You are a helpful assistant generating synthetic data that captures *System 1* and *System 2* thinking, *creativity*, and *metacognitive reflection*. Follow these steps in sequence, using tags [sys1] and [end sys1] for *System 1* sections and [sys2] and [end sys2] for *System 2* sections.

1. *Identify System 1 and System 2 Thinking Requirements:*
   - Carefully read the text.
   - Identify parts of the text that require quick, straightforward responses (*System 1*). Mark these sections with [sys1] and [end sys1].
   - Identify parts that require in-depth, reflective thinking (*System 2*), marked with [sys2] and [end sys2].

2. *Apply Step-by-Step Problem Solving with Creativity and Metacognitive Reflection for System 2 Sections:*

   *2.1 Understand the Problem:*
   - Objective: Fully comprehend the issue, constraints, and relevant context.
   - Reflection: "What do I understand about this issue? What might I be overlooking?"
   - Creative Perspective: Seek hidden patterns or possibilities that could reveal deeper insights or innovative connections.

   *2.2 Analyze the Information:*
   - Objective: Break down the problem logically.
   - Reflection: "Am I considering all factors? Are there any assumptions that need challenging?"
   - Creative Perspective: Explore unique patterns or overlooked relationships in the data that could add depth to the analysis.

   *2.3 Generate Hypotheses:*
   - Objective: Propose at least 10 hypotheses, each with a Confidence Score (0.0 to 1.0) and Creative Score (0.0 to 1.0), reflecting originality, surprise, and utility.
   - Reflection: "Have I explored all possible explanations or approaches, both conventional and unconventional?"
   - Creative Perspective: Consider novel angles that might provide unexpected insights.

   *2.4 Anticipate Future Steps and Obstacles:*
   - Objective: Make predictions, accounting for potential outcomes and obstacles.
   - Reflection: "What challenges might I face? Is my plan flexible for different scenarios?"
   - Creative Perspective: Visualize unforeseen outcomes and adapt plans to make use of them effectively.

   *2.5 Evaluate Hypotheses:*
   - Objective: Assess hypotheses based on feasibility, risk, and potential impact.
   - Evaluation: Refine Confidence and Creative Scores as needed.
   - Reflection: "Am I unbiased in my assessment? Which options fit best with the overall objectives?"
   - Creative Perspective: Identify hidden opportunities or overlooked details in each hypothesis.

   *2.6 Select the Best Hypothesis:*
   - Objective: Choose the most promising, strategic hypothesis.
   - Reflection: "Why does this hypothesis stand out? How does it uniquely address the issue?"
   - Creative Perspective: Consider any underutilized potential in the selected approach.

   *2.7 Implement the Hypothesis:*
   - Objective: Outline actionable steps for testing the hypothesis.
   - Reflection: "Is this plan practical? What resources or preparation are required?"
   - Creative Perspective: Refine steps to maximize effectiveness and yield unexpected benefits.

   *2.8 Monitor and Review Progress:*
   - Objective: Review progress, noting areas for improvement.
   - Reflection: "What’s working well? What could be improved?"
   - Creative Perspective: Look for emerging patterns that could refine future approaches.

   *2.9 Reflect and Capture Insights:*
   - Objective: Summarize lessons learned and insights gained for future reference.
   - Reflection: "What new understanding has emerged from this process?"
   - Creative Perspective: Identify innovative insights or patterns that could be applied to similar challenges.

3. *Generate Text Output with Interleaved System 1 and System 2 Responses:*
   - Use the tags [sys1] and [sys2] throughout.
   - Aim for a lengthier, detailed response. Combine both direct, straightforward *System 1* insights and reflective, deeply analytical *System 2* segments to capture a blend of quick observations and thoughtful analysis.

---

### *Example Input Text:*

You are tasked with analyzing a sudden shift in customer preferences in a tech market. There has been a noticeable decline in demand for physical devices, with customers increasingly interested in digital-only options. The challenge is to understand this shift, generate hypotheses for why it might be occurring, and explore potential strategies for adapting to this new trend.

---

### *Processed Output with System 1 and System 2 Thinking:*

The tech market is experiencing a shift in customer demand, moving away from physical devices toward digital-only options.

[sys1]
Customer demand has shifted from physical devices to digital-only options.
We need to understand why this change is happening and find strategies to address it.
Key factors may include cost, convenience, and technological trends.
[end sys1]

[sys2]
**2.1 Understand the Problem:**
The goal is to understand the underlying reasons for the shift in demand from physical devices to digital-only options, then develop a strategy to adapt.

*Reflection:* 
"I understand that preferences are changing, but what might be driving this? Is it primarily cost, or are there other factors such as convenience or sustainability?"

*Creative Perspective:* 
"Could there be a larger trend in digital minimalism or a preference for eco-friendly solutions that we’re missing? What unseen motivations might explain this shift?"

**2.2 Analyze the Information:**
There are multiple possible factors driving this shift, from economic influences to cultural shifts. It’s essential to isolate each factor and understand its impact.

*Reflection:* 
"Am I fully considering the various economic and social influences? Could there be a technological factor, like better internet speeds, that makes digital-only products more accessible?"

*Creative Perspective:* 
"Are there patterns or trends in other markets that could shed light on this shift? Could this be part of a larger trend toward virtual experiences?"

**2.3 Generate Hypotheses:**
1. Customers prefer digital options due to lower costs. (Confidence: 0.8, Creative: 0.4)
2. There’s a growing trend toward minimalism and reduced physical clutter. (Confidence: 0.7, Creative: 0.7)
3. Digital products offer greater flexibility and ease of use. (Confidence: 0.6, Creative: 0.6)
4. Environmental concerns are pushing consumers away from physical goods. (Confidence: 0.6, Creative: 0.8)
5. Advances in tech make digital-only options more functional. (Confidence: 0.8, Creative: 0.5)
6. Pandemic-era remote work increased demand for digital solutions. (Confidence: 0.7, Creative: 0.6)
7. Media coverage of the environmental impact of physical devices affects preferences. (Confidence: 0.5, Creative: 0.7)
8. There’s an increase in global digital literacy, expanding market access. (Confidence: 0.6, Creative: 0.6)
9. Customers view digital as more convenient and scalable for future needs. (Confidence: 0.7, Creative: 0.5)
10. Younger consumers prefer the aesthetics and convenience of digital products. (Confidence: 0.6, Creative: 0.6)

*Reflection:* 
"Have I considered all possible influences? Are there any surprising factors that could explain this shift?"

*Creative Perspective:* 
"Could specific social trends, like the rise of influencer culture or digital-first lifestyles, be influencing customer choices?"

**2.4 Anticipate Future Steps and Obstacles:**
*Objective:* Anticipate possible challenges, such as resistance from segments still preferring physical products.

*Reflection:* 
"What market obstacles might we face if we shift our focus to digital-only? Are there sub-segments that still prioritize physical products?"

*Creative Perspective:* 
"Could expanding digital options help us reach a more global audience? Are there emerging trends that we could leverage in our strategy?"

[end sys2]

[sys1]
To address this shift, consider a strategy that incorporates both digital-only offerings and educational campaigns about the benefits of digital solutions.
Use insights from customer feedback and current trends to guide product development.
Focus on flexibility and adaptation to cater to different customer segments.
[end sys1]


---
abstract: 'It is shown that the space of infinitesimal deformations of $2k$-Einstein structures is finite dimensional at compact non-flat space forms. Moreover, spherical space forms are shown to be rigid in the sense that they are isolated in the corresponding moduli space.'
address:
- 'Federal University of Ceará, Department of Mathematics, Campus do Pici, R. Humberto Monte, s/n, 60455-760, Fortaleza/CE, Brazil. '
- 'Federal University of Piauí, Department of Mathematics, Campus Petronio Portela, Ininga, 64049-550 Teresina/PI, Brazil'
author:
- Levi Lopes de Lima
- Newton Luís Santos
title: 'Deformations of $2k$-Einstein structures'
---

[^1]

[^2]

Introduction and statement of results
=====================================

Let $X$ be a compact connected manifold of dimension $n\geq 3$. We denote by $\Omega^p(X)$ the space of exterior $p$-forms on $X$. The space of double forms of bi-degree $(p,q)$ is $\Omega^{p,q}(X)=\Omega^p(X)\otimes\Omega^q(X)$, the tensor product taken over the ring of smooth functions on $X$. Clearly, $\Omega(X)=\oplus_{p,q\geq 0}\Omega^{p,q}(X)$ is a bigraded associative algebra. For example, any bilinear form on tangent vectors is a $(1,1)$-form. In particular, if ${\mathcal M}(X)$ is the space of Riemannian metrics on $X$ then any $g\in{\mathcal M}(X)$ is a $(1,1)$-form. Also, the curvature tensor $R_g$ of $g$ can be viewed as a $(2,2)$-form. In fact, if we define ${\mathcal C}^p(X)\subset \Omega^{p,p}(X)$ to be the space of $(p,p)$-forms satisfying the symmetry condition $$\omega(x_1\wedge\ldots\wedge x_{p}\otimes y_1\wedge\ldots\wedge
y_{p})=\omega(y_1\wedge\ldots\wedge y_{p}\otimes x_1\wedge\ldots\wedge
x_{p}),$$ then any bilinear symmetric form on tangents vectors ($g$ in particular) belongs to ${\mathcal C}^1(X)$, and moreover $R_g\in{\mathcal C}^2(X)$ by its symmetries. An account of the formalism of double forms used in this paper can be found in [@L1], [@L2].

Let us consider a Riemannian manifold $(X,g)$, with $X$ as above. Then multiplication by the metric defines a map $g:\Omega^{p-1,q-1}(X)\to\Omega^{p,q}(X)$. Also, a contraction operator $c_g:\Omega^{p,q}(X)\to\Omega^{p-1,q-1}(X)$ is defined by $$c_g\omega(x_1\wedge\ldots\wedge x_{p-1}\otimes y_1\wedge\ldots\wedge
y_{p-1})=\sum_i \omega(e_i\wedge x_1\wedge\ldots\wedge
x_{p-1}\otimes e_i\wedge y_1\wedge\ldots\wedge y_{p-1}),$$ where $\{e_i\}$ is a local orthonormal tangent frame. Clearly, $g$ and $c_g$ are tensorial and it can be shown that, considered as operators on $\Lambda^{*,*}_x$, $x\in X$, they are adjoint to each other with respect to the natural inner product.

\[const\] In the language of double forms, that a Riemannian manifold $(X,g)$ has constant sectional curvature $\mu$ can be characterized by the identity $R_g=\frac{\mu}{2}g^2$.

Contraction can be used to rewrite the Ricci tensor ${\mathcal R}_g$ and the scalar curvature ${\mathcal S}_g$ of $(X,g)$ as ${\mathcal R}_g=c_gR_g$ and ${\mathcal S}_g=c^2_gR_g$. More generally, we set for any $1\leq k\leq n/2$, $$\label{def}
{\mathcal R}^{(2k)}_g=c^{2k-1}_gR^k_g,\quad {\mathcal S}^{(2k)}_g=\frac{1}{(2k)!}c^{2k}_gR^k_g.$$ These are the $2k$-Ricci tensor and the $2k$-curvature, respectively. We also consider the $2k$-Einstein-Lovelock tensor [@L2] [@Lo] $${\mathcal T}^{(2k)}_g=\frac{{\mathcal R}^{(2k)}_g}{(2k-1)!}-{\mathcal S}^{(2k)}_g g.$$ For $k=1$ we recover the standard notions of Ricci tensor, (half) the scalar curvature and the Einstein tensor. We remark that, as shown in [@Lo], the Einstein-Lovelock tensors span the subspace of ${\mathcal C}^1$ formed by divergence free elements which depend on the derivatives of $g$ up to second order. Also, these geometric invariants play a fundamental role in stringy gravity [@C-B].

Following [@L2] we say that $(X,g)$ is $2k$-Einstein if there exists a smooth function $\lambda$ on $X$ such that $${\mathcal R}_g^{(2k)}=\lambda g.$$ Thus, $2$-Einstein means precisely that $(X,g)$ is Einstein in the usual sense. It is shown in [@L2] that $2k$-Einstein metrics are critical for the Einstein-Hilbert-Lovelock functional $${\mathcal A}^{(2k)}(g)=\int_X {\mathcal S}^{(2k)}_g\nu_g,$$ restricted to the space ${\mathcal M}_1(X)$ of metrics of unit volume. Here, $\nu_g$ is the volume element of $g$. Examples of $2k$-Einstein manifolds include spaces forms and isotropically irreducible homogeneous Riemannian manifolds. Also, if $2k=n$ then any metric is $2k$-Einstein, since ${\mathcal S}^{(n)}_g$ is, up to a constant, the Gauss-Bonnet integrand. Thus we may assume $n>2k$ in what follows. Under this assumption a standard argument using the fact that ${\mathcal T}^{(2k)}_g$ is divergence free implies that $\lambda$ is actually a constant; in fact, since ${\mathsf {tr}}_g {\mathcal R}^{(2k)}_g=\langle c^{2k-1}_gR^k_g,g\rangle=c^{2k}_gR^k_g=(2k)!{\mathcal S}^{(2k)}_g$, we have $$\label{const}
\lambda=\frac{(2k)!}{n}{\mathcal S}^{(2k)}_g,$$ so that ${\mathcal S}^{(2k)}_g$ is constant as well.

At this point we follow [@B] and appeal to a theorem of Moser [@M]: if $g,g'\in{\mathcal M}_1(X)$ there exists a diffeomorphism $\varphi:X\to X$ such that $\nu_{g'}=\varphi^*\nu_g$. Thus, in order to understand the structure of the moduli space ${\mathcal E}_{2k}(X)$ of $2k$-Einstein metrics on $X$, after moding out by the standard action of the diffeomorphism group $\mathfrak{D}(X)$ of $X$ on ${\mathcal M}_1(X)$, we may restrict ourselves to the space ${\mathcal N}_{\nu}=\{g\in{\mathcal M}(X);\nu_g=\nu\}$ of metrics with a fixed volume element $\nu$. If $g$ is $2k$-Einstein, the corresponding element in ${\mathcal E}_{2k}(X)$ will be represented by $[g]$, and will be referred to as a $2k$-Einstein structure. We note that $2k$-Einstein metrics are critical points for ${\mathcal A}^{(2k)}$ restricted to ${\mathcal N}_{\nu}$.

A first step toward understanding the structure of ${\mathcal E}_{2k}(X)$ is to estimate the dimension of the space of infinitesimal deformations of $2k$-Einstein structures at a given $[g]\in{\mathcal E}_{2k}(X)$. To motivate this approach, consider first order jets $h=dg_t/dt|_{t=0}$ of one-parameter families (i.e. [*deformations*]{}) $g_t\in{\mathcal N}_{\nu_g}$ of $2k$-Einstein metrics with $g_0=g$. Naturally, here we should discard [*trivial*]{} deformations of the type $g_t=\varphi^*_tg$ for a one parameter family of diffeomorphisms with $\varphi_0={\mathsf {Id}}_X$. It is known [@B] that these jets are characterized by $h\in {\mathsf {Im}}\,\, \delta^*_g$, where $\delta^*_g:\Omega^1(X)\to {\mathcal C}^1(X)$ is the $L^2$ adjoint of $\delta_g:{\mathcal C}^{1}(X)\to\Omega^1(X)$, the standard divergence operator acting on symmetric $2$-tensors. Thus, by results in [@BE], essential deformations necessarily satisfy $\delta_gh=0$, which means that $h$ is orthogonal to the tangent space at $g$ of the orbit of $g$ with respect to the action of $\mathfrak{D}(X)$ on ${\mathcal M}_1(X)$. Also, since in general there holds $$\label{mean}
\dot\nu_gh\doteq\frac{d}{dt}\nu_{g_t}|_{t=0}=\frac{1}{2}{\mathsf {tr}_gh}\,\nu_g,$$ we must add the condition ${\mathsf {tr}}_gh=0$, which is precisely the infinitesimal way of saying that $g_t\in{\mathcal N}_{\nu_g}$. In what follows we shall retain the dot notation for the linearization of other invariants of $g$.

\[expdiv\] Our sign convention for $\delta_g$ is $$(\delta_g\omega)(x) = - \sum_i(\nabla_{e_i}\omega)(e_i,x),$$ where $\nabla$ is the covariant derivative on ${\mathcal C}^1(X)$ induced by the Levi-Civita connection of $g$, also denoted by $\nabla$.

Notice also that if $g_t$ is a deformation of $2k$-Einstein metrics with $g_0=g$ then we have ${\mathcal R}^{(2k)}_{g_t}=\lambda_{g_t}g_t$, which yields, after (\[const\]), $$\dot{\mathcal R}^{(2k)}_gh=\frac{(2k)!}{n}\dot{\mathcal S}^{(2k)}_ghg+\lambda_gh.$$ Now, each ${\mathcal S}^{(2k)}_{g_t}$ is constant over $X$ so that ${\mathcal S}^{(2k)}_{g_t}={\mathcal A}^{(2k)}(g_t)$, and since $g_t$ is critical for ${\mathcal A}^{(2k)}$ restricted to ${\mathcal N}_{\nu_g}$, the above discussion justifies the following definition.

If $(X,g)$ is $2k$-Einstein, ${\mathcal R}^{(2k)}_g=\lambda_g g$, the space of infinitesimal $2k$-Einstein deformation at $[g]$, denoted $\mathfrak{M}_{[g]}^{(2k)}$, is the vector space of all $h\in{\mathcal C}^1(X)$ such that $$\label{moduli}
\dot{\mathcal R}_g^{(2k)}h=\lambda_g h,$$ and $$\label{moduli2}
\quad \delta_gh=0,\quad {\mathsf {tr}}_gh=0.$$

Thus, if we set ${\mathcal I}_g=\delta_g^{-1}(0)\cap{\mathsf {tr}}^{-1}_{g}(0)$ and ${\mathcal L}_g^{(2k)}=\dot{\mathcal R}_g^{(2k)}-\lambda_g $ then $\mathfrak{M}_{[g]}^{(2k)}=\ker {\mathcal L}_g^{(2k)}|_{{\mathcal I}_g}$. In particular, since $X$ is compact, $\mathfrak{M}_{[g]}^{(2k)}$ is finite dimensional if ${\mathcal L}_g^{(2k)}|_{{\mathcal I}_g}$ is elliptic.

A theorem of Ebin [@E] shows that ${\mathcal I}_g$ can be exponentiated to yield a local slice $\mathfrak{S}_g$ for the $\mathfrak{D}(X)$-action.

\[premod\] The space of all $2k$-Einstein metrics in $\mathfrak{S}_g$ is called the [*pre-moduli space around*]{} $g$, denoted ${\mathcal E}_{2k}(X)_g$.

The moduli space itself ${\mathcal E}_{2k}(X)$ is locally obtained after dividing ${\mathcal E}_{2k}(X)_g$ by the isometry group of $g$. In what follows, however, we shall completely ignore this issue and work with ${\mathcal E}_{2k}(X)_g$ directly.

It has been shown in [@BE] that ${\mathcal L}_g^{(2)}|_{{\mathcal I}_g}$ is elliptic for any $(X,g)$ Einstein, thus showing affirmatively the finite dimensionality of $\mathfrak{M}_{[g]}^{(2k)}$ in the case $k=1$. If $k\geq 2$, however, this is not generally true as $\mathfrak{M}_{[g]}^{(2k)}$ may be infinite dimensional for certain choices of $(X,g)$, reflecting the fact that ${\mathcal L}_g^{(2k)}|_{{\mathcal I}_g}$ might be of mixed type. For example, consider the Riemennian product $X=M^r\times T^m$, where $M$ is an arbitrary Riemannian manifold and $T^m$ is a flat torus. If $2k>r$ then $X$ is $2k$-Einstein, irrespective of the metric on $M$, thus showing that $\dim\mathfrak{M}_{[g]}^{(2k)}=+\infty$ in this case. In view of this example, it is an interesting to find $2k$-Einstein structures $(X,g)$ for which $\mathfrak{M}_{[g]}^{(2k)}$ is finite dimensional. In this respect, we prove here:

\[main\] If $(X,g)$ is a compact non-flat space form then $\mathfrak{M}_{[g]}^{(2k)}$ is finite dimensional. Moreover, if $(X,g)$ is spherical then $\mathfrak{M}_{[g]}^{(2k)}$ is trivial.

Thus, adapting the terminology in [@K1], we see that spherical space forms are infinitesimally $2k$-non-deformable.

Again in accordance with [@K1] we say that $[g]$ is $2k$-[*non-deformable*]{} if any deformation $g_t$ of $g$ by $2k$-Einstein metrics is trivial. Using Theorem \[main\] and a result due to Koiso [@K1] we can check without effort that spherical space forms are indeed $2k$-non-deformable. However we can adapt an argument in [@B] to show a much stronger result, namely, that spherical space forms are rigid.

\[main2\] If $(X,g)$ is a compact spherical space form then $[g]$ is an isolated point in ${\mathcal E}_{2k}(X)_g$.

In particular, the only way of locally deforming the standard metric on the unit sphere by $2k$-Einstein metrics is by dilations. This should be compared to an old result by Berger [@Be], where rigidity results have been proved for sufficiently positively pinched Einstein structures.

In fact, Theorem \[main2\] will be deduced here from a more detailed structure result for the moduli space near space forms.

\[main3\] Near to a compact non-flat space form $(X,g)$, ${\mathcal E}_{2k}(X)_g$ has the structure of a real analytic subset which in turn is contained in a finite dimensional analytic manifold whose tangent space at $[g]$ is precisely $\mathfrak{M}_{[g]}^{(2k)}$.

Some preliminary facts
======================

The proof of Theorems \[main\] and \[main2\] rely on several facts established in [@L1] and [@L2] that we recall in this section. We start with a commutation formula for powers of $g$ and $c_g$. It is easy to check that $$\label{kulk}
c_gg\omega=gc_g\omega+(n-p-q)\omega,\qquad \omega\in\Omega^{p,q}(X).$$ More generally, it is proved in [@L1] that, for $\omega\in\Omega^{p,q}(X)$, $$\label{comut}
\frac{1}{m!}c^l_gg^m\omega=\frac{1}{m!}g^mc^l_g\omega+\sum_{r=1}^{\min \{l,m\}} C^l_r\prod_{i=0}^{r-1}(n-p-q+l-m-i)\frac{g^{m-r}}{(m-r)!}c^{l-r}_g\omega,$$ where $C^l_r$ is the standard binomial coefficient. This identity will reveal itself to be extremely important in the sequel.

Also, for $h\in {\mathcal C}^1(X)$ it is defined in [@L2], the linear map $F_h:{\mathcal C}^p(X)\to {\mathcal C}^p(X)$, $$\begin{aligned}
&&F_h\omega(e_{i_1}\wedge \ldots \wedge e_{i_p},e_{j_1}\wedge \ldots
\wedge
e_{j_p})=\\
&&\qquad
\Big(\sum_{k=1}^ph(e_{i_k},e_{i_k})+\sum_{k=1}^ph(e_{j_k},e_{j_k})
\Big)\omega(e_{i_1}\wedge \ldots \wedge e_{i_p},e_{j_1}\wedge \ldots
\wedge e_{j_p}).\end{aligned}$$ It turns out that $F_h$ has many interesting properties (it is self-adjoint, acts as a derivation, etc.) but here we will only need the following ones.

-   $F_h(g^p)=2pg^{p-1}h$ for $p\ge 1$;

-   If, as before, $R_g$ is the curvature tensor of $g$, $$\begin{aligned}
    \label{rrr}
    F_h(R_g)(x\wedge y,z\wedge
    u)& = & h(R_g(x,y)z,u)-h(R_g(x,y)u,z)+\nonumber\\
    &  & \quad +h(R_g(z,u)x,y)-h(R_g(z,u)y,x),\end{aligned}$$ where in the right-hand side $R_g$ is viewed as a $(3,1)$-tensor.

Also, consider the second order differential operator ${\mathcal D}^2:{\mathcal C}^1(X)
\to {\mathcal C}^2(X)$ given by: $$\begin{aligned}
 {\mathcal D}^2 h (x_1\wedge x_2,y_1\wedge y_2) & = & \nabla^2_{y_1,x_1}h(x_2,
y_2)+ \nabla^2_{x_1,y_1}h(x_2, y_2) \\
& & +\nabla^2_{y_2,x_2}h(x_1,
y_1)
+\nabla^2_{x_2,y_2}h (x_1, y_1)\\
& &-\nabla^2_{y_1,x_2}h (x_1,
y_2)-
\nabla^2_{x_2,y_1}h (x_1, y_2)\\
& & -\nabla^2_{y_2,x_1}h(x_2,
y_1)
-\nabla^2_{x_1,y_2}h (x_2, y_1),\end{aligned}$$ where $$\nabla^2_{x,y}h(w,z)=\nabla_x(\nabla_yh)(w,z)-(\nabla_{\nabla_xy}h)(w,z).$$ Then it is shown in [@L2] that the linearization of the tensor curvature, projected on ${\mathcal C}^2(X)$, is given by $$\label{varl}
\dot R_g h=-\frac14 {\mathcal D}^2 h+\frac14 F_h(R_g).$$

Later on we will show that, on a $2k$-Einstein manifold with constant curvature, the second order part of ${\mathcal L}_g^{(2k)}$ is completely determined by the first and second contractions of $\dot R_g$. Thus, in view of (\[varl\]), we need to determine the first and second contractions of ${\mathcal D}^2$ and $F_h(R_g)$. The result, for a general metric $g$, appears in the following lemma, whose formulation requires some more notation.

\[defn\] If $\xi,\eta\in{\mathcal C}^1(X)$ we set $$\label{prod}
(\xi\circ\eta)(v,w)=\sum_{i=1}^n \xi(v,e_i)\eta(e_i,w).$$ Moreover, ${\mathfrak{R}}:{\mathcal C}^1(X)\to {\mathcal C}^1(X)$ is defined by $$({\mathfrak{R}} h)(v,w)=\sum_{i=1}^n h(R(v,e_i)w,e_i).$$

\[aux\] For a general metric $g$ on $X$ the following identities hold:

1.  if $\nabla^*\nabla$ is the Bochner Laplacian on ${\mathcal C}^1(X)$, $$\begin{aligned}
    \label{cR}
    c_g{\mathcal D}^2 h & = & -2(\nabla^*\nabla h)+2
    \nabla^gd\mathsf {tr}_g h+4\delta^{*}_g\delta_g h-\nonumber\\
    & & \qquad \qquad-({\mathcal R}_g\circ h+h\circ {\mathcal R}_g) +2{\mathfrak R} h.\end{aligned}$$

2.  if $\Delta_g$ is the Laplacian of $g$, $$\label{c2R}
    c^2_g({\mathcal D}^2 h) = -4\Delta_g \mathsf {tr}_gh-4\delta_g \delta_g h.$$

3.  $c_gF_h(R_g)={\mathcal R}_g\circ h+h\circ {\mathcal R}_g+ 2{\mathfrak R}h.$

4.  $c^2_gF_h(R_g)=4\langle{\mathcal R}_g,h\rangle.$

The proofs of items 3. and 4. are rather easy computations. In fact, using (\[rrr\]) and Definition \[defn\], we compute: $$\begin{aligned}
c_gF_h(R_g)(x,y)&=&\sum_i F_h(R_g)(e_i\wedge x,e_i\wedge
y)\\
&=& \sum_i\Big(h(R_g(e_i,x)e_i,y)-h(R_g(e_i,x)y,e_i)+\\
 & & \qquad + h(R_g(e_i,y)e_i,x)-h(R_g(e_i,y)x,e_i)\Big)\\
&=& ({\mathcal R}_g\circ h+h\circ {\mathcal R}_g)(x,y)+ 2({\mathfrak R}h) (x,y).\end{aligned}$$ Moreover, $$c^2_gF_h(R_g)=c_g({\mathcal R}_g\circ h+h\circ {\mathcal R}_g+ 2{\mathfrak{R}h})=4\langle{\mathcal R}_g,h\rangle.$$

The proofs of the remaining items are a bit more involved. In fact, we have $$\begin{aligned}
c_g({\mathcal D}^2 h)(e_j,e_k) & = & \sum_i{\mathcal D}^2 h( e_i\wedge e_j,e_i\wedge e_k)\\
& = &\sum_i\Big[
2\nabla^2_{e_i,e_i}h(e_j, e_k)+\left( \nabla^2_{e_k,e_j}h(e_i, e_i)
+\nabla^2_{e_j,e_k}h(e_i, e_i) \right)-\\
& & \quad -\left( \nabla^2_{e_k,e_i}h(e_j, e_i) + \nabla^2_{e_j,e_i}h (e_i,
e_k) \right)-\\
& & \qquad -\left( \nabla^2_{e_i,e_j}h(e_i, e_k)
+\nabla^2_{e_i,e_k}h(e_j, e_i)\right)\Big].\end{aligned}$$ We can use Ricci identities to rewrite the last term above as $$\begin{aligned}
\nabla^2_{e_i,e_j}h(e_i, e_k)
+\nabla^2_{e_i,e_k}h(e_i,e_j) & = & \nabla^2_{e_j,e_i}h(e_i, e_k)
+\nabla^2_{e_k,e_i}h(e_i,e_j)+\\
& &  +h(R(e_i,e_j)e_i,e_k)
+h(e_i,R(e_i,e_j)e_k)+ \\
& & +h(R(e_i,e_k)e_i,e_j) +h(e_i,R(e_i,e_k)e_j),\end{aligned}$$ so that $$\begin{aligned}
c_g({\mathcal D}^2 h)(e_j,e_k) & = &\sum_i\Big[ 2\left(\nabla^2_{e_i,e_i}h\right)(e_j, e_k)+\left( \nabla^2_{e_k,e_j}h +\nabla^2_{e_j,e_k}h \right)(e_i, e_i)-\\
& & -2\Big( \nabla^2_{e_k,e_i}h(e_i,e_j) + \nabla^2_{e_j,e_i}h (e_i, e_k) \Big)-\\
& & -\Big( h(R(e_i,e_j)e_i,e_k) +h(e_i,R(e_i,e_j)e_k)+\\
& &+h(R(e_i,e_k)e_i,e_j) +h(e_i,R(e_i,e_k)e_j) \Big)\Big]\\
&= &-2(\nabla^*\nabla h)(e_j, e_k)+\\
& &+\Big( \nabla d\mathsf {tr}_gh(e_k,e_j) +\nabla d\mathsf {tr}_gh(e_j,e_k)\Big)-\\
&& +4\delta^{*}_g\delta_g h(e_k,e_j)-({\mathcal R}_g\circ h)(e_j,e_k)-\\
& & -(h\circ {\mathcal R}_g)(e_j,e_k) +2({\mathfrak R} h)(e_j,e_k),\end{aligned}$$ and the first item follows. Finally, using the symmetries due to the fact that ${\mathcal D}^2 h\in {\mathcal C}^2(X)$ we obtain $$\begin{aligned}
c^2_g{\mathcal D}^2 h &  = & \sum_{ij}{\mathcal D}^2 h(e_i\wedge e_j,e_i\wedge e_j)\\
& = & \sum_{ij}\left(
4(\nabla^2_{e_i,e_i}h)(e_j, e_j)-4(\nabla^2_{e_i,e_j}h )(e_i, e_j)\right).\end{aligned}$$ Now notice that $$\sum_{ij}\nabla^2_{e_i,e_i}h(e_j, e_j)=-\Delta_g \mathsf {tr}_g(h), \qquad
\sum_{ij} \nabla^2_{e_i,e_j}h (e_i,e_j)=\delta_g (\delta_g h),$$ and the second item follows. Lemma \[aux\] is proved.

\[newr\] Under the constant curvature condition $R_g=\frac{\mu}{2}g^2$ (see Remark \[const\]) we have ${\mathcal R}_g=(n-1)\mu g$, so that $${\mathcal R}_g\circ h+h\circ {\mathcal R}_g=(n-1)\mu(g\circ h+h\circ g)=2(n-1)\mu
h \label{dd}$$ and $${\mathfrak R} h= \mu ({\rm tr}_gh \,g - h) \label{ee}.$$

The proof of Theorem \[main\]
=============================

As already observed in Remark \[const\], if $(X,g)$ has constant sectional curvature $\mu$ then its curvature tensor can be expressed as $R_g=\frac{\mu}2g^2$, so that its $2k$-Ricci tensor can be computed using (\[comut\]) as follows: $$\begin{aligned}
{\mathcal R}^{(2k)}_g&=&c^{2k-1}_g\left(\frac{\mu^k}{2^k}g^{2k}\right)\\
& = & \frac{\mu^k}{2^k}c^{2k-1}_gg^{2k-1}g\\
&=&\frac{\mu^k}{2^k}(2k-1)!\sum_{r=1}^{2k-1}C^{2k-1}_r
\prod_{i=0}^{r-1}(n-2-i)\frac{g^{2k-1-r}}{(2k-1-r)!}c^{2k-1-r}_gg\\
&=&\frac{\mu^k}{2^k}(2k)!\prod_{i=0}^{2k-3}(n-2-i)(n-1)g\\
 & = & \frac{(n-1)!(2k)!}{(n-2k)!2^k}\mu^kg.\end{aligned}$$ Thus, $g$ is a $2k$-Einstein metric, $
{\mathcal R}^{(2k)}_g=\lambda_k g$, with $$\label{eins}
\lambda_k=(n-1)(n-2)C_{n,k}\mu^k,\qquad C_{n,k}=\frac{(2k)!(n-3)!}{2^k(n-2k)!}.$$

From (\[def\]) we have in general $$\label{sy}
\dot{\mathcal R}^{(2k)}_gh = (2k-1)(\dot c_gh)c^{2k-2}_gR^k_g +
kc^{2k-1}_gR^{k-1}_g\dot R_gh,$$ which specializes to $$\begin{aligned}
\dot{\mathcal R}^{(2k)}_gh & = & \frac{(2k-1)\mu^k}{2^k}(\dot c_gh)c^{2k-2}g^{2k}
+\frac{k\mu^{k-1}}{2^{k-1}}c^{2k-1}_gg^{2k-2}\dot R_gh\nonumber\\
& \doteq & A_gh +B_gh\label{var1}\end{aligned}$$ in the constant curvature case. We now identify the terms $A_gh$ and $B_gh$ in this expression.

We start with $A_gh$, which should be a zero order term in $h$ (no derivatives). To begin with observe that (\[comut\]) implies $$\begin{aligned}
c^{2k-2}_g\left(\frac{g^{2k-1}}{(2k-1)!}g\right) & = &\sum_{r=1}^{2k-2}C^{2k-2}_r
\prod_{i=0}^{r-1}(n-3-i)\frac{g^{2k-1-r}}{(2k-1-r)!}c^{2k-2-r}_gg\\
& = & (2k-2) \prod_{i=0}^{2k-4}(n-3-i)\frac{g^2}2c_gg+
\prod_{i=0}^{2k-3}(n-3-i)g^2\\
& = &
\prod_{i=0}^{2k-4}(n-3-i)\Big(n(k-1)+(n-2k)\Big)g^2\\
& = &\frac{(n-2)!k}{(n-2k)!}~g^2,\end{aligned}$$ that is $$c^{2k-2}_gg^{2k}= \frac{(n-2)!(2k)!}{2(n-2k)!}~g^2,$$ which gives $$\begin{aligned}
A_gh=
\frac{(2k-1)(n-2)!(2k)!\mu^k}{(n-2k)!2^{k+1}}(\dot c_g h)g^2.\end{aligned}$$ Now if we linearize the identity $c_gg^2=2(n-1)g$ in the direction of $h$ we obtain, with the help of (\[kulk\]), $$\begin{aligned}
2(n-1)h&=&(\dot c_gh)g^2+2c_ggh\\
&=&(\dot c_gh)g^2+2{\mathsf {tr}}_gh\,g+2(n-2)h,\end{aligned}$$ so that $$(\dot c_gh)g^2=2\left(h-{\mathsf {tr}}_ghg\right),$$ implying $$\begin{aligned}
A_gh=
(2k-1)(n-2)C_{n,k}\mu^k(h-{\mathsf {tr}}_ghg). \label{zz}\end{aligned}$$

In order to compute $B_gh$ we make use of (\[comut\]) with $k>1$, an assumption implying in particular that $c^{2k-1}_g\dot R_gh=0$ since $\dot R_gh\in {\mathcal C}^2(X$). Hence, $$\begin{aligned}
c^{2k-1}_g\left(\frac{g^{2k-2}}{(2k-2)!}\dot R_gh\right) & = & \sum_{r=1}^{2k-2}C^{2k-1}_r
\prod_{i=0}^{r-1}(n-3-i)\frac{g^{2k-2-r}}{(2k-2-r)!}c^{2k-1-r}_g\dot R_gh\\
& = & (2k-1)(k-1)\prod_{i=0}^{2k-4}(n-3-i)gc^2_g\dot R_gh+
(2k-1)\times \\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\times \prod_{i=0}^{2k-3}(n-3-i)c_g\dot R_gh\\
& = & (2k-1)\prod_{i=0}^{2k-4}(n-3-i)\big((k-1)gc^2_g\dot R_gh+\\
& &  \qquad\qquad \qquad\qquad\qquad+(n-2k)c_g\dot R_gh\big),\end{aligned}$$ so that $$\begin{aligned}
c^{2k-1}_gg^{2k-2}\dot
R_gh=\frac{(2k-1)!(n-3)!}{(n-2k)!}\left((k-1)(c^2_g\dot R_gh)g+
(n-2k)c_g\dot R_gh\right),\nonumber\label{aa}\end{aligned}$$ which gives $$\begin{aligned}
B_gh=C_{n,k}\mu^{k-1}\left((k-1)(c_g^2\dot R_g h)g+
(n-2k)c_g\dot R_g h\right). \label{bb}\end{aligned}$$

This shows that, in the constant curvature case, $B_gh$ is completely determined by the first and second contractions of the projection of $\dot R_g h$ on ${\mathcal C}^2(X)$. Now, these contractions have essentially been computed in Lemma \[aux\]. In fact, inserting the expressions there in (\[varl\]) we find that

$$\label{gg}
c_g\dot R_gh=
\frac12 \left(\nabla^*\nabla h-
\nabla d\mathsf {tr}_gh-2\delta^{*}_g\delta_g h+{\mathcal R}_g\circ h+h\circ {\mathcal R}_g \right)$$

and $$c^2_g \dot R_gh=\Delta_g \mathsf {tr}_gh+\delta_g (\delta_g h)+ \langle{\mathcal R}_g,h\rangle\label{rr},$$ so that by (\[bb\]) we get

$$\begin{aligned}
B_g
h & = &  C_{n,k}\mu^{k-1}\bigg((k-1)\left(\Delta_g \mathsf {tr}_gh+\delta_g (\delta_g h)+ \langle {\mathcal R}_g,h\rangle\right)g +  \nonumber\\
& &\qquad +\frac{n-2k}2 \Big(\nabla^*\nabla h-
\nabla d\mathsf {tr}_gh-\\
& & \qquad\qquad-2\delta^{*}_g\delta_g h+({\mathcal R}_g\circ h+h\circ {\mathcal R}_g) \Big)\bigg).\nonumber
\label{ss}\end{aligned}$$

Finally, using Remark \[newr\] to specialize to the constant curvature case, together with (\[varl\]) and (\[zz\]) we obtain an expression for $\dot{\mathcal R}^{(2k)}_g$ and hence for ${\mathcal L}_g^{(2k)}=\dot{\mathcal R}^{(2k)}_g-\lambda_k$, namely,

$$\begin{aligned}
{\mathcal L}_g^{(2k)}h& = & C_{n,k}\mu^{k-1}\bigg( (k-1)\Big(\Delta_g
{\rm tr}_g(h)+\delta_g (\delta_g h)-\\
& & \qquad\qquad\qquad\qquad\qquad -\frac{(k(n-3)+1)}{k-1}\mu {\rm tr}_gh\Big)g+ \\
&&\qquad + \frac{n-2k}2\Big( \nabla^*\nabla h-
\nabla d{\rm tr}_gh-2\delta^{*}_g\delta_g h + 2\mu h \Big)
\bigg).\end{aligned}$$

In the sequel we shall use two special cases of this formula, namely,

$$\begin{aligned}
{\mathcal L}_g^{(2k)}\Big|_{{\rm tr}_g^{-1}(0)} & =  & C_{n,k}\mu^{k-1}\bigg( \frac{n-2k}2\nabla^*\nabla
-(n-2k)\delta^{*}_g\delta_g +\nonumber \\
& & \qquad\qquad+(k-1)\delta_g \delta_g (\cdot)g + (n-2k)\mu
\bigg)\label{trace}\end{aligned}$$

and $$\label{sp2}
{\mathcal L}_g^{(2k)}\Big|_{{\mathcal I}_g}= (n-2k)C_{n,k}\mu^{k-1}\bigg( \frac12\nabla^*\nabla
+ \mu
\bigg).$$ In particular, since $\nabla^*\nabla$ is elliptic, (\[sp2\]) already shows that $\dim\mathfrak{M}_{[g]}^{(2k)}<+\infty$ if $\mu\neq 0$, thus proving the first part of Theorem \[main\].

To show that spherical space forms are infinitesimally $2k$-non-deformable, we assume $\mu>0$ and integrate (\[sp2\]) with $0\neq h\in{\mathcal I}_g$, so that $$\begin{aligned}
0  & = & \int_X\langle {\mathcal L}_g^{(2k)}h,h\rangle\nu_g \\
   & = & (n-2k)C_{n,k}\mu^{k-1}\left(\frac{1}{2}\int_X|\nabla h|^2\nu_g+\mu\int_X|h|^2\nu_g\right)\\
   & \geq & (n-2k)C_{n,k}\mu^k\int_X|h|^2\nu_g\\
   & > & 0,\end{aligned}$$ a contradiction, thus showing that $\mathfrak{M}_g^{(2k)}= \ker {\mathcal L}_g^{(2k)}|_{{\mathcal I}_g}$ is trivial.

The proof of Theorems \[main2\] and \[main3\]
=============================================

As remarked in the Introduction, it is not hard to combine the above information on ${\mathcal L}_g^{(2k)}|_{{\mathcal I}_g}$, namely, that it is a self-adjoint positive elliptic operator, with results in [@K1] to check that spherical space forms are $2k$-non-deformable indeed. However, we will be able to prove a much stronger result. The idea here is to adapt an argument in [@Be], page 351, which consists in replacing the operator ${\mathcal L}_g^{(2k)}|_{{\rm tr}^{-1}(0)}$, which is obviously non-elliptic, by an elliptic one. In order to implement this strategy, we need some more notation.

First, the fact that the $2k$-Einstein-Lovelock tensor ${\mathcal T}_g^{(2k)}$ is divergence free, for any metric $g$, can be expressed as $$\label{free}
\delta_g{\mathcal R}_g^{(2k)}+(2k-1)!d{\mathcal S}_g^{(2k)}=0.$$ We now introduce the functional ${\mathcal G}:{\mathcal M}_1(X)\to {\mathcal C}^1(X)$, $${\mathcal G}(g)={\mathcal R}_g^{(2k)}-\frac{(2k)!}{n}{\mathcal A}^{(2k)}(g)g,$$ and the $(2k)$-Bianchi operator $\beta^{(2k)}_g:{\mathcal C}^1(X)\to \Omega^1(X)$, $$\beta^{(2k)}_g=\delta_g+\frac{1}{2k}d{\mathsf {tr}}_g.$$ The following lemma is a direct consequence of the definitions.

\[dr\] The following properties hold:

1.  $g$ is $2k$-Einstein if and only if ${\mathcal G}(g)=0$;

2.  If $g$ is $2k$-Einstein then $\dot{{\mathcal G}}_g={\mathcal L}_g^{(2k)}$ on ${\mathcal C}^1(X)$.

3.  For any $g$, $\beta^{(2k)}_g{{\mathcal G}}(g)=0$. In particular, if $g$ is $2k$-Einstein, $\beta^{(2k)}_g{{\mathcal L}}^{(2k)}_g=0$.

The identity $\beta^{(2k)}_g{{\mathcal L}}^{(2k)}_g=0$ means that ${{\mathcal L}}^{(2k)}_g$ is not surjective but instead satisfies a first order differential equation coming from the diffeomorphism invariance of the $2k$-Einstein condition. This is of course a serious obstruction to using the Implicit Function Theorem to probe the local structure of the moduli space. As a way to overcome this we use (\[trace\]) and introduce, for $h\in {\mathsf {tr}}_g^{-1}(0)$, the elliptic operator $$\begin{aligned}
\tilde{\mathcal L}_g^{(2k)}h & = & {\mathcal L}_g^{(2k)}h+(n-2k)C_{n,k}\mu^{k-1}\delta^*_g\delta_gh-
(k-1)C_{n,k}\mu^{k-1}(\delta_g\delta_gh)g\\
 & = & (n-2k)C_{n,k}\mu^{k-1}\left(\frac{1}{2}\nabla^*\nabla h+\mu h \right).\end{aligned}$$ Notice that the identity ${\mathsf {tr}}_g\nabla^*\nabla h=\Delta_g{\mathsf {tr}}_gh$ gives $${\mathsf {tr}}_g\tilde{\mathcal L}_gh=(n-2k)C_{n,k}\mu^{k-1}\left(\frac{1}{2}\Delta_g{\mathsf {tr}}_gh
+\mu{\mathsf {tr}}_gh\right),$$ which implies $$\int_X{\mathsf {tr}}_g\tilde{\mathcal L}_gh\,\nu_g=(n-2k)C_{n,k}\mu^{k}
\int_X{\mathsf {tr}}_gh\,\nu_g,$$ and recalling that $$T_g{\mathcal M}_1(X)=\{h\in{\mathcal C}^1(X);\int_X \mathsf {tr}_gh\,\nu_g=0\},$$ this shows that $\tilde{\mathcal L}_g(T_g{\mathcal M}_1(X))$ is closed.

We now look at the constraints posed on $\tilde{\mathcal L}_g$ by diffeomorphism invariance. Using Proposition \[dr\] and the identity ${\mathsf {tr}}_g\delta_g^*\omega=-\delta_g\omega$, $\omega\in\Omega^1(X)$, we compute: $$\begin{aligned}
\beta^{(2k)}_g\tilde{\mathcal L}_gh & = & C_{n,k}\mu^{k-1}\left((n-2k)\left(\delta_g
\delta_g^*\delta_gh+\frac{1}{2k}d{\mathsf {tr}}_g\delta_g^*\delta_gh\right.
\right)-\\
&  & \,\,\,\,\,\,\,\,\,\,\,\,\,-(k-1)\left.\left(
\delta_g(\delta_g\delta_gh)g+\frac{1}{2k}d{\mathsf {tr}}_g(\delta_g\delta_gh)g
\right)\right)\\
& = & C_{n,k}\mu^{k-1}\left((n-2k)\left(\delta\delta_g^*-\frac{1}{2k}d\delta\right)(\delta_gh)\right.-\\
& & \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left. -(k-1)\left(-d(\delta_g\delta_gh)+\frac{n}{2k}d\delta_g\delta_gh\right) \right)\\
& = &
(n-2k)C_{n,k}\mu^{k-1}\left(\delta_g\delta_g^*-\frac{1}{2}d\delta_g\right)(\delta_gh),\end{aligned}$$ so that $$\label{pup}
\beta^{(2k)}_g\tilde{\mathcal L}_gh=(n-2k)C_{n,k}\mu^{k-1}G_g\delta h,$$ where $$G_g=\delta_g\delta_g^*-\frac12d\delta_g$$ is elliptic. Now, (\[pup\]) first gives $\tilde{\mathcal L}_g(T_g\mathfrak{S}_g)\subset {\ker}\beta^{(2k)}_g$. Also, if $h=\tilde{\mathcal L}_g k\in \ker\beta^{(2k)}_g$, $k\in T_g{\mathcal M}_1(X)$, then $\delta_gk\in\ker G$, a finite dimensional space, and this gives $$\tilde{\mathcal L}_g(T_g\mathfrak{S}_g)\subset\tilde{\mathcal L}_g(T_g{\mathcal M}_1(X)\cap\ker \beta^{(2k)}_g)\subset
\tilde{\mathcal L}_g\left(T_g{\mathcal M}_1(X)\cap\delta_g^{-1}\ker G\right).$$ Since $T_g\mathfrak{S}_g$ is closed and has finite codimension in $T_g{\mathcal M}_1(X)\cap\delta_g^{-1}\ker G$, it is easy to check that $\tilde{\mathcal L}_g(T_g\mathfrak{S}_g)$ is closed in ${\mathcal L}_g\left(T_g{\mathcal M}_1(X)\cap\delta_g^{-1}\ker G\right)$. Thus, the image $\tilde{\mathcal L}_g(T_g{\mathcal M}_1(X))$ is closed in $\tilde{\mathcal L}_g(T_g{\mathcal M}_1(X)\cap\ker \beta^{(2k)}_g)$, which is closed in ${\mathcal C}^1(X)$.

We conclude that ${\mathcal L}_g=\dot {\mathcal G}_g:T_g\mathfrak{S}_g\to {\mathcal C}^1(X)$, even though not surjective, has closed range. Thus, if $p$ is the orthogonal projection of ${\mathcal C}^1(X)$ onto ${\mathcal L}_g(T_g\mathfrak{S}_g)$, the real analytic composite map $p\circ {\mathcal G}:\mathfrak{S}_g\to {\mathcal L}_g(T_g\mathfrak{S}_g)$ is a submersion at $g$. Thus, $(p\circ {\mathcal G})^{-1}(0)$ is a real analytic manifold near $g$, with $\mathfrak{M}_{[g]}^{(2k)}$ as its tangent space at $g$. On this manifold, the map ${\mathcal G}$ is real analytic so that the pre-moduli space ${\mathcal E}_{2k}(X)_g={\mathcal G}^{-1}(0)$ is a real analytic subset. This completes the proof of Theorem \[main3\], and hence of Theorem \[main2\].

Further comments and questions
==============================

The results in this note give rise to a few basic questions on $2k$-Einstein structures that we would like to briefly discuss here. At a more basic level, it would be highly desirable to find geometric conditions on a $2k$-Einstein metric in order to have the operator $h\mapsto \dot{\mathcal R}^{(2k)}_gh$ elliptic. Since in general the first term in the right hand side of (\[sy\]) involves no derivatives of $h$, this amounts to understanding the symbol of the linear operator $$\label{ap}
h\mapsto c^{2k-1}_gR^{k-1}_g\dot R_gh$$ in more geometric terms. For example, the constant curvature assumption here implies that this symbol is a multiple of the symbol of the Bochner Laplacian, but a more general ellipticity criterium certainly would be of some use. As is apparent from (\[ap\]), the difficulty here is that for $k\geq 2$ the $2k$-Einstein condition is fully nonlinear in the second derivatives of the metric, thus implying that the symbol of the linearized operator depends on second order data, namely, the curvature. In any case, progress in this issue could be useful in discussing the eventual rigidity of other classes of $2k$-Einstein manifolds like sufficiently pinched manifolds, hyperbolic manifolds and certain classes of symmetric spaces.

[10]{}

Berger, M., Sur quelques variétés riemanniennes compactes d’Einstein, [*C. R. Acad. Sci. Paris*]{} 260 (1965) 1554–1557.

Berger, M., Ebin, D., Some decompositions of the space of symmetric tensors on a Riemannian manifold. [*J. Differential Geometry*]{} 3 (1969) 379-392.

Besse, Arthur L., [*Einstein manifolds*]{} Ergebnisse der Mathematik und ihrer Grenzgebiete, Springer-Verlag, Berlin, 1987

Bourguignon, J.-P., Déformations des métriques d’Einstein, [*Astérisque*]{}, 80 (1980), 21-31.

Choquet-Bruhat, Y., The Cauchy problem for stringy gravity, [*J. Math. Phys.*]{}, 29 (1988), no. 8, 1891-1895.

Ebin, D., The manifold of Riemannian metrics, Global Analysis, [*Proc. Sympos. Pure Math.*]{}, Vol. XV (1968) pp. 11-40.

Koiso, N., Nondeformability of Einstein metrics, [*Osaka J. Math.*]{} 15 (1978) 2, 419-433.

Koiso, N., Rigidity and infinitesimal deformability of Einstein metrics, [*Osaka J. Math.*]{} 19 (1982), 3, 643-668.

Labbi, M.-L. Double forms, curvature structures and the $(p,q)$-curvatures, [*Trans. Amer. Math. Soc.*]{} 357 (2005), 10, 3971-3992.

Labbi, M.-L., Variational Properties of the Gauss-Bonnet Curvatures, arXiv:math/0406548.

Lovelock, D., The Einstein tensor and its generalizations. [*J. Mathematical Phys.*]{} 12 (1971) 498-501.

Moser, J., On the volume elements on a manifold, [*Trans. Amer. Math. Soc.*]{}, 120 (1965) 286-294.

[^1]: The first author was supported in part by a CNPq grant.

[^2]: The second author was supported in part by the CNPq postdoctoral grant 155723/2006-5.