Sphere Theorems for σ_k-Einstein Manifolds

Abstract

A problem that geometers have always been concerned with is when a closed manifold is isometric to a round sphere. A classical result shows that a closed locally conformally flat Einstein manifold is always isometric to a quotient of a round sphere. In this note, we provide the definitions of ${\sigma }_k$-curvatures and ${\sigma }_k$-Einstein manifolds, and we show that a closed ${\sigma }_k$-Einstein manifold under certain pinching conditions of a Weyl curvature and Einstein curvature is isometric to a quotient of a round sphere.

1. Introduction

In geometry, a problem that mathematicians have always been concerned with is the so-called sphere theorem, which asks under what conditions a closed manifold is isometric to a round sphere. A classic result (one can find in Besse’s book [1]) is as follows:

Theorem 1.

In a connected and complete Riemannian manifold M, if the metric is Einstein and conformally flat, then the manifold is isometric to a quotient of a sphere of the same dimension.

It is well known that a manifold is conformally flat if and only if its Weyl curvature tensor $W_{ijkl}=0$, while a manifold is Einstein if and only if its Einstein curvature tensor $E_{ij}=0$ (definitions of these curvature tensors can be found in Section 2). Over the decades, mathematicians have attempted to improve these two conditions. In 1993, Michael A. Singer discovered that the condition $W_{ijkl}=0$ could be improved; he proposed the following theorem in [2].

Theorem 2.

Let $(M^n,g)$ be a compact, oriented Riemannian manifold of even dimension with a non-zero Euler characteristic $\chi \left(M\right)$, and let g be a positive Einstein metric with Weyl curvature. Then, there exists a constant $\epsilon >0$, which depends only on n and $\chi \left(M\right)$, such that if ${\parallel W\parallel }_{L^{n/2}}<\epsilon$, then $W_{ijkl}=0$, meaning that the manifold is isometric to a quotient space of the sphere $S^n$.

Singer’s result improved the condition $W_{ijkl}=0$ to a pinching condition of Weyl curvature tensor.

In 2019, the condition $E_{ij}=0$ was also improved. Yi Fang and Wei Yuan further generalized Singer’s result in [3]; they showed that Bach flat manifolds under certain conditions could be Einstein, and thus are locally spherical. They provided two versions of results and called them $L^{\infty }$-sphere theorem and $L^{{\displaystyle \frac{n}{2}}}$-sphere theorem.

Theorem 3

($L^{\infty }$-sphere theorem). Suppose $(M^n,g)$ is a closed Bach flat Riemannian manifold with constant scalar curvature $R=n(n-1)$. If

$${\parallel W\parallel }_{L^{\infty }}+{\displaystyle \frac{2n}{3(n-2)}}{\parallel E\parallel }_{L^{\infty }}<{\epsilon }_0\left(n\right):={\displaystyle \frac{n-1}{4}},$$

then the manifold $(M^n,g)$ is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

Theorem 4

($L^{{\displaystyle \frac{n}{2}}}$-sphere theorem). Suppose $(M^n,g)$ is a closed Bach-flat Riemannian manifold with constant scalar curvature $R=n(n-1)$. Assume that there is a constant ${\alpha }_0$ such that its Yamabe constant satisfies that $Y(M,\left[g\right])\ge {\alpha }_0>0$. If the following inequality holds

$${\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}+{\displaystyle \frac{n}{2(n-2)}}{\parallel E\parallel }_{L^{{\displaystyle \frac{n}{2}}}}<{\displaystyle \frac{(n-2){\alpha }_0}{4max\left\{8\right(n-1),n(n-2\left)\right\}}},$$

then $(M^n,g)$ is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

In these two theorems, the condition of closed manifolds being Einstein has been improved to Bach flatness, constant scalar curvature, and certain pinching conditions of Weyl curvatures and Einstein curvature tensors.

Our work in this note is to generalize these two sphere theorems, so we improved these conditions in Theorems 3 and 4 to ${\sigma }_k$-Einstein, constant ${\sigma }_{k-1}$-curvature and related pinching conditions. One can find specific definitions and properties about ${\sigma }_k$-curvature and ${\sigma }_k$-Einstein manifolds in Section 2 and Section 4. Roughly speaking, ${\sigma }_k$-Einstein manifolds generalize the idea of Einstein manifolds and Bach flat manifolds. In conclusion, we have the following results:

Theorem 5

(Main theorem A, $L^{\infty }$-sphere theorem). Let $(M^n,g)$ be a closed ${\sigma }_k$-Einstein manifold in the positive light cone with $n\ge 3$, where ${\sigma }_{k-1}$ is a constant. If ${\left[T^{(k-1)}\right]}_{ij,}^j=0$, and the following inequality holds

$$\begin{array}{c}\hfill {\parallel W\parallel }_{L^{\infty }}+{\displaystyle \frac{n(2n-1)}{3(n-1)(n-2)}}{\parallel E\parallel }_{L^{\infty }}<{\displaystyle \frac{R}{4n}},\end{array}$$

(1)

then the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

Here, $T^{(k-1)}$ is the Newton tensor associated with Schouten tensor; one can find definition in Section 2.

Theorem 6

(Main theorem B, $L^{{\displaystyle \frac{n}{2}}}$-sphere theorem). Let $(M^n,g)$ be a closed ${\sigma }_k$-Einstein manifold in the positive light cone with $n\ge 3$, where ${\sigma }_{k-1}$ is a constant. Assume that the Yamabe constant has a lower bound satisfying $Y(M,\left[g\right])\ge {\alpha }_0>0$. If ${\left[T^{(k-1)}\right]}_{ij,}^j=0$ and the following inequality holds

$$\begin{array}{c}\hfill {\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}+{\displaystyle \frac{n^2}{2(n-1)(n-2)}}{\parallel E\parallel }_{L^{{\displaystyle \frac{n}{2}}}}<{\displaystyle \frac{(n-2){\alpha }_0}{4max\left\{8\right(n-1),n(n-2\left)\right\}}},\end{array}$$

(2)

then the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

This note is organized as follows. In Section 2, we list all the curvature-related notation that will be used in this note, including ${\sigma }_k$-curvature, Newton tensor, and $\theta$-Codazzi tensor. In Section 3, we calculate the variation formula of ${\sigma }_k$-curvature then derive the formula for its $L^2$-adjoint $d{\sigma }_k^{*}$. In Section 4, we discuss the definition and properties of ${\sigma }_k$-Einstein manifolds; we will show that ${\sigma }_k$-Einstein manifolds generalize the idea of Einstein manifolds and Bach flat manifolds. Finally, we will prove our main theorems in Section 5 and Section 6.

2. Notation and Conventions

In this section, we will introduce a series of concepts and lemmas that will be used throughout this paper. Let us start with curvature tensors.

2.1. Basic Conventions

For a Riemannian manifold $(M^n,g)$, the definition of the Riemannian curvature tensor is

$$\begin{array}{c}\hfill R_{kij}^l={\partial }_i{\Gamma }_{jk}^l-{\partial }_j{\Gamma }_{ik}^l+{\Gamma }_{jk}^p{\Gamma }_{ip}^l-{\Gamma }_{ik}^p{\Gamma }_{jp}^l,\end{array}$$

(3)

where the Christoffel symbols ${\Gamma }_{ijk}={\displaystyle \frac{1}{2}}\left({\partial }_j{g}_{ik}+{\partial }_k{g}_{ij}-{\partial }_i{g}_{jk}\right)$ and ${\Gamma }_{ij}^k=g^{kl}{\Gamma }_{lij}$.

The Ricci curvature tensor is defined by contraction as follows:

$$\begin{array}{c}\hfill R_{jk}=R_{ijkl}{g}^{il}.\end{array}$$

(4)

Further contraction yields the scalar curvature $R=g^{ij}{R}_{ij}$.

The Schouten curvature tensor is defined as

$$\begin{array}{c}\hfill P_{ij}=R_{ij}-{\displaystyle \frac{R}{2(n-1)}}{g}_{ij}.\end{array}$$

(5)

It is well known that one can decompose the Riemann curvature tensor by

$$\begin{array}{c}\hfill R_{ijkl}=W_{ijkl}+{\displaystyle \frac{1}{n-2}}{(P \overline{)\wedge } g)}_{ijkl},\end{array}$$

(6)

where $W_{ijkl}$ is the Weyl curvature tensor, and

$$\begin{array}{c}\hfill {(P \overline{)\wedge } g)}_{ijkl}=P_{il}{g}_{jk}+P_{jk}{g}_{il}-P_{ik}{g}_{jl}-P_{jl}{g}_{ik}.\end{array}$$

(7)

Here, $\overline{)\wedge }$ is the Kulkarni–Nomizu product.

Here, we introduce the Einstein tensor $E_{ij}=R_{ij}-{\displaystyle \frac{R}{n}}{g}_{ij}$, which gives another way to represent the Schouten curvature tensor by

$$\begin{array}{c}\hfill P_{ij}=E_{ij}+{\displaystyle \frac{n-2}{2n(n-1)}}R{g}_{ij};\end{array}$$

(8)

then, Formula (6) could also be written as

$$\begin{array}{c}\hfill R_{ijkl}=W_{ijkl}+{\displaystyle \frac{1}{n-2}}{(E \overline{)\wedge } g)}_{ijkl}+{\displaystyle \frac{R}{2n(n-1)}}{(g \overline{)\wedge } g)}_{ijkl}.\end{array}$$

(9)

For any function $f\in C^{\infty }\left(M\right)$, the Ricci identity is as follows:

$$\begin{array}{c}\hfill f_{i,kj}^j-f_{i,jk}^j=R_{ikj}^m{f}_m^j-R_{mkj}^j{f}_i^m.\end{array}$$

(10)

All of the above definitions will be used throughout this paper.

2.2. Newton Tensor

The Newton tensor is an important tool for calculating ${\sigma }_k$-curvature. One can see the following definitions and properties given by Reilly in his papers [4,5].

Definition 1.

Let $(M,g)$ be an m-dimensional Riemannian manifold with $m\ge 3$, and consider the Schouten tensors $P_{ij}$ as an $n\times n$ matrix. Then, denote the eigenvalues of $P_{ij}$ as ${\lambda }_{i_1},\dots ,{\lambda }_{i_m}$. For $0\le k\le m$, the symmetric polynomial of $P_{ij}$ can be expressed as ${\sigma }_k\left(P\right)={\displaystyle \sum _{1\le i_1<\cdots <i_k\le m}}{\lambda }_{i_1}\dots {\lambda }_{i_k}$. For ${\sigma }_k$ and the Newton tensor, we have the following formulae:

$$\begin{array}{c}\hfill {\sigma }_k\left(P\right)={\displaystyle \frac{1}{k!}}\sum \delta \left(\begin{array}{ccc}{i}_1& \dots & i_k\\ j_1& \dots & j_k\end{array}\right)P_{i_1}^{j_1}\cdots P_{i_k}^{j_k},\end{array}$$

(11)

$$\begin{array}{c}\hfill {\left(T^{\left(k\right)}\left(P\right)\right)}_j^i={\displaystyle \frac{1}{k!}}\sum \delta \left(\begin{array}{ccc}{i}_1& \dots & i_{k}i\\ j_1& \dots & j_{k}j\end{array}\right)P_{i_1}^{j_1}\cdots P_{i_k}^{j_k}.\end{array}$$

(12)

Here, $\delta \left(\begin{array}{ccc}{i}_1& \dots & i_k\\ j_1& \dots & j_k\end{array}\right)$ represents the Kronecker delta, which for $0\le k\le m$, $1\le i_1,\dots$, $i_k,j_1,\dots ,j_k\le m$, takes the value 1 if $i_1,\dots ,i_k$ and $j_1,\cdots ,j_k$ form an even permutation, −1 if they form an odd permutation, and 0 otherwise.

Proposition 1.

The Newton tensor has following properties:

(1) $T_{ij}^{\left(k\right)}={\displaystyle \sum _{m=0}^k}{(-1)}^m{P}_{ij}^m{\sigma }_{k-m}$;

(2) $\mathrm{tr}\left(T_{ij}^{\left(k\right)}\right)=g^{ij}{T}_{ij}^{\left(k\right)}=(n-k){\sigma }_k$;

(3) $P^{ij}{T}_{ij}^{\left(k\right)}=(k+1){\sigma }_{k+1}$.

One can find details in Reilly’s papers [4,5]. Lastly, we denote ${\stackrel{\circ }{T}}_{ij}^{(k-1)}$ as the traceless part of the Newton tensor $T_{ij}$. One can confirm that

$$\begin{array}{c}\hfill T_{ij}^{(k-1)}={\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}_{ij}.\end{array}$$

(13)

2.3. ${\sigma }_k$ and Its $L^2$-Adjoint d${\sigma }_k^{*}$

One way to understand ${\sigma }_k$ is to consider ${\sigma }_k$ as a nonlinear map between vector spaces of M:

$$\begin{array}{cc}\hfill S^2\left(M\right)& \to C^{\infty }\left(M\right)\hfill \\ \hfill {\sigma }_k:g& \mapsto {\sigma }_k\left(P_g\right),\hfill \end{array}$$

where $S^2\left(M\right)$ is the space of symmetric 2-tensor on M. We denote $d{\sigma }_k$ as the linearization of ${\sigma }_k$, which is also an operator between vector spaces, as follows:

$$\begin{array}{cc}\hfill S^2\left(M\right)& \to C^{\infty }\left(M\right)\hfill \\ \hfill d{\sigma }_k:h& \mapsto \dot{{\sigma }_k}\left(h\right),\hfill \end{array}$$

where $\dot{{\sigma }_k}$ is the variation of the ${\sigma }_k$-curvature; we will provide the details of this in Section 3. $d{\sigma }_k^{*}$ is denoted as the $L^2$-adjoint operator of ${\sigma }_k$:

$$\begin{array}{cc}\hfill C^{\infty }\left(M\right)& \to S^2\left(M\right)\hfill \\ \hfill d{\sigma }_k^{*}:f& \mapsto {\left[d{\sigma }_k^{*}\left(f\right)\right]}_{ij},\hfill \end{array}$$

satisfying the formula ${\langle d{\sigma }_k\left(h\right),f\rangle }_{L^2}={\langle d{\sigma }_k^{*}\left(f\right),h\rangle }_{L^2}$.

2.4. $\theta$-Codazzi Tensor

Definition 2.

Let $(M^n,g)$ be an n-dimensional Riemannian manifold with $n\ge 3$. $C_{ijk}^{\theta }$ is a θ-Codazzi Tensor, if it satisfies

$$\begin{array}{c}\hfill C_{ijk}^{\theta }={\stackrel{\circ }{T}}_{ij,k}-\theta {\stackrel{\circ }{T}}_{ik,j},\end{array}$$

(14)

where $\stackrel{\circ }{T}$ is any symmetric 2-tensor on $M^n$ and θ is any real number.

Lemma 1.

If $\stackrel{\circ }{T}$ is a symmetric tensor, then we have

$$\begin{array}{c}\hfill {\int }_M|\nabla \stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g\ge {\displaystyle \frac{2\theta }{1+{\theta }^2}}{\int }_M[|\delta \stackrel{\circ }{T}{|}^2+W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}{|T|}^2]\mathrm{d}{v}_g.\end{array}$$

Here, $\mathrm{d}{v}_g$ is the volume element of $M^n$. Particularly, let $\theta =-1$; then, we have

$$\begin{array}{c}\hfill {\int }_M|\nabla \stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g\ge -{\int }_M[|\delta \stackrel{\circ }{T}{|}^2+W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.\end{array}$$

(15)

Proof. 

According to Definition 2, it is easy to obtain

$$|C^{\theta }{|}^2=(1+{\theta }^2){|\nabla \stackrel{\circ }{T}|}^2-2\theta {\stackrel{\circ }{T}}_{ij,k}{\stackrel{\circ }{T}}^{ik,j}.$$

Integrating both sides gives

$$\begin{array}{c}\hfill {\int }_M{|\nabla \stackrel{\circ }{T}|}^2\mathrm{d}{v}_g={\int }_M{\displaystyle \frac{|C^{\theta }{|}^2}{1+{\theta }^2}}\mathrm{d}{v}_g+{\displaystyle \frac{2\theta }{1+{\theta }^2}}\int {\stackrel{\circ }{T}}_{ij,k}{\stackrel{\circ }{T}}^{ik,j}\mathrm{d}{v}_g.\end{array}$$

(16)

It is known that ${\int }_M{\stackrel{\circ }{T}}_{ij,k}{\stackrel{\circ }{T}}^{ik,j}\mathrm{d}{v}_g=-{\int }_M{\stackrel{\circ }{T}}_{i,kj}^j{\stackrel{\circ }{T}}^{ik}\mathrm{d}{v}_g$, and among them

$$\begin{array}{cc}\hfill {\stackrel{\circ }{T}}_{i,kj}^j-{\stackrel{\circ }{T}}_{i,jk}^j& =R_{ikj}^m{\stackrel{\circ }{T}}_m^j-R_{mkj}^j{\stackrel{\circ }{T}}_i^m=-R_{likj}{\stackrel{\circ }{T}}^{jl}+R_{mk}{\stackrel{\circ }{T}}_i^m\hfill \\ \hfill & =-W_{likj}{\stackrel{\circ }{T}}^{jl}-{\displaystyle \frac{1}{n-2}}(E_{jl}{g}_{ik}+E_{ik}{g}_{jl}-E_{kl}{g}_{ij}-E_{ij}{g}_{kl}){\stackrel{\circ }{T}}^{jl}\hfill \\ \hfill & -{\displaystyle \frac{R}{n(n-1)}}(g_{jl}{g}_{ik}-g_{kl}{g}_{ij}){\stackrel{\circ }{T}}^{jl}+E_{mk}{\stackrel{\circ }{T}}_i^m+{\displaystyle \frac{R}{n}}{g}_{mk}{\stackrel{\circ }{T}}_i^m\hfill \\ \hfill & =-W_{likj}{\stackrel{\circ }{T}}^{jl}-{\displaystyle \frac{1}{n-2}}(E·\stackrel{\circ }{T})g_{ik}+{\displaystyle \frac{2}{n-2}}{E}_k^l{\stackrel{\circ }{T}}_{il}+{\displaystyle \frac{R}{n(n-1)}}{\stackrel{\circ }{T}}_{ik}+E_k^m{\stackrel{\circ }{T}}_{mi}+{\displaystyle \frac{R}{n}}{\stackrel{\circ }{T}}_{ik}.\hfill \end{array}$$

That is,

$${\stackrel{\circ }{T}}_{i,kj}^j={\stackrel{\circ }{T}}_{i,jk}^j-W_{likj}{\stackrel{\circ }{T}}^{jl}-{\displaystyle \frac{1}{n-2}}(E·\stackrel{\circ }{T})g_{ik}+{\displaystyle \frac{n}{n-2}}{E}_k^l{\stackrel{\circ }{T}}_{il}+{\displaystyle \frac{R}{n-1}}{\stackrel{\circ }{T}}_{ik}.$$

Therefore,

$${\int }_M{\stackrel{\circ }{T}}_{ij,k}{\stackrel{\circ }{T}}^{ik,j}\mathrm{d}{v}_g=-{\int }_M{\stackrel{\circ }{T}}_{i,kj}^j{\stackrel{\circ }{T}}^{ik}\mathrm{d}{v}_g={\int }_M[|\delta \stackrel{\circ }{T}{|}^2+W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.$$

Substituting this equation into (16) yields

$$\begin{array}{cc}\hfill {\int }_M{|\nabla \stackrel{\circ }{T}|}^2\mathrm{d}{v}_g& ={\int }_M{\displaystyle \frac{|C^{\theta }{|}^2}{1+{\theta }^2}}\mathrm{d}{v}_g+{\displaystyle \frac{2\theta }{1+{\theta }^2}}{\int }_M[|\delta \stackrel{\circ }{T}{|}^2+W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g\hfill \\ \hfill & \ge {\displaystyle \frac{2\theta }{1+{\theta }^2}}{\int }_M[|\delta \stackrel{\circ }{T}{|}^2+W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.\hfill \end{array}$$

Taking $\theta =-1$, we obtain inequality (15). □

2.5. Newton’s Inequality

Newton’s inequality is an inequality related to the eigenvalues of tensors. Its significance lies in the conditions under which the equality holds. The proof relies entirely on linear algebra, and, thus, we will skip the details.

Lemma 2.

Let the eigenvalues of the Schouten tensor $P_{ij}$ be denoted as ${\lambda }_1,{\lambda }_2,\cdots ,{\lambda }_n$. The k-th elementary symmetric polynomial of the eigenvalues is ${\sigma }_k$, and, thus, the basic symmetric means can be expressed as $S_k={\displaystyle \frac{{\sigma }_k}{\left(\genfrac{}{}{0pt}{}{n}{k}\right)}}$, where $S_k^2\ge S_{k-1}·S_{k+1}$ is known as Newton’s inequality. The inequality holds with equality if and only if ${\lambda }_1={\lambda }_2=\cdots ={\lambda }_n\ge 0$. Specifically, when all eigenvalues of the Schouten tensor are positive, we have ${\displaystyle \frac{S_k}{S_{k-1}}}\ge {\displaystyle \frac{S_{k+1}}{S_k}}$. Let $\Lambda ={\displaystyle \frac{(n-2)}{2n(n-1)}}R$; then, we obtain

$${\displaystyle \frac{{\sigma }_1/\left(\genfrac{}{}{0pt}{}{n}{1}\right)}{{\sigma }_0/\left(\genfrac{}{}{0pt}{}{n}{0}\right)}}={\displaystyle \frac{n\Lambda /n}{1/1}}=\Lambda \ge {\displaystyle \frac{{\sigma }_k/\left(\genfrac{}{}{0pt}{}{n}{k}\right)}{{\sigma }_{k-1}/\left(\genfrac{}{}{0pt}{}{n}{k-1}\right)}},$$

which simplifies to the inequality

$$k{\sigma }_k-(n-k+1)\Lambda {\sigma }_{k-1}\le 0,$$

with equality if and only if ${\lambda }_1={\lambda }_2=\cdots ={\lambda }_n\ge 0$.

2.6. The Yamabe Constant

The Yamabe curvature plays a role in the proof of $L^{{\displaystyle \frac{n}{2}}}$-sphere theorem. We list its definition and one lemma that will be used later.

Definition 3.

Let $(M^n,g)$ be an n-dimensional Riemannian manifold with $n\ge 3$, and the Yamabe constant is defined as

$$Y(M,\left[g\right])=\underset{0\ne \mu \in C^{\infty }}{inf}{\displaystyle \frac{{\int }_M(\frac{4(n-1)}{n-2}{|\nabla \mu |}^2+R{\mu }^2)\mathrm{d}{v}_g}{{\left({\int }_M{\mu }^{\frac{2n}{n-2}}\mathrm{d}{v}_g\right)}^{\frac{n-2}{n}}}}.$$

Lemma 3.

Let $(M^n,g)$ be an n-dimensional Riemannian manifold with $n\ge 3$, and the Yamabe constant satisfies $Y(M,\left[g\right])\ge {\alpha }_0>0$. Take $C_S={\displaystyle \frac{4(n-1)}{(n-2){\alpha }_0}}$: for $\forall \mu \ne 0\in C^{\infty }$; then, we have

$$\begin{array}{c}\hfill {\parallel \mu \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2\le C_S{[\parallel \nabla \mu \parallel }_{L^2}^2+{\displaystyle \frac{n-2}{4(n-1)}}{\int }_{M}R{\mu }^2\mathrm{d}{v}_g].\end{array}$$

(17)

Proof. 

According to the formula for the Yamabe constant in Definition 3, we have

$${\displaystyle \frac{4(n-1)}{n-2}}{\int }_M{|\nabla \mu |}^2\mathrm{d}{v}_g+{\int }_{M}R{\mu }^2\mathrm{d}{v}_g\ge {\alpha }_0[({\int }_M{\mu }^{{\displaystyle \frac{2n}{n-2}}}\mathrm{d}{v}_g{)}^{{\displaystyle \frac{n-2}{2n}}}{]}^2={\alpha }_0{\parallel \mu \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2,$$

which implies that

$${\displaystyle \frac{4(n-1)}{n-2}}{\parallel \nabla \mu \parallel }_{L^2}^2+{\int }_{M}R{\mu }^2\mathrm{d}{v}_g\ge {\alpha }_0{\parallel \mu \parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2.$$

Rearranging the terms yields Equation (17). □

2.7. Weyl Tensor-Related Lemmas

In this section, we show several lemmas that have come from Singer’s work in [2] and Fang-Yuan’s work in [3]. These lemmas show that Einstein manifolds with tiny Weyl curvature are also locally conformally flat.

Lemma 4.

Let $(M^n,g)$ be an Einstein manifold with constant scalar curvature R; then, its Weyl tensor satisfies

$$\begin{array}{c}\hfill \Delta W-{\displaystyle \frac{2R}{n}}W-2\mathcal{Q}\left(W\right)=0,\end{array}$$

(18)

where $\mathcal{Q}\left(W\right):=B_{ijkl}-B_{jikl}+B_{ikjl}-B_{jkil}$ is a quadratic combination of Weyl tensors with $B_{ijkl}:=g^{pq}{g}^{rs}{W}_{pijr}{W}_{qkls}$.

One can find details in Singer’s work [2].

Lemma 5.

Let $(M^n,g)$ be an Einstein manifold with constant scalar curvature R. If

$${\parallel W\parallel }_{L^{\infty }}<{\displaystyle \frac{R}{4n}},$$

the Weyl tensor vanishes.

Proof. 

From the definition and Equation (18) in Lemma 4, we obtain

$$-{\int }_M<\Delta W-{\displaystyle \frac{2R}{n}}W,W>\mathrm{d}{v}_g=-2{\int }_M<\mathcal{Q}\left(W\right),W>\mathrm{d}{v}_g\le 8{\int }_M{|W|}^3\mathrm{d}{v}_g.$$

That is,

$$\begin{array}{c}\hfill {\int }_M{(|\nabla W|}^2+{\displaystyle \frac{2R}{n}}{|W|}^2)\mathrm{d}{v}_g\le 8{\int }_M{|W|}^3\mathrm{d}{v}_g.\end{array}$$

(19)

By the Hölder inequality, ${\int }_M{|W|}^3\mathrm{d}{v}_g\le {\parallel W\parallel }_{L^{\infty }}{\int }_M{|W|}^2\mathrm{d}{v}_g$; thus, we have

$${\displaystyle \frac{2R}{n}}{\int }_M{|W|}^2\mathrm{d}{v}_g\le 8{\int }_M{|W|}^3\mathrm{d}{v}_g\le {8\parallel W\parallel }_{L^{\infty }}{\int }_M{|W|}^2\mathrm{d}{v}_g.$$

That is,

$${(8\parallel W\parallel }_{L^{\infty }}-{\displaystyle \frac{2R}{n}}){\int }_M{|W|}^2\mathrm{d}{v}_g\ge 0.$$

Since ${\parallel W\parallel }_{L^{\infty }}<{\displaystyle \frac{R}{4n}}$, the Weyl tensor vanishes. □

Lemma 6.

Let $(M^n,g)$ be an Einstein manifold, and the Yamabe constant satisfies $Y(M,\left[g\right])\ge {\alpha }_0>0$. If

$${\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}<{\tau }_0={\displaystyle \frac{(n-2){\alpha }_0}{4max\{8(n-1),n(n-2)\}}},$$

the Weyl tensor vanishes.

Proof. 

From the Formula (17) in Lemma 3 and Formula (19) in Lemma 5, we can obtain

$$\begin{array}{cc}\hfill {\parallel W\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2& \le C_S{\int }_M{(|\nabla W|}^2+{\displaystyle \frac{n-2}{4(n-1)}}{R|W|}^2)\mathrm{d}{v}_g\hfill \\ \hfill & \le C_{S}max\{1,{\displaystyle \frac{n(n-2)}{8(n-1)}}\}{\int }_M{(|\nabla W|}^2+{\displaystyle \frac{2R}{n}}{|W|}^2)\mathrm{d}{v}_g\hfill \\ \hfill & \le {\displaystyle \frac{4}{(n-2){\alpha }_0}}max\{8(n-1),n(n-2)\}{\int }_M{|W|}^3\mathrm{d}{v}_g.\hfill \end{array}$$

By the Hölder inequality, ${\int }_M{|W|}^3\mathrm{d}{v}_g\le {\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}{\parallel W\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2$; thus, we have

$$(1-{\displaystyle \frac{4}{(n-2){\alpha }_0}}{max\{8(n-1),n(n-2)\}\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}{)\parallel W\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2\le 0.$$

Since ${\parallel W\parallel }_{L^{\infty }}<{\tau }_0$, the Weyl tensor vanishes. □

3. Variation of ${\mathit{\sigma }}_{\mathit{k}}$ and Explicit Formula of $\mathit{d}{\mathit{\sigma }}_{\mathit{k}}^{*}$

In this section, our aim is to show the variation formula of ${\sigma }_k$-curvature. Just as in the definition of classic curvature variation, we consider $g\left(t\right)$ to be a one-parameter family of metrics on the n-dimensional closed Riemannian manifold $(M^n,g)$, with $g\left(t\right)=g+th$, and define ${\dot{g}}_{ij}=h_{ij}$. Firstly, we introduce several variational results for the Ricci curvature tensor and scalar curvature; details can be found in [6].

Proposition 2.

The variation of the Ricci curvature tensor is given by

$$\begin{array}{c}\hfill {\dot{R}}_{ij}=-{\displaystyle \frac{1}{2}}({\Delta }_L{h}_{ij}+{\nabla }_{ij}^2\left(\mathrm{tr}h\right)-{\nabla }_i{\nabla }_k{h}_j^k-{\nabla }_j{\nabla }_k{h}_i^k),\end{array}$$

(20)

where the Lichnerowicz Laplacian acting on h is defined as ${\Delta }_L{h}_{ij}=\Delta h_{ij}+2R_{kijl}{h}^{kl}-R_{ip}{h}_j^p-R_{jp}{h}_i^p$. The variation of the scalar curvature is

$$\begin{array}{c}\hfill \dot{R}=-h^{ij}{R}_{ij}-\Delta \left(\mathrm{tr}h\right)+{\nabla }_i{\nabla }_j{h}^{ij}.\end{array}$$

(21)

Based on the above formulae, we present the variational formula for the Schouten curvature.

Proposition 3.

The variational formula for the Schouten tensor is as follows:

$$\begin{array}{cc}\hfill {\dot{P}}_{ij}& =-{\displaystyle \frac{1}{2}}\Delta h_{ij}-W_{pijq}{h}^{pq}-{\displaystyle \frac{1}{n-2}}{E}_{ij}\left(\mathrm{tr}h\right)+{\displaystyle \frac{n}{n-2}}{E}_i^p{h}_{pj}+{\displaystyle \frac{R}{2(n-1)}}{h}_{ij}-{\displaystyle \frac{1}{2}}{\nabla }_{ij}^2\left(\mathrm{tr}h\right)\hfill \\ \hfill & +{\nabla }_{ip}^2{h}_j^p-[{\displaystyle \frac{1}{n-2}}{E}_{pq}{h}^{pq}+{\displaystyle \frac{R}{n(n-1)}}\left(\mathrm{tr}h\right)+{\displaystyle \frac{\dot{R}}{2(n-1)}}]g_{ij}.\hfill \end{array}$$

(22)

Proof. 

According to Equation (8), the variation of the Schouten tensor $P_{ij}$ is given by

$$\begin{array}{c}\hfill {\dot{P}}_{ij}=\dot{(E_{ij}+{\displaystyle \frac{(n-2)R}{2n(n-1)}}{g}_{ij})}={\dot{E}}_{ij}+{\displaystyle \frac{(n-2)}{2n(n-1)}}\dot{R}{g}_{ij}+{\displaystyle \frac{(n-2)R}{2n(n-1)}}{h}_{ij}.\end{array}$$

Furthermore, using Equations (8), (9), (20), and (21), we find the variation of the Einstein tensor $E_{ij}$ as follows:

$$\begin{array}{cc}\hfill {\dot{E}}_{ij}& ={\dot{R}}_{ij}-{\displaystyle \frac{\dot{R}}{n}}{g}_{ij}-{\displaystyle \frac{R}{n}}{\dot{g}}_{ij}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}(\Delta h_{ij}+2R_{pijq}{h}^{pq}-R_{ip}{h}_j^p-R_{jp}{h}_i^p)-{\displaystyle \frac{1}{2}}{\nabla }_{ij}^2\left(\mathrm{tr}h\right)+{\displaystyle \frac{1}{2}}({\nabla }_i{\nabla }_p{h}_j^p+{\nabla }_j{\nabla }_p{h}_i^p)\hfill \\ \hfill & -{\displaystyle \frac{\dot{R}}{n}}{g}_{ij}-{\displaystyle \frac{R}{n}}{h}_{ij}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}\Delta h_{ij}-[W_{pijq}{h}^{pq}+{\displaystyle \frac{1}{n-2}}{(E\wedge g)}_{pijq}{h}^{pq}+{\displaystyle \frac{R}{2n(n-1)}}{(g\wedge g)}_{pijq}{h}^{pq}]+{\displaystyle \frac{1}{2}}(E_{ip}\hfill \\ \hfill & +{\displaystyle \frac{R}{n}}{g}_{ip})h_j^p+{\displaystyle \frac{1}{2}}(E_{jp}+{\displaystyle \frac{R}{n}}{g}_{jp})h_i^p-{\displaystyle \frac{1}{2}}{\nabla }_{ij}^2\left(\mathrm{tr}h\right)+{\displaystyle \frac{1}{2}}{\nabla }_i{\nabla }_p{h}_j^p+{\displaystyle \frac{1}{2}}{\nabla }_j{\nabla }_p{h}_i^p-{\displaystyle \frac{\dot{R}}{n}}{g}_{ij}-{\displaystyle \frac{R}{n}}{h}_{ij}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}\Delta h_{ij}-W_{pijq}{h}^{pq}-{\displaystyle \frac{1}{n-2}}{E}_{ij}\left(\mathrm{tr}h\right)+{\displaystyle \frac{n}{n-2}}{E}_i^p{h}_{pj}+{\displaystyle \frac{R}{n(n-1)}}{h}_{ij}-{\displaystyle \frac{1}{2}}{\nabla }_{ij}^2\left(\mathrm{tr}h\right)\hfill \\ \hfill & +{\nabla }_{ip}^2{h}_j^p-[{\displaystyle \frac{1}{n-2}}{E}_{pq}{h}^{pq}+{\displaystyle \frac{R}{n(n-1)}}\left(\mathrm{tr}h\right)+{\displaystyle \frac{\dot{R}}{n}}]g_{ij}.\hfill \end{array}$$

Combining the above, we present the variational Formula (22). □

Note that both ${\sigma }_k$-curvature and the Newton tensor are defined based on the Schouten tensor; therefore, we can further calculate their variational formulae.

Proposition 4.

The variational formula for ${\sigma }_k$ is given by

$$\begin{array}{cc}\hfill \dot{{\sigma }_k}& ={\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}{\sigma }_{k-1}[{\nabla }_i{\nabla }_j{h}^{ij}-\Delta \left(\mathrm{tr}h\right)-h^{ij}{R}_{ij}]-{\displaystyle \frac{1}{2}}\Delta h_{ij}({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}\hfill \\ \hfill & -W_{pijq}{h}^{pq}({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}-{\displaystyle \frac{1}{n-2}}\left(\mathrm{tr}h\right)E_{ij}({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}+{\displaystyle \frac{2}{n-2}}{E}_i^p{h}_{pj}({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}\hfill \\ \hfill & +{\displaystyle \frac{R}{n(n-1)}}{h}_{ij}({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}-{\displaystyle \frac{1}{2}}{\nabla }_{ij}^2\left(\mathrm{tr}h\right)({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}+{\nabla }_i\left({\nabla }^l{h}_{jl}\right)({\stackrel{\circ }{T}}^{(k-1)}{)}^{ij}.\hfill \end{array}$$

(23)

Proof. 

The calculation is quite cumbersome, so we break it down into several steps.

Step one: Showing that

$$\begin{array}{c}\hfill \dot{{\sigma }_k}=-h^{jl}{P}_l^i{T}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{(T^{(k-1)})}^{il}.\end{array}$$

(24)

According to the Formula (11) for ${\sigma }_k$, taking the variation of ${\sigma }_k$ yields

$$\dot{{\sigma }_k}={\displaystyle \frac{1}{k!}}\sum \delta \left(\begin{array}{ccc}{i}_1& \dots & i_k\\ j_1& \dots & j_k\end{array}\right)P_{i_1}^{j_1}\cdots \dot{\left(P_{i_t}^{j_t}\right)}\cdots P_{i_k}^{j_k},$$

where the results of taking the variation on each of the k $P_i^j$ are equal. Furthermore, due to the definition of the Kronecker symbol, moving $i_t$ and $j_t$ to the last positions in the two rows requires an even number of permutations; thus, the sign remains unchanged after the permutation. Therefore, using the formula for the Newton tensor, we obtain

$$\dot{{\sigma }_k}={\displaystyle \frac{1}{(k-1)!}}\sum \delta \left(\begin{array}{cccc}{i}_1& \dots & i_k& i_t\\ j_1& \dots & j_k& j_t\end{array}\right)P_{i_1}^{j_1}\cdots \dot{\left(P_{i_t}^{j_t}\right)}\cdots P_{i_k}^{j_k}={\left(T^{(k-1)}\right)}_{j_t}^{i_t}\dot{\left(P_{i_t}^{j_t}\right)}.$$

Thus,

$$\dot{{\sigma }_k}=\dot{\left(P_i^j\right)}{\left(T^{(k-1)}\right)}_j^i=\dot{\left(g^{jl}{P}_{il}\right)}{\left(T^{(k-1)}\right)}_j^i=-h^{jl}{P}_l^i{T}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{\left(T^{(k-1)}\right)}^{il}.$$

Step two: Showing that

$$\begin{array}{c}\hfill \dot{{\sigma }_k}=-h^{jl}{P}_l^i{\stackrel{\circ }{T}}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{({\stackrel{\circ }{T}}^{(k-1)})}^{il}+{\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}{\sigma }_{k-1}\left(\dot{R}\right).\end{array}$$

(25)

Recall Proposition 1 for the Newton tensor; the variation of ${\sigma }_k$ can be expressed as

$$\begin{array}{cc}\hfill \dot{{\sigma }_k}& =-h^{jl}{P}_l^i{T}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{\left(T^{(k-1)}\right)}^{il}\hfill \\ \hfill & =-h^{jl}{P}_l^i({\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}_{ij})+\dot{\left(P_{il}\right)}[{({\stackrel{\circ }{T}}^{(k-1)})}^{il}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}^{il}]\hfill \\ \hfill & =-h^{jl}{P}_l^i{\stackrel{\circ }{T}}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{({\stackrel{\circ }{T}}^{(k-1)})}^{il}-{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{P}_{ij}{h}^{ij}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}[\dot{\left(P_{il}{g}^{il}\right)}-P_{il}\dot{g^{il}}]\hfill \\ \hfill & =-h^{jl}{P}_l^i{\stackrel{\circ }{T}}_{ij}^{(k-1)}+\dot{\left(P_{il}\right)}{\left({\stackrel{\circ }{T}}^{(k-1)}\right)}^{il}+{\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}{\sigma }_{k-1}\left(\dot{R}\right).\hfill \end{array}$$

Step three: Showing Formula (23). In fact, this is derived from Equations (21), (22) and (25), and we finish the proof. □

Corollary 1.

The explicit formula of $d{\sigma }_k^{*}\left(f\right)$ is given by

$$\begin{array}{cc}\hfill & d{\sigma }_k^{*}{\left(f\right)}_{ij}={\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}{\sigma }_{k-1}({\nabla }_i{\nabla }_{j}f-\Delta f{g}_{ij}-R_{ij}f)-{\displaystyle \frac{1}{2}}\Delta (f{\stackrel{\circ }{T}}_{ij}^{(k-1)})\hfill \\ \hfill & -f{W}_{pijq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}-{\displaystyle \frac{1}{n-2}}f{E}_{pq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}{g}_{ij}+{\displaystyle \frac{2}{n-2}}f{E}_i^p{\stackrel{\circ }{T}}_{pj}^{(k-1)}+{\displaystyle \frac{R}{n(n-1)}}f{\stackrel{\circ }{T}}_{ij}^{(k-1)}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\nabla }_{pq}^2(f{({\stackrel{\circ }{T}}^{(k-1)})}^{pq})g_{ij}+{\nabla }_i({\nabla }^l(f{\stackrel{\circ }{T}}_{jl}^{(k-1)}))+{\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}(f{\nabla }_{ij}^2{\sigma }_{k-1}-f\Delta {\sigma }_{k-1}{g}_{ij}).\hfill \end{array}$$

(26)

Proof. 

This is a derived result from the variational formula of the ${\sigma }_k$-curvature, according to the equation ${\langle d{\sigma }_k\left(h\right),f\rangle }_{L^2}={\langle d{\sigma }_k^{*}\left(f\right),h\rangle }_{L^2}$. Notice that the manifold is closed; by applying the divergence theorem, one can compare each term to obtain the formula. □

Furthermore, when ${\sigma }_{k-1}$ is constant, letting $f=1$, one can deduce the following:

$$\begin{array}{cc}\hfill & d{\sigma }_k^{*}{\left(1\right)}_{ij}=-{\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}{\sigma }_{k-1}{R}_{ij}-{\displaystyle \frac{1}{2}}\Delta ({\stackrel{\circ }{T}}_{ij}^{(k-1)})-W_{pijq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}\hfill \\ \hfill & +{\displaystyle \frac{2}{n-2}}{E}_i^p{\stackrel{\circ }{T}}_{pj}^{(k-1)}+{\displaystyle \frac{R}{n(n-1)}}{\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\nabla }_i({\nabla }^l{\stackrel{\circ }{T}}_{jl}^{(k-1)})-{\displaystyle \frac{1}{n-2}}{E}_{pq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}{g}_{ij}\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\nabla }_{pq}^2{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}{g}_{ij}.\hfill \end{array}$$

(27)

4. ${\mathit{\sigma }}_{\mathit{k}}$-Einstein Manifolds

Definition 4.

An n-dimensional closed Riemannian manifold $(M^n,g)$ with $n\ge 3$ is said to be ${\sigma }_k$-Einstein if it satisfies the following equation:

$$\begin{array}{c}\hfill d{\sigma }_k^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{k{\sigma }_k}{n}}{g}_{ij}-{\displaystyle \frac{n-2}{2(n-1)}}{\stackrel{\circ }{T}}_{ij}^{\left(k\right)}=0.\end{array}$$

(28)

In this paper, the definition of a ${\sigma }_k$-Einstein manifold is given because, for Equation (28), when $k=1$, it is naturally established; when $k=2$, it yields a result approximately equivalent to Bach flatness.

Proof. 

When $k=1$, for the scalar curvature R, the variational Formula (21) easily yields $d{R}^{*}{\left(1\right)}_{ij}=-R_{ij}$, and from ${\sigma }_1=trP={\displaystyle \frac{n-2}{2(n-1))}}R$, we can derive

$$d{\sigma }_1^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{{\sigma }_1}{n}}{g}_{ij}=-{\displaystyle \frac{n-2}{2(n-1)}}{R}_{ij}+{\displaystyle \frac{n-2}{2n(n-1)}}R{g}_{ij}=-{\displaystyle \frac{n-2}{2(n-1)}}{E}_{ij}.$$

By definition, we have

$${\stackrel{\circ }{T}}_{ij}^{\left(1\right)}={\sigma }_1{g}_{ij}-P_{ij}-{\displaystyle \frac{n-1}{n}}{\sigma }_1{g}_{ij}=-{\displaystyle \frac{n-2}{2(n-1)}}{E}_{ij}=d{\sigma }_1^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{{\sigma }_1}{n}}{g}_{ij}.$$

This gives us

$$d{\sigma }_1^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{{\sigma }_1}{n}}{g}_{ij}-{\displaystyle \frac{n-2}{2(n-1)}}{\stackrel{\circ }{T}}_{ij}^{\left(1\right)}=0.$$

When $k=2$, by definition, we can separately derive

$$\begin{array}{cc}\hfill {\sigma }_2& =\sum _{i\ne j}{\lambda }_i{\lambda }_j={\displaystyle \frac{1}{2}}[{(\sum {\lambda }_i)}^2-\sum {\left({\lambda }_i\right)}^2]={\displaystyle \frac{1}{2}}(t{r}^{2}P-P_{ij}{P}^{ij})\hfill \\ \hfill & ={\displaystyle \frac{{(n-2)}^2}{8{(n-1)}^2}}{R}^2-{\displaystyle \frac{1}{2}}{(|Ric|}^2-{\displaystyle \frac{R^2}{(n-1)}}+{\displaystyle \frac{n}{4{(n-1)}^2}}{R}^2)={\displaystyle \frac{{(n-2)}^2}{8n(n-1)}}{R}^2-{\displaystyle \frac{1}{2}}{|E|}^2.\hfill \end{array}$$

$$\begin{array}{cc}\hfill & {\stackrel{\circ }{T}}_{ij}^{\left(2\right)}=T_{ij}^{\left(2\right)}-{\displaystyle \frac{n-2}{n}}{\sigma }_2{g}_{ij}={\sigma }_2{g}_{ij}-{\sigma }_1{P}_{ij}+P_i^l{P}_{lj}-{\displaystyle \frac{n-2}{n}}{\sigma }_2{g}_{ij}\hfill \\ \hfill & ={\displaystyle \frac{2}{n}}{\sigma }_2{g}_{ij}-{\displaystyle \frac{n-2}{2(n-1)}}R(E_{ij}+{\displaystyle \frac{n-2}{2n(n-1)}}R{g}_{ij})+(E_i^l+{\displaystyle \frac{n-2}{2n(n-1)}}R{g}_i^l)(E_{lj}+{\displaystyle \frac{n-2}{2n(n-1)}}R{g}_{lj})\hfill \\ \hfill & ={\displaystyle \frac{2}{n}}{\sigma }_2{g}_{ij}-{\displaystyle \frac{{(n-2)}^2}{2n(n-1)}}R{E}_{ij}-{\displaystyle \frac{{(n-2)}^2}{4n^2(n-1)}}{R}^2{g}_{ij}+E_i^l{E}_{lj}.\hfill \end{array}$$

Furthermore, from reference [3], we obtain the formula for the Bach tensor:

$$\begin{array}{cc}\hfill B_{ij}& ={\displaystyle \frac{1}{n-2}}\Delta E_{ij}-{\displaystyle \frac{1}{2(n-1)}}({\nabla }_{ij}^{2}R-{\displaystyle \frac{1}{n}}\Delta R{g}_{ij})+{\displaystyle \frac{2}{n-2}}{W}_{ipqj}{E}^{pq}-{\displaystyle \frac{n}{{(n-2)}^2}}{E}_{ip}{E}_j^p\hfill \\ \hfill & +{\displaystyle \frac{1}{{(n-2)}^2}}{|E|}^2{g}_{ij}-{\displaystyle \frac{R}{(n-1)(n-2)}}{E}_{ij}.\hfill \end{array}$$

Additionally, taking the variation of ${\sigma }_2$ easily yields

$$\begin{array}{cc}\hfill d{\sigma }_2^{*}{\left(1\right)}_{ij}& ={\displaystyle \frac{1}{2}}\Delta R_{ij}+R_{ipqj}{R}^{pq}-{\displaystyle \frac{1}{4(n-1)}}\Delta R{g}_{ij}+{\displaystyle \frac{2-n}{4(n-1)}}{\nabla }_{ij}^{2}R-{\displaystyle \frac{n}{4(n-1)}}{R}_{ij}R\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\Delta E_{ij}+{\displaystyle \frac{1}{2n}}\Delta R{g}_{ij}+W_{ipqj}{E}^{pq}+{\displaystyle \frac{1}{n-2}}(E_{ij}{g}_{pq}+E_{pq}{g}_{ij}-E_{iq}{g}_{pj}-E_{pj}{g}_{iq})E^{pq}\hfill \\ \hfill & +{\displaystyle \frac{R}{n(n-1)}}(g_{ij}{g}_{pq}-g_{iq}{g}_{pj})E^{pq}+(E_{ij}+{\displaystyle \frac{R}{n}}{g}_{ij}){\displaystyle \frac{R}{n}}-{\displaystyle \frac{1}{4(n-1)}}\Delta R{g}_{ij}+{\displaystyle \frac{2-n}{4(n-1)}}{\nabla }_{ij}^{2}R\hfill \\ \hfill & -{\displaystyle \frac{n}{4(n-1)}}{E}_{ij}R-{\displaystyle \frac{1}{4(n-1)}}{R}^2{g}_{ij}\hfill \\ \hfill & ={\displaystyle \frac{1}{2}}\Delta E_{ij}+{\displaystyle \frac{n-2}{4n(n-1)}}\Delta R{g}_{ij}+W_{ipqj}{E}^{pq}+{\displaystyle \frac{1}{n-2}}{|E|}^2{g}_{ij}-{\displaystyle \frac{2}{n-2}}{E}_{iq}{E}_j^q\hfill \\ \hfill & -{\displaystyle \frac{{(n-2)}^2+4}{4n(n-1)}}{E}_{ij}R-{\displaystyle \frac{{(n-2)}^2}{4n^2(n-1)}}{R}^2{g}_{ij}+{\displaystyle \frac{2-n}{4(n-1)}}{\nabla }_{ij}^{2}R.\hfill \end{array}$$

Thus, combining the above, we can derive

$$d{\sigma }_2^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{2{\sigma }_2}{n}}{g}_{ij}-{\displaystyle \frac{n-2}{2}}{B}_{ij}={\displaystyle \frac{n-4}{2(n-2)}}{\stackrel{\circ }{T}}_{ij}^{\left(2\right)}.$$

That is,

$$d{\sigma }_2^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{2{\sigma }_2}{n}}{g}_{ij}-{\displaystyle \frac{n-2}{2(n-1)}}{\stackrel{\circ }{T}}_{ij}^{\left(2\right)}={\displaystyle \frac{n-2}{2}}{B}_{ij}-{\displaystyle \frac{n}{2(n-1)(n-2)}}{\stackrel{\circ }{T}}_{ij}^{\left(2\right)}=0,$$

which is approximately Bach flat. □

5. ${\mathit{L}}^{\infty }$-Sphere Theorem

Now, we utilize the definitions and properties previously discussed to establish the proof of the $L^{\infty }$-sphere theorem.

Theorem 7.

Let $(M^n,g)$ be a closed ${\sigma }_k$-Einstein manifold in the positive light cone with $n\ge 3$, where ${\sigma }_{k-1}$ is a constant. If ${\left[T^{(k-1)}\right]}_{ij,}^j=0$, and the following inequality holds:

$$\begin{array}{c}\hfill {\parallel W\parallel }_{L^{\infty }}+{\displaystyle \frac{n(2n-1)}{3(n-1)(n-2)}}{\parallel E\parallel }_{L^{\infty }}<{ϵ}_0:=min\{{\Lambda }_n,{\displaystyle \frac{R}{4n}}\}={\displaystyle \frac{R}{4n}},\end{array}$$

(29)

where ${\Lambda }_n={\displaystyle \frac{3n-2}{6{(n-1)}^2}}R$; then, the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

Proof. 

From the definition of $T_{ij}^{\left(k\right)}$, we have

$$\begin{array}{cc}\hfill {\stackrel{\circ }{T}}_{ij}^{\left(k\right)}& =T_{ij}^{\left(k\right)}-{\displaystyle \frac{n-k}{n}}{\sigma }_k{g}_{ij}=({\sigma }_k{g}_{ij}-T_{il}^{(k-1)}{P}_j^l)-{\displaystyle \frac{n-k}{n}}{\sigma }_k{g}_{ij}\hfill \\ \hfill & ={\displaystyle \frac{k}{n}}{\sigma }_k{g}_{ij}-{\stackrel{\circ }{T}}_{il}^{(k-1)}{P}_j^l-{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{P}_{ij}\hfill \\ \hfill & =-{\stackrel{\circ }{T}}_{il}^{(k-1)}{E}_j^l-{\displaystyle \frac{n-2}{2n(n-1)}}{\stackrel{\circ }{T}}_{ij}^{(k-1)}R-{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{R}_{ij}+{\displaystyle \frac{1}{n}}[k{\sigma }_k+{\displaystyle \frac{(n-k+1)R{\sigma }_{k-1}}{2(n-1)}}]g_{ij}.\hfill \end{array}$$

Furthermore, according to Equation (27), where ${\sigma }_{k-1}$ is a constant, we obtain

$$\begin{array}{cc}\hfill 0& =d{\sigma }_k^{*}{\left(1\right)}_{ij}+{\displaystyle \frac{k{\sigma }_k}{n}}{g}_{ij}-{\displaystyle \frac{n-2}{2(n-1)}}{\stackrel{\circ }{T}}_{ij}^{\left(k\right)}\hfill \\ \hfill & =-{\displaystyle \frac{1}{2}}(\Delta {\stackrel{\circ }{T}}_{ij}^{(k-1)})-W_{pijq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}+[{\displaystyle \frac{2}{n-2}}+{\displaystyle \frac{n-2}{2(n-1)}}]E_i^p{\stackrel{\circ }{T}}_{pj}^{(k-1)}+[{\displaystyle \frac{1}{n(n-1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(n-2)}^2}{4n{(n-1)}^2}}]R{\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\nabla }_i({\nabla }^l{\stackrel{\circ }{T}}_{jl}^{(k-1)})+[{\displaystyle \frac{k{\sigma }_k}{2(n-1)}}-{\displaystyle \frac{(n-2)(n-k+1)}{4n{(n-1)}^2}}R{\sigma }_{k-1}\hfill \\ \hfill & -{\displaystyle \frac{1}{n-2}}{E}_{pq}{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}-{\displaystyle \frac{1}{2}}{\nabla }_{pq}^2{({\stackrel{\circ }{T}}^{(k-1)})}^{pq}]g_{ij}.\hfill \end{array}$$

By contracting both sides with ${\stackrel{\circ }{T}}_{ij}^{(k-1)}$ and integrating, we obtain

$$\begin{array}{c}\hfill 0={\int }_M[{\displaystyle \frac{1}{2}}|\nabla \stackrel{\circ }{T}{|}^2-W(\stackrel{\circ }{T},\stackrel{\circ }{T})+{\displaystyle \frac{n^2}{2(n-1)(n-2)}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})+{\displaystyle \frac{n}{4{(n-1)}^2}}R|\stackrel{\circ }{T}{|}^2-|\delta \stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.\end{array}$$

Since $[T^{(k-1)}{]}_{ij,}^j=0$, and

$$T_{ij}^{(k-1)}={\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}_{ij},$$

when ${\sigma }_{k-1}=const$, it follows that $|\delta \stackrel{\circ }{T}{|}^2=0$.

Combining Lemma 1’s Equation (15), we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{1}{2}}{\int }_M|\nabla \stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g& ={\int }_M[W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n^2}{2(n-1)(n-2)}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{4{(n-1)}^2}}R|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g\hfill \\ \hfill & \ge -{\displaystyle \frac{1}{2}}{\int }_M[W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{n-2}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{R}{n-1}}|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.\hfill \end{array}$$

After rearranging, we obtain

$$\begin{array}{c}\hfill {\int }_M[{\displaystyle \frac{3}{2}}W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n(2n-1)}{2(n-1)(n-2)}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{3n-2}{4{(n-1)}^2}}R|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g\ge 0.\end{array}$$

Furthermore, since

$$\begin{array}{c}\hfill {\int }_{M}W(\stackrel{\circ }{T},\stackrel{\circ }{T})\mathrm{d}{v}_g\le {\parallel W\parallel }_{L^{\infty }}{\int }_M{|\stackrel{\circ }{T}|}^2\mathrm{d}{v}_g,\end{array}$$

and

$$\begin{array}{c}\hfill {\int }_{M}E(\stackrel{\circ }{T},\stackrel{\circ }{T})\mathrm{d}{v}_g\le {\parallel E\parallel }_{L^{\infty }}{\int }_M{|\stackrel{\circ }{T}|}^2\mathrm{d}{v}_g,\end{array}$$

we arrive at the inequality

$$({\displaystyle \frac{3}{2}}{\parallel W\parallel }_{L^{\infty }}+{\displaystyle \frac{n(2n-1)}{2(n-1)(n-2)}}{\parallel E\parallel }_{L^{\infty }}-{\displaystyle \frac{3n-2}{4{(n-1)}^2}}R){\int }_M|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g\ge 0.$$

Therefore, when the inequality

$${\parallel W\parallel }_{L^{\infty }}+{\displaystyle \frac{n(2n-1)}{3(n-1)(n-2)}}{\parallel E\parallel }_{L^{\infty }}<{\Lambda }_n={\displaystyle \frac{3n-2}{6{(n-1)}^2}}R$$

is satisfied, this implies that $\stackrel{\circ }{T}=0$.

Thus, we have $T_{ij}^{(k-1)}={\stackrel{\circ }{T}}_{ij}^{(k-1)}+{\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}_{ij}={\displaystyle \frac{n-k+1}{n}}{\sigma }_{k-1}{g}_{ij}$. Taking the inner product with $P_{ij}$ on both sides, we obtain

$$k{\sigma }_k={\displaystyle \frac{(n-k+1)(n-2)}{2n(n-1)}}R{\sigma }_{k-1}=(n-k+1)\Lambda {\sigma }_{k-1}.$$

According to Lemma 2, when the equality holds, we have ${\lambda }_1=\cdots ={\lambda }_k=\lambda \ge 0$; hence, $P_{ij}=\lambda g_{ij}$, which implies $E_{ij}=0$. Therefore, the manifold is an Einstein manifold.

At this point, since the manifold is an Einstein manifold, according to Formula (29), we obtain ${\parallel W\parallel }_{L^{\infty }}<{ϵ}_0={\displaystyle \frac{R}{4n}}$. According to Lemma 6, the Weyl tensor vanishes. Therefore, the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$. □

6. ${\mathit{L}}^{{\displaystyle \frac{\mathit{n}}{\mathbf{2}}}}$-Sphere Theorem

Theorem 8.

Let $(M^n,g)$ be a closed ${\sigma }_k$-Einstein manifold in the positive light cone with $n\ge 3$, where ${\sigma }_{k-1}$ is a constant. Assume that the Yamabe constant has a lower bound satisfying $Y(M,\left[g\right])\ge {\alpha }_0>0$. If ${\left[T^{(k-1)}\right]}_{ij,}^j=0$ and the following inequality holds

$$\begin{array}{c}\hfill {\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}+{\displaystyle \frac{n^2}{2(n-1)(n-2)}}{\parallel E\parallel }_{L^{{\displaystyle \frac{n}{2}}}}\le min\{{\delta }_0,{\tau }_0\}={\tau }_0,\end{array}$$

(30)

where ${\delta }_0={\displaystyle \frac{(n-2){\alpha }_0}{8(n-1)}}min\{1,{\displaystyle \frac{2n}{(n-1)(n-2)}}\}$ and ${\tau }_0={\displaystyle \frac{(n-2){\alpha }_0}{4max\left\{8\right(n-1),n(n-2\left)\right\}}}$, then the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$.

Proof. 

As in Theorem 7, we derive the equation

$$\begin{array}{c}\hfill {\int }_M|\nabla \stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g={\int }_M[2W(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n^2}{(n-1)(n-2)}}E(\stackrel{\circ }{T},\stackrel{\circ }{T})-{\displaystyle \frac{n}{2{(n-1)}^2}}R|\stackrel{\circ }{T}{|}^2]\mathrm{d}{v}_g.\end{array}$$

By the Hölder inequality, we have

$${\int }_{M}W(\stackrel{\circ }{T},\stackrel{\circ }{T})\mathrm{d}{v}_g\le {\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}·{\parallel |T|}^2{\parallel }_{L^{{\displaystyle \frac{n}{n-2}}}}\le {\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}·{\parallel \stackrel{\circ }{T}\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2,$$

$${\int }_{M}E(\stackrel{\circ }{T},\stackrel{\circ }{T})\mathrm{d}{v}_g\le {\parallel E\parallel }_{L^{{\displaystyle \frac{n}{2}}}}·{\parallel \stackrel{\circ }{T}\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2.$$

Thus, we have

$$\begin{array}{cc}\hfill {\int }_M{|\nabla \stackrel{\circ }{T}|}^2\mathrm{d}{v}_g& \le {(2\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}+{\displaystyle \frac{n^2}{(n-1)(n-2)}}{\parallel E\parallel }_{L^{{\displaystyle \frac{n}{2}}}})·\parallel \stackrel{\circ }{T}{\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2-{\displaystyle \frac{nR}{2{(n-1)}^2}}{\int }_M|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g\hfill \\ \hfill & \le 2{\delta }_0\parallel \stackrel{\circ }{T}{\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2-{\displaystyle \frac{n}{2{(n-1)}^2}}{\int }_{M}R{|\stackrel{\circ }{T}|}^2\mathrm{d}{v}_g.\hfill \end{array}$$

This gives us the formula

$$\begin{array}{c}\hfill \parallel \stackrel{\circ }{T}{\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2\ge {\displaystyle \frac{1}{2{\delta }_0}}(\parallel \nabla \stackrel{\circ }{T}{\parallel }_{L^2}^2+{\displaystyle \frac{n}{2{(n-1)}^2}}{\int }_{M}R|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g).\end{array}$$

(31)

Since $Y(M,\left[g\right])\ge {\alpha }_0>0$, taking $u=|\stackrel{\circ }{T}|$ in Equation (15) from Lemma 1, we obtain

$$\begin{array}{c}\hfill \parallel \stackrel{\circ }{T}{\parallel }_{L^{{\displaystyle \frac{2n}{n-2}}}}^2\le C_S[\parallel \nabla \stackrel{\circ }{T}{\parallel }_{L^2}^2+{\displaystyle \frac{n-2}{4(n-1)}}{\int }_{M}R|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g],\end{array}$$

(32)

where the Kato inequality is used, i.e., $\parallel \nabla |\stackrel{\circ }{T}{|\parallel }_{L^n}^2\le {\parallel \nabla \stackrel{\circ }{T}\parallel }_{L^n}^2$.

Combining Equations (31) and (32), we obtain

$$\begin{array}{c}\hfill C_S[\parallel \nabla \stackrel{\circ }{T}{\parallel }_{L^2}^2+{\displaystyle \frac{n-2}{4(n-1)}}{\int }_{M}R|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g]\ge {\displaystyle \frac{1}{2{\delta }_0}}(\parallel \nabla \stackrel{\circ }{T}{\parallel }_{L^2}^2+{\displaystyle \frac{n}{2{(n-1)}^2}}{\int }_{M}R|\stackrel{\circ }{T}{|}^2\mathrm{d}{v}_g).\end{array}$$

That is,

$$\begin{array}{c}\hfill ({\displaystyle \frac{1}{2{\delta }_0}}-C_S)\parallel \nabla \stackrel{\circ }{T}{\parallel }_{L^2}^2+({\displaystyle \frac{n}{4{\delta }_0{(n-1)}^2}}-{\displaystyle \frac{n-2}{4(n-1)}}{C}_S){\int }_{M}R{|\stackrel{\circ }{T}|}^2\mathrm{d}{v}_g\le 0.\end{array}$$

(33)

By the condition ${\delta }_0={\displaystyle \frac{(n-2){\alpha }_0}{8(n-1)}}min\{1,{\displaystyle \frac{2n}{(n-1)(n-2)}}\}$, two coefficients ${\displaystyle \frac{1}{{\delta }_0}}-C_S$ and ${\displaystyle \frac{n}{4{\delta }_0{(n-1)}^2}}-{\displaystyle \frac{n-2}{4(n-1)}}{C}_S$ are both positive, which shows that $\stackrel{\circ }{T}=0$. Therefore, the manifold is an Einstein manifold.

At this point, since the manifold is an Einstein manifold, according to Formula (30), we can obtain ${\parallel W\parallel }_{L^{{\displaystyle \frac{n}{2}}}}<{\tau }_0={\displaystyle \frac{(n-2){\alpha }_0}{4max\left\{8\right(n-1),n(n-2\left)\right\}}}$. According to Lemma 5, the Weyl tensor vanishes. Therefore, the manifold is isometric to a quotient of the round sphere ${\mathbb{S}}^n$. □

7. Remarks

It is worth noting that, when $k=2$ in Theorem 7 and 8, these two sphere theorems are consistent with the conclusions from Fang-Yuan’s results [3] by adding an additional ${\displaystyle \frac{n}{2(n-1)(n-2)}}{\stackrel{\circ }{T}}_{ij}^{\left(2\right)}$-term.

A well-known conjecture is that any Bach flat manifold is also Einstein. Inspired by the relation between ${\sigma }_2$-Einstein and Bach flat, we assume that any ${\sigma }_k$-Einstein manifold is also Einstein.