-2)}{n+k-3}|i(Z)\phi|^{2}\leq|\phi|^{2}.$

Together (3.18) and (7.4) give

(7.5)
$\displaystyle|\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)|^{2}\leq\tfrac{1}{k+1}|\phi|^{2}+\tfrac{k(n+k-3)}{(k+1)(n+2(k-1))}|\phi|^{2}=\tfrac{n+k-2}{n+2(k-1)}|\phi|^{2}.$

Contracting (3.12) with $\mathsf{\sigma}_{D}(Z)(\omega)$ and using the Cauchy-
Schwarz inequality shows

(7.6)
$\displaystyle\begin{split}&\tfrac{1}{2}Z^{i}d_{i}|\phi|^{2}=Z^{i}\phi^{i_{1}\dots
i_{k}}D_{i}\phi_{i_{1}\dots
i_{k}}=\langle\mathsf{\sigma}_{D}(Z)(\phi),D\phi\rangle\\\
&=\langle\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi),\mathscr{L}(\phi)\rangle+\tfrac{2k}{k+1}\langle\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi),\mathscr{K}(\phi)\rangle+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}\langle
i(Z)\phi,\operatorname{\mathsf{div}}(\phi)\rangle,\\\
&\leq|\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)||\mathscr{L}(\phi)|+\tfrac{2k}{k+1}|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)||\mathscr{K}(\phi)|+\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}|i(Z)\phi||\operatorname{\mathsf{div}}(\phi)|.\end{split}$

Suppose
$\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$.
By (3.13), $|D\phi|^{2}=|\mathscr{L}(\phi)|^{2}$. Substituting this and (7.5)
into (7.6) gives

(7.7)
$\displaystyle\begin{split}|\phi|^{2}|Z^{i}d_{i}|\phi||^{2}=\tfrac{1}{4}|Z^{i}d_{i}|\phi|^{2}|^{2}&\leq|\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)|^{2}|\mathscr{L}(\phi)|^{2}=|\mathsf{\sigma}_{\mathscr{L}}(Z)(\phi)|^{2}|D\phi|^{2}\leq\tfrac{n+k-2}{n+2(k-1)}|\phi|^{2}|D\phi|^{2}.\end{split}$

This holds for all unit norm $Z^{i}$, so shows (7.1).

Suppose
$\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{L}\cap\ker\operatorname{\mathsf{div}}$.
By (3.13), $|D\phi|^{2}=\tfrac{2k}{k+1}|\mathscr{K}(\phi)|^{2}$, and, by
(3.19), $2|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)|^{2}\leq|\phi|^{2}$. In
(7.6) these give

(7.8)
$\displaystyle\begin{split}|\phi|^{2}|Z^{i}d_{i}|\phi||^{2}=\tfrac{1}{4}|Z^{i}d_{i}|\phi|^{2}|^{2}&\leq(\tfrac{2k}{k+1})^{2}|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)|^{2}|\mathscr{K}(\phi)|^{2}\\\
&=\tfrac{2k}{k+1}|\mathsf{\sigma}_{\mathscr{K}}(Z)(\phi)|^{2}|D\phi|^{2}\leq\tfrac{k}{n+1}|\phi|^{2}|D\phi|^{2}.\end{split}$

This holds for all unit norm $Z^{i}$, so shows (7.2).

Suppose
$\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{L}\cap\ker\mathscr{K}$.
By (3.13),
$|D\phi|^{2}=\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}|\operatorname{\mathsf{div}}(\phi)|^{2}$.
With (7.4) in (7.6) this gives

(7.9)
$\displaystyle\begin{split}|\phi|^{2}|Z^{i}d_{i}|\phi||^{2}&=\tfrac{1}{4}|Z^{i}d_{i}|\phi|^{2}|^{2}\leq\left(\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}\right)^{2}|i(Z)\phi|^{2}|\operatorname{\mathsf{div}}(\phi)|^{2}\\\
&=\tfrac{k(n+2(k-2))}{(n+k-3)(n+2(k-1))}|i(Z)\phi|^{2}|D\phi|^{2}\leq\tfrac{k}{n+2(k-1)}|\phi|^{2}|D\phi|^{2}.\end{split}$

This holds for all unit norm $Z^{i}$, so shows (7.3). ∎

######  Remark 7.2.

When $n=2$ the inequalities (7.1), (7.2), and (7.3) are in fact equalities;
see section $3$ of [27]. Since here attention is focused on the case $n>2$,
further discussion is omitted.

######  Remark 7.3.

The proof of Lemma 7.1 shows that if $\phi\in\Gamma(S^{k}_{0}(T^{\ast}M))$
then the largest eigenvalue $\mu$ of the symmetric two-tensor
$\operatorname{\rho}(\phi\owedge\phi)_{ij}$ satisfies
$\mu\leq\tfrac{n+k-3}{n+2(k-2)}|\phi|^{2}$. By (7.4) for any vector field
$X^{i}$ there holds
$X^{i}X^{j}\operatorname{\rho}(\phi\owedge\phi)_{ij}=|i(X)\operatorname{\rho}(\phi)|^{2}\leq\tfrac{n+k-3}{n+2(k-2)}|\phi|^{2}|X|^{2}$,
which suffices to show the claim. This means
$\operatorname{\rho}(\phi\owedge\phi)_{ij}\leq\tfrac{n+k-3}{n+2(k-2)}|\phi|^{2}h_{ij}$.

######  Lemma 7.4.

Let $h$ be a Riemannian metric on a manifold $M$ of dimension $n>2$ and let
$\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$. Wherever $\omega\neq 0$ there hold

(7.10)
$\displaystyle|\omega|^{(n+2(k-1))/(n+k-2)}\Delta_{h}|\omega|^{(n-2)/(n+k-2)}\geq\tfrac{n-2}{n+k-2}\mathscr{Q}_{\mathscr{R}}(\omega),$
$\displaystyle\text{if}\,\,\omega\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}},$
(7.11)
$\displaystyle|\omega|^{(k+1)/k}\Delta_{h}|\omega|^{(k-1)/k}\geq-\tfrac{2(k-1)}{k+1}\langle\omega,\mathscr{K}^{\ast}\mathscr{K}(\omega)\rangle,$
$\displaystyle\text{if}\,\,\omega\in\ker\mathscr{L}\cap\ker\operatorname{\mathsf{div}},$
(7.12)
$\displaystyle|\omega|^{(n+2(k-1))/k}\Delta_{h}|\omega|^{(2-n)/k}\leq\tfrac{n-2}{n+k-2}\mathscr{Q}_{\mathscr{R}}(\omega),$
$\displaystyle\text{if}\,\,\omega\in\ker\mathscr{K}\cap\ker\mathscr{L}.$

###### Proof.

Let $\omega\in\Gamma(S^{k}_{0}(T^{\ast}M))$. Wherever $|\omega|>0$ there holds

(7.13)
$\displaystyle\tfrac{1}{2\lambda}|\omega|^{2(1-\lambda)}\Delta_{h}|\omega|^{2\lambda}=\tfrac{1}{2}\Delta_{h}|\omega|^{2}+2(\lambda-1)|d|\omega||^{2}.$

Combining (7.13) with (7.1), (3.28), (3.33), (3.34), and Lemma 7.1 yields
(7.10)-(7.12). ∎

######  Remark 7.5.

If $h$ is flat and $f\in C^{\infty}(M)$ is harmonic, then $\omega_{i_{1}\dots
i_{k}}=D_{i_{1}}\dots
D_{i_{k}}f\in\Gamma(S^{k}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$.
By Lemma 7.4 the function $|\omega|^{p}$ is subharmonic for all
$p\geq(n-2)/(n+k-2)$. For the flat Euclidean connection on $\mathbb{R}^{n}$
and $k=1$ this is [54, Theorem $A$], and for $k>1$ it is [9, Theorem $1$]. In
the opposite direction, [55, Theorem $2(b)$] shows that on flat Euclidean
space the best $p$ for which $|\omega|^{p}$ is subharmonic is $(n-2)/(n+k-2)$,
and [55, Theorem $2(a)$] shows that on flat Euclidean space, given any section
$\omega$ of $S^{k}_{0}(T^{\ast}M)$ there is around every point a neighborhood
$U$ and a harmonic function $f\in C^{\infty}(U)$ such that on $U$ there holds
$\omega_{i_{1}\dots i_{k}}=D_{i_{1}}\dots D_{i_{k}}f$.

If $f\in C^{\infty}(M)$ is harmonic then
$df\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$, and Lemma 7.4 shows
that

(7.14)
$\displaystyle(n-1)|df|^{n/(n-1)}\Delta_{h}|df|^{(n-2)/(n-1)}\geq(n-2)\operatorname{\rho}(\mathscr{R})(df,df).$

If $h$ has nonnegative Ricci curvature it follows that $|df|^{p}$ is
subharmonic for any $p\geq(n-2)/(n-1)$.

######  Remark 7.6.

Suppose
$\omega_{ij}\in\Gamma(S^{2}_{0}(T^{\ast}M))\cap\ker\mathscr{K}\cap\operatorname{\mathsf{div}}$.
If the sectional curvature is nonnegative then $\mathscr{Q}_{\mathscr{R}}$ is
nonnegative on $S^{2}_{0}(T^{\ast}M)$ so $|\omega|^{(n-2)/n}$ is subharmonic
by (7.10); if $M$ is compact this means $|\omega|$ is constant, and by (3.34)
this implies $\omega$ is parallel. Moreover, if the sectional curvature is
positive at some point, then $\mathscr{Q}_{\mathscr{R}}$ is positive on
$S^{2}_{0}(T^{\ast}M)$ at that point and $\omega$ must be identically zero.
This recovers Theorem 7.7.

######  Theorem 7.7 (M. Berger - D. Ebin, [1]).

On a compact Riemannian manifold with nonnegative sectional curvature, a
$b_{ij}\in\Gamma(S^{2}(T^{\ast}M))$ such that $D_{[i}b_{j]k}=0$ and
$D_{i}b_{p}\,^{p}=0$ is parallel; if the sectional curvature is somewhere
positive, then $b_{ij}$ is a constant multiple of the metric.

######  Corollary 7.8.

The following are equivalent for a Riemannian manifold $(M,h)$ of dimension
$n\geq 3$.

  1. (1)

The curvature $\mathscr{R}_{ijkl}$ of $h$ is harmonic, meaning
$D_{p}\mathscr{R}_{ijk}\,^{p}=0$.

  2. (2)

The trace-free part $\operatorname{\mathsf{tf}}(\mathscr{Ric})_{ij}$ of its
Ricci tensor is in $\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$, so is
a Codazzi tensor.

  3. (3)

Its Ricci tensor is a Codazzi tensor and its scalar curvature is constant.

If $n\geq 4$, then these conditions are equivalent to

  1. (4)

The Weyl curvature $\mathscr{W}_{ijkl}$ of $h$ is harmonic, meaning
$D_{p}\mathscr{W}_{ijk}\,^{p}=0$.

If $h$ satisfies any of these equivalent conditions and $M$ is compact, then
that $h$ have nonnegative sectional curvature which is somewhere positive
implies $h$ is Einstein.

###### Proof.

For any Riemannian metric,
$\operatorname{\mathsf{div}}(\operatorname{\mathsf{tf}}(\mathscr{Ric}))_{i}=\tfrac{n-2}{2n}D_{i}\mathscr{R}$
and

(7.15) $\displaystyle
2\mathscr{K}(\operatorname{\mathsf{tf}}(\mathscr{Ric}))_{ijk}=D_{p}\mathscr{R}_{ijk}\,^{p}+\tfrac{2}{n(n-1)}h_{k[i}D_{j]}\mathscr{R}=2D_{i}\mathscr{Ric}_{j]k}+\tfrac{2}{n(n-1)}h_{k[i}D_{j]}\mathscr{R}.$

It follows that, if $n>2$, then
$\operatorname{\mathsf{tf}}(\mathscr{Ric})_{ij}\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$
if and only if $D_{p}\mathscr{R}_{ijk}\,^{p}=0$. In this case
$D_{i}\mathscr{R}=0$, so
$\mathscr{Ric}\in\ker\mathscr{K}\cap\ker\operatorname{\mathsf{div}}$ and
$\mathscr{R}$ is constant. That (3) implies (2) is immediate. When $n>3$, the
identity

(7.16) $\displaystyle
D_{p}\mathscr{W}_{ijk}\,^{p}=\tfrac{2(n-3)}{n-2}\left(\mathscr{K}(\operatorname{\mathsf{tf}}(\mathscr{Ric}))_{ijk}+\tfrac{n-2}{2n(n-1)}h_{k[i}D_{j]}\mathscr{R}\right),$

implies the equivalence of (3) and (4). Applying Theorem 7.7 to
$\operatorname{\mathsf{tf}}(\mathscr{R})_{ij}$ yields the final claim. ∎

The equivalencies of Corollary 7.8 are essentially [21, Lemma $1$]. The final
claim (4) might be new.

##  8\. Constraints on solutions

This section obtains a priori constraints on the growth of solutions of
(5.22). This requires bounds on complicated tensorial equations. The bounds
presented are sharp for $k\leq 3$ but could be improved for $k>3$. It would
also be interesting to extend, even partially, such constraints to solutions
of (5.21).

######  Lemma 8.1.

Let $(\mathbb{V},h)$ be a Euclidean vector space of dimension $n\geq 2$. For
$k\geq 2$ and $\omega\in S^{k}_{0}\mathbb{V}^{\ast}$ there hold

(8.1)
$\displaystyle\tfrac{n+k-3}{n+2(k-2)}|\omega|^{4}_{h}\geq|\operatorname{\rho}(\omega\owedge\omega)|^{2}_{h}\geq\tfrac{1}{n}|\omega|^{4}_{h},$
(8.2)
$\displaystyle\tfrac{(n-2)(n+k-2)}{n(n+2(k-2))}|\omega|^{4}_{h}\geq|\operatorname{\mathsf{tf}}\operatorname{\rho}(\omega\owedge\omega)|^{2}_{h},$
(8.3) $\displaystyle
4|\omega|_{h}^{4}\geq|\omega\owedge\omega|^{2}_{h}\geq\tfrac{2}{n(n-1)}|\omega|_{h}^{4},$
(8.4)
$\displaystyle\left(4-\tfrac{2}{n(n-1)}\right)|\omega|_{h}^{4}\geq|\operatorname{\mathsf{tf}}(\omega\owedge\omega)|^{2}_{h}+\tfrac{4}{n-2}|\operatorname{\mathsf{tf}}(\operatorname{\rho}(\omega\owedge\omega))|_{h}^{2}\geq|\operatorname{\mathsf{tf}}(\omega\owedge\omega)|^{2}_{h}.$

###### Proof.

For $x_{i}\in\mathbb{V}^{\ast}$, the nonnegativity of the squared norm of

(8.5) $\displaystyle x_{[i}\omega_{j]i_{1}\dots
i_{k-1}}-\tfrac{1}{n+k-3}\sum_{s=1}^{k-1}h_{i_{s}[i}\omega_{j]i_{1}\dots\hat{i}_{s}\dots
i_{k-1}p}x^{p},$

(where $\hat{i}_{s}$ denotes the omission of the index $i_{s}$) yields

(8.6) $\displaystyle\begin{split}0&\leq 2x^{i}\omega^{ji_{1}\dots
i_{k-1}}\left(x_{[i}\omega_{j]i_{1}\dots
i_{k-1}}-\tfrac{1}{n+k-3}\sum_{s=1}^{k-1}h_{i_{s}[i}\omega_{j]i_{1}\dots\hat{i}_{s}\dots
i_{k-1}p}x^{p}\right)\\\
&=|x|^{2}_{h}|\omega|^{2}_{h}-|\iota(x)\omega|^{2}_{h}-\tfrac{k-1}{n+k-3}|\iota(x)\omega|^{2}_{h}=|x|^{2}_{h}|\omega|^{2}_{h}-\tfrac{n+2(k-2)}{n+k-3}|\iota(x)\omega|^{2}_{h},\end{split}$

so that

(8.7)
$\displaystyle\operatorname{\rho}(\omega\owedge\omega)_{ij}x^{i}x^{j}=|\iota(x)\omega|^{2}_{h}\leq\tfrac{n+k-3}{n+2(k-2)}|x|^{2}_{h}|\omega|^{2}_{h}.$

This shows
$\tfrac{n+k-3}{n+2(k-2)}|x|^{2}_{h}|\omega|^{2}_{h}h_{ij}-\operatorname{\rho}(\omega\owedge\omega)_{ij}$
is positive semidefinite. Because
$\operatorname{\rho}(\omega\owedge\omega)_{ij}$ is also positive semidefinite
and the endomorphisms
$\tfrac{n+k-3}{n+2(k-2)}|\omega|^{2}_{h}\delta_{i}\,^{j}-\operatorname{\rho}(\omega\owedge\omega)_{i}\,^{j}$
and $\operatorname{\rho}(\omega\owedge\omega)_{i}\,^{j}$ commute, contracting
with $\operatorname{\rho}(\omega\owedge\omega)^{ij}$ yields

(8.8)
$\displaystyle\tfrac{n+k-3}{n+2(k-2)}|\omega|^{4}_{h}\geq|\operatorname{\rho}(\omega\owedge\omega)|^{2}_{h}=|\operatorname{\mathsf{tf}}\operatorname{\rho}(\omega\owedge\omega)|^{2}_{h}+\tfrac{1}{n}|\omega|^{4}_{h},$

from which the inequalities (8.1) and (8.2) follow.

From (2.6) there follows

(8.9)
$\displaystyle\begin{split}|\omega\owedge\omega|^{2}_{h}&=|\operatorname{\mathsf{tf}}(\omega\owedge\omega)|^{2}_{h}+\tfrac{4}{n-2}|\operatorname{\mathsf{tf}}(\operatorname{\rho}(\omega\owedge\omega))|_{h}^{2}+\tfrac{2}{n(n-1)}|\omega|^{4}_{h}.\end{split}$

Using the nonnegativity of
$|\operatorname{\mathsf{tf}}(\omega\owedge\omega)|^{2}_{h}$ and
$|\operatorname{\mathsf{tf}}(\operatorname{\rho}(\omega\owedge\omega))|_{h}^{2}$
yields the second inequality in (8.3). There remains to show the leftmost
inequality of (8.3). For $\tau_{ij}\in S^{2}_{0}\mathbb{V}^{\ast}$,

(8.10) $\displaystyle\begin{split}0&\leq(\tau_{ij}\omega_{klq_{1}\dots
q_{k-2}}-\tau_{kl}\omega_{ijq_{1}\dots q_{k-2}})(\tau^{ij}\omega^{klq_{1}\dots
q_{k-2}}-\tau^{kl}\omega^{ijq_{1}\dots
q_{k-2}})=|\tau|^{2}_{h}|\omega|^{2}_{h}-|\iota(\tau)\omega|^{2}_{h},\end{split}$