The Lichnerowicz Laplacian Acting on Symmetric Tensor Fields — The Bochner Technique Point of View

Abstract

In this paper, we prove vanishing theorems for the null space of the Laplacian admitting the Weizenböck decomposition and acting on the space of smooth sections of a Riemannian bundle and, in particular, the space of smooth sections of a bundle of symmetric tensors over complete and closed Riemannian manifolds. In addition, we give a general estimate for eigenvalues of the Laplacian admitting the Weizenböck decomposition and acting on the space of smooth sections of a bundle of symmetric tensors over closed Riemannian manifolds.

The Bochner technique is considered part

of the basis vocabulary of every geometer.

Wu H.-H.

Introduction

The Lichnerowicz Laplacian acting on smooth sections of a tensor bundle over a Riemannian manifold differs from the usual Laplacian acting on functions by the Weitzenböck decomposition formula involving the Riemann curvature tensor (see, for example, ([14], p. 344); ([4], pp. 54-58); [11]). The Laplacian of this type is the simplest elliptic operator and is at the core of Hodge theory as well as the results of de Rham cohomology.

On the other hand, one of the oldest and most important techniques in modern Riemannian geometry is that of the Bochner technique (see, for example, ([14], pp. 333-364); [15, 19]). Furthermore, in the well-known monograph ([4], p. 53) the following was written: the Bochner technique is a method of proving vanishing theorems for null space of a Laplace operator admitting a Weitzenböck decomposition and further of estimating its lowest eigenvalue. Therefore, the Bochner technique has a broad scope of applications.

Our paper is structured as follows. The second section of the article discusses the basic information about the Laplacian admitting the Weizenböck decomposition and acting on the space of smooth sections of a Riemannian bundle (see ([15], p.104) and [9]). In particular, we prove several new vanishing theorems for the null space of the Laplace operator acting on these sections and obtain new estimates for its smallest eigenvalue. As an application of the obtained results, in the third section of the article, we consider the kernel and give a general estimate for eigenvalues of the Laplacian admitting the Weizenböck decomposition and acting on traceless q-tensors (special cases were considered in [6]). In addition, in these paper sections, we will show how the Bochner technique works for these operators. This topic is of exceptional importance (see, for example, [13] and its References).

In the presented paper, we will continue the research begun in our article [17].

Preliminary information about a Laplace operator admitting a Weizenböck decomposition

2.1. Let (M, g) be an n-dimensional connected complete Riemannian manifold with the Levi-Civita connection $\nabla$. Further, let E be a Riemannian fiber bundle of rank r over (M, g), i.e., a fiber bundle equipped with a scalar product and a compatible connection (see ([2], p. 378)). At the same time, the scalar product and compatible connection of E (as well as of M) will be denoted by the same symbols g and $\nabla$. Let P(TM) denote the principal O(n)-bundle of orthonormal frames on (M, g) or, in particular, SO(n)-bundle if (M, g) has an orientation. In this case, we can consider $E \rightarrow M$ as a Riemannian vector bundle, associated with P(TM) via an orthogonal representation $\rho \,{:}\ O(n)\rightarrow O(E)$.

Moreover, we can define the $L^{2}$-global scalar product on $C^{\infty }$-sections of E by the formula $\langle \theta ,\eta \rangle ={\int _{M}} g(\theta ,\eta ){d\nu _{g}}$ for $\theta ,\eta \in {C^{\infty }(E)}$ and we introduce the associated Hilbert space $L^{2}(E)$. Then, using the $L^{2}$-structures on ${C^{\infty }(E)}$, we can define the Bochner (or connection) Laplacian by the formula $\bar{\Delta }=\nabla ^*\nabla$ where $\nabla ^*$ is the self-adjoint operator with respect to the $L^{2}$-global scalar product of the compatible connection $\nabla$ : ${C^{\infty }(E)} \rightarrow C^{\infty }(T^*M\otimes E)$ (see ([14], p. 378) and ([2], p. 378)). Due to the above definitions and notations, we can define the natural Laplacian $\Delta _E$: ${C^{\infty }(E)}\rightarrow {C^{\infty }(E)}$ which satisfies the Weitzenböck decomposition formula (see [9])

$$\begin{aligned} \Delta _E=\bar{\Delta }+t\ \mathfrak {R}, \end{aligned}$$

(1)

where $t\in \mathbb {R}$ is a suitable constant and $\mathfrak {R}$ is the Weitzenböck curvature operator that defines a symmetric endomorphism $\mathfrak {R}_x$: $E_x\rightarrow E_x$ for all $x\in M$ (see also ([2], p. 378-379)). The Weitzenböck curvature operator $\mathfrak {R}$ is defined as follows (see [9]): the Riemann curvature tensor R of (M, g) at each its point lies in Sym$^2 (\mathfrak {s}o(n))$ for the Lie algebra $\mathfrak {s}o(n) \cong \Lambda ^2 \mathbb {R}^n$. Applying the representation $\rho \,{:}\ \mathfrak {s}o(n)\rightarrow \text {End}\, E$, we get the Weitzenböck curvature operator $\mathfrak {R}$ in Sym$^2 (\text {End}\, E)$ and then composition of endomorphisms gives a self-adjoint endomorphism of E.

Various geometrical problems give rise to different values of $t\in \mathbb {R}$, (see [9]). Namely, the Laplacian $\Delta _E$ is a second-order elliptic linear differential operator on ${C^{\infty }(E)}$ which is symmetric with respect to the $L^{2}$-global scalar product. On a closed (i.e., compact without boundary) manifold (M, g), for fixed $t\in \mathbb {R}$, we have the orthogonal (with respect to the $L^{2}$-global scalar product) decomposition formula (see ([2], p. 464))

$$\begin{aligned} {C^{\infty }(E)}=\ker \Delta _E\oplus \textrm{Im}\Delta _E, \end{aligned}$$

(2)

where the first component $\ker \Delta _E$ is the kernel of the Laplacian $\Delta _E$. Since $\Delta _E$ is elliptic, the dimension of the kernel of $\Delta _E$ is finite. In accordance with ([15], p. 104), its smooth sections will be called $\Delta _E$ -harmonic. Moreover, we define the vector space of $\Delta _E$-harmonic $C^{\infty }$-sections of $E\rightarrow M$ by $\ker \Delta _E=\{\varphi \in {C^{\infty }(E)}{:}\ \Delta _E\, \varphi =0\}$ for an arbitrary $t\ne 0$. On the other hand, for $t=0$ any smooth $\Delta _E$-harmonic section satisfies the equation $\nabla \varphi =0$.

If (M, g) is a closed manifold, then multiplying the Weitzenböck formula (1) by $\varphi$ and integrating over M, gives (see also ([2], p. 389))

$$\begin{aligned} \begin{array}{l} {\int _{M}} g(\Delta _E\varphi ,\varphi ){d\nu _{g}}= {\int _{M}} \{g(\bar{\Delta }\varphi ,\varphi )+t\,g(\mathfrak {R}(\varphi ),\varphi )\}{d\nu _{g}}= \\ {\int _{M}} \Vert \nabla \varphi \Vert ^2{d\nu _{g}} +t{\int _{M}} g(\mathfrak {R}(\varphi ),\varphi )\}{d\nu _{g}} \ge t{\int _{M}} g(\mathfrak {R}(\varphi ),\varphi )\}{d\nu _{g}} , \end{array} \end{aligned}$$

(3)

where $\varphi \in {C^{\infty }(E)}$. In this case, if $t>0$ and $\mathfrak {R}$ is non-negative endomorphism at each point of M, then the right side of inequality (3) is non-negative. Hence $\Delta _E\varphi =0$ implies both $\nabla \varphi =0$ and $\mathfrak {R}(\varphi )=0$. Furthermore, if $\mathfrak {R}$ is strictly positive at some point of M, then clearly $\varphi =0$. Therefore, we can formulate the following theorem.

Theorem 2.1

Let $\Delta _E$ be the Laplacian on $C^{\infty }$-sections of a Riemannian fiber bundle $E\rightarrow M$ of rank r over a closed Riemannian manifold (M, g), satisfying (1).

• (i) If $t>0$ and the Weitzenböck curvature operator $\mathfrak {R}$ satisfies $\mathfrak {R}\ge 0$, then $\nabla \varphi =0$ for an arbitrary $\varphi \in \ker \Delta _E$ and $\dim _{\mathbb R}\ker \Delta _E\le r$. In particular, if $\mathfrak {R}\ge 0$, and there exists an x in M such that $\mathfrak {R}>0$, then $\dim _{\mathbb R}\ker \Delta _E=0$.

• (ii) On the other hand, if $t<0$ and the Weitzenböck curvature operator $\mathfrak {R}$ satisfies $\mathfrak {R}\le 0$, then $\nabla \varphi =0$ for an arbitrary $\varphi \in \ker \Delta _E$ and $\dim _{\mathbb R}\ker \Delta _L\le r$. In particular, if $\mathfrak {R}\le 0$, and there exists an x in M such that $\mathfrak {R}<0$, then $\dim _{\mathbb R}\ker \Delta _E=0$.

The Weitzenböck curvature operator $\mathfrak {R}$ has the following representation (see ([5], p. 1214), [9])

$$\begin{aligned} \mathfrak {R}=- \sum \limits _{a,b}R_{ab} \text {d}\varrho (E_a ) \text {d}\varrho (E_b), \end{aligned}$$

where $\{E_a \}$ is an orthonormal basis of the Lie algebra $\mathfrak {s}o(n) \cong \Lambda ^2 \mathbb {R}^n$, such that the curvature operator of the first kind $\hat{R}\,{:}\ \Lambda ^2 M\rightarrow \Lambda ^2 M$ of (M, g) is identified by $\hat{R} = \sum _{a,b}R_{ab} E_a \otimes E_b$ (see ([14], p. 345); [5, 9]). As observed by Hitchin (see [9]), the curvature operator of the first kind $\hat{R}\,{:}\ \Lambda ^2 M\rightarrow \Lambda ^2 M$ of (M, g) is positive semidefinite (resp. positive) if and only if the Weitzenböck curvature operator $\mathfrak {R}$ is positive semidefinite (resp. positive) for all irreducible representations $\varrho \,{:}\ SO(n)\rightarrow O(E)$. Or in other words, if $\varrho \,{:}\ \mathfrak {s}o(n)\rightarrow \text {End}\,E$ has no trivial factors and $\hat{R}$ is positive definite, then $\mathfrak {R}$ is also positive definite. In particular, if $\hat{R}$ is positive semidefinite, then so is $\mathfrak {R}$ (see [5, 9]). Due to the above definitions and notations, we can formulate a corollary.

Corollary 2.1

Let $\Delta _E$ be the Laplacian on $C^\infty$-sections of a Riemannian fiber bundle $E\rightarrow M$ of rank r over a closed Riemannian manifold (M, g), satisfying (1) for $t>0$. If the representation $\varrho \,{:}\ \mathfrak {s}o(n)\rightarrow \text {End}\,E$ has no trivial factors and the curvature operator of the first kind $\hat{R}\,{:}\ \Lambda ^2 M\rightarrow \Lambda ^2 M$ of (M, g) is positive semidefinite, then $\nabla \varphi =0$ for an arbitrary $\varphi \in ker \Delta _E$ and dim$_\mathbb {R} ker\Delta _E\le r$. In particular, if $\hat{R}$ is positive definite, dim$_\mathbb {R} ker\Delta _E=0$.

The above arguments belong to the classical Bochner technique (see, for example, ([14], pp. 333-364)). In what follows, we will assume that (M, g) is a complete noncompact Riemannian manifold. First of all, we define the vector space of $L^q(E)$-sections of $E\rightarrow M$ by the condition $L^q(\ker \Delta _E)=\{\varphi \in \ker \Delta _E$: $\Vert \varphi \Vert \in L^q(M)\}$, where $\Vert \varphi \Vert ^2=g(\varphi ,\varphi )$.

Let $\varphi \in \ker \Delta _E$, then from the second Kato inequality (see ([2], p. 380))

$$\begin{aligned} \Vert \varphi \Vert \cdot \Delta \Vert \varphi \Vert \ge -g(\bar{\Delta }\varphi ,\varphi ), \end{aligned}$$

where $\Delta =div\circ grad$ is the Beltrami Laplacian on function and Weitzenböck decomposition (1), we deduce

$$\begin{aligned} \Vert \varphi \Vert \cdot \Delta \Vert \varphi \Vert \ge t\ g(\mathfrak {R}(\varphi ),\varphi ). \end{aligned}$$

(4)

In this case, if $t>0$ and $\mathfrak {R}_x$ is a non-negative symmetric endomorphism at each point x of M, then the right side of the above inequality is non-negative. In this case, by the S.-T. Yau theorem in the above inequality, we conclude that for any positive number $q>1$ either ${\int _{M}}\Vert \varphi \Vert ^q{d\nu _{g}}=\infty$ or $\Vert \varphi \Vert =C$ for some constant $C\ge 0$ (see [20, 21]). In particular, if $\Vert \varphi \Vert \in L^q(M)$ for some the positive number $q>1$ and the volume of (M, g) is infinite, then the constant C is zero. On the other hand, for the case when $\Vert \varphi \Vert =C$, $t>0$ and $\mathfrak {R}$ is non-negative, we conclude from (4) that $\nabla \varphi =0$. Then, the following statement holds.

Theorem 2.2

Let (M, g) be a complete noncompact Riemannian manifold and $\Delta _E$ be the natural Laplacian on $C^{\infty }$-sections of a Riemannian fiber bundle $E\rightarrow M$ over (M, g), satisfying (1).

• (i) If $t>0$ and $\mathfrak {R}_x$ is a non-negative symmetric endomorphism of $E_x$ at each point x of M, then $\nabla \varphi =0$ for an arbitrary $\varphi \in L^q(\ker \Delta _E)$ and for any positive number $q>1$. In particular, if (M, g) has infinite volume, then $L^q(\ker \Delta _E)$ is trivial for any number $q>1$.

• (ii) On the other hand, if $t<0$ and $\mathfrak {R}_x$ is a non-negative symmetric endomorphism of $E_x$ at each point x of M, then $\nabla \varphi =0$ for an arbitrary $\varphi \in L^q(\ker \Delta _E)$ and for any positive number $q>1$. In particular, if (M, g) has infinite volume, then $L^q(\ker \Delta _E)$ is trivial for any number $q>1$.

2.2. Let (M, g) be an n-dimensional $(n\ge 2)$ closed Riemannian manifold. An eigensection of the natural Laplacian $\Delta _E$ is $\varphi \in {C^{\infty }(E)}$ satisfying the condition $\Delta _L\varphi _x=\lambda \,\varphi _x$ for some $\lambda \in \mathbb {R}$ at each point $x\in M$, where the constant $\lambda$ is called the eigenvalue of $\Delta _E$ corresponding to $\varphi$. Since $\Delta _E$ is an elliptic operator, we can conclude that on a closed manifold (M, g) the space $E(\lambda )$ of $\Delta _E$ eigensection of associated with the eigenvalue $\lambda$ has a finite dimension. An estimate for the dimension of this space can be found in ([2], p. 389). Moreover, the set of all eigenvalues of the Laplacian $\Delta _E$ is discrete and forms a nondecreasing sequence of the eigenvalues $\{\lambda _a\}_{a\ge 1}$ counted with multiplicities (see [2]).

We define the real numbers $\mathfrak {R}_{\min }=\inf \{\mathfrak {R}_x{:}\,x\in M\}$ and $\mathfrak {R}_{\max }=\sup \{\mathfrak {R}_x{:}$  $x\in M\}$, then $\mathfrak {R}_{\min }\le \mathfrak {R}_x\le \mathfrak {R}_{\max }$ for any $x\in M$. Next, let t be a fixed positive number, then integrating the Weitzenböck formula (1), we obtain (see also ([2], p. 394))

$$\begin{aligned} {\int _{M}} g(\Delta _L\varphi ,\varphi ){d\nu _{g}} \ge t\,\mathfrak {R}_{\min }{\int _{M}} \Vert \varphi \Vert ^2{d\nu _{g}} , \end{aligned}$$

Then, from the above inequality, we deduce $\lambda \ge t\,\mathfrak {R}_{\min }$ for any $\varphi \in E(\lambda )$. As a result, we can formulate the following

Theorem 2.3

Let $\Delta _L$: ${C^{\infty }(E)}\rightarrow {C^{\infty }(E)}$ be the natural Laplacian on a closed Riemannian manifold (M, g) satisfying the Weitzenbock decomposition formula $\Delta _L=\bar{\Delta }+t\, \mathfrak {R}$ for $t\in \mathbb {R}$. Then, for all $a\ge 1$ the eigenvalues $\lambda _a$ of $\Delta _L$ satisfy the inequalities

• (i) $\lambda _a \ge t\,\mathfrak {R}_{\min }$ for a fixed positive number t;

• (ii) $\lambda _a \ge t\,\mathfrak {R}_{\max }$ for a fixed negative number t.

2.3. Let $t>0$ and $E=\otimes ^p T^*M$, then the natural Laplacian on $C^{\infty }$-sections of E satisfying (1) is the Lichnerowicz Laplacian defined in ([14], p. 344). In special cases when $t=1$ the Lichnerowicz-type Laplacian is the ordinary Lichnerowicz Laplacian defined in ([11], p. 315) and ([4], p. 54) by the Weitzenböck decomposition formula $\Delta _L=\bar{\Delta }+\mathfrak {R}$, where the Weitzenböck curvature operator $\mathfrak {R}$ depends linearly on the Riemann curvature tensor R and the Ricci tensor Ric of (M, g).

Remark In particular, if $E=C^{\infty }(\Lambda ^qM)$ for an arbitrary $1\le q\le n-1$, where $\Lambda ^qM$ is the vector bundle of differential q-forms on M, then the Lichnerowicz Laplacian $\Delta _L$ is the well-studied Hodge Laplacian $\Delta _H$ acting on $C^{\infty }$-sections of $\Lambda ^qM$ on (M, g) (see ([14], p. 237)). It admits decomposition (1) with $t=1$, where $\mathfrak {R}$ is an algebraic operator, which is the restriction of the Weitzenböck curvature operator to differential forms. Further in this paper, we will consider the Lichnerowicz Laplacian $\Delta _L$ acting on $C^{\infty }$-sections of the bundle $S^qM$ of covariant symmetric q-tensors (see ([3], pp. 387-388)). The theory of such Lichnerovich Laplacians has hardly been studied for the case $q>2$. An exception is the studies of Boucetta (see, for example, [6]).

On the kernel and estimate for the eigenvalues of the Lichnerowicz Laplacian acting on symmetric tensors

3.1. In this section, we consider the Lichnerowicz Laplacian $\Delta _L$: $C^{\infty }(S^qM)\!\rightarrow {C^{\infty }}(S^qM)$. In this case, we have the Weitzenböck decomposition formula (1) for a suitable constant $t>0$ and the curvature operator $\mathfrak {R}$ given by the formula (see ([4], p. 54); ([11], p. 315))

$$\begin{aligned} \begin{array}{cl} \mathfrak {R}(\varphi )_{i_1\dots i_q}=&{}\displaystyle \sum\limits ^q_{a=1} g^{jk}R_{i_a j} \varphi _{i_1\dots i_{a-1}k i_{a+1}\dots i_q} - \\ \ {} &{} \displaystyle \ \ \ \ \sum\limits ^q_{a,b=1, a\ne b} g^{jk}g^{lm}R_{i_a ji_b l} \varphi _{i_1\dots i_{a-1}k i_{a+1}\dots i_{b-1}m i_{b+1}\dots i_q} \end{array} \end{aligned}$$

for any $i,j,k,l,\dots \in \{1,2,\dots ,n\}$ and for local components $\varphi _{i_1\dots i_q}$, $R_{ik}$ and $R_{ikjl}$ of $\varphi \in {C^{\infty }}(S^qM)$ the Ricci tensor Ric and the curvature tensor R, respectively. These components are defined by the formulas (see ([10], pp. 203; 249))

$$\begin{aligned} \varphi _{i_1\dots i_q}=\varphi (e_{i_1},\dots , e_{i_q}), \ \ \ R^l_{kji} e_l=R(e_i,e_j)e_k, \ \ \ R_{ijkl}=g_{im}R^m_{jkl}, \end{aligned}$$

where $g_{ij}=g(e_i,e_j)$ for any frame $\{e_1,\dots ,e_n\}$ of $T_xM$ at an arbitrary point x. In addition, we have that $R_{lkij}+R_{ijkl}=0$, $R_{kl}=R^i_{kil}$, $R^k_j=g^{ki}R_{ij}$, where $(g_{ij})=(g^{ij})^{-1}$.

Remark Some writers denote our $R_{ijkl}=g_{im}R^m_{jkl}$ by $R_{jkli}=g_{im}R_{jkl}{}^m$ for $R_{ijk}{}^me_m=R(e_i,e_j)e_k$.

In this case, the Weitzenböck curvature operator $\mathfrak {R}$: $\otimes ^qT^*M\rightarrow \otimes ^qT^*M$ satisfies the following identities (see ([11], p. 315)):

$$\begin{aligned} g(\mathfrak {R}(\varphi ),\varphi ')=g(\varphi ,\mathfrak {R}(\varphi ')), \ \ \ trace_g\mathfrak {R}(\varphi )=\mathfrak {R}(trace_g(\varphi )) \end{aligned}$$

for any $\varphi ,\varphi '\in {C^{\infty }}(S^qM)$.

By direct calculations from the Weitzenböck decomposition formula (1), we obtain the following formula:

$$\begin{aligned} \frac{1}{2}\ \Delta \Vert \varphi \Vert ^2=-g(\Delta _L\varphi ,\varphi )+\Vert \nabla \varphi \Vert ^2 + t\,Q(\varphi ), \end{aligned}$$

(5)

where $Q(\varphi )=g(\mathfrak {R}(\varphi ),\varphi )$ is a quadratic form Q: $S^qM\otimes S^qM \rightarrow \mathbb {R}$ such that (see also [16])

$$\begin{aligned} \begin{array}{cl} Q(\varphi )=&{}\mathfrak {R}(\varphi )_{i_1,\dots ,i_q}\varphi ^{i_1,\dots ,i_q}=\\ q\cdot R_{ij}\varphi ^{ikk_3\dots k_q}\varphi ^j_{kk_3\dots k_q}- q(q-1)R_{ijkl}\varphi ^{ikk_3\dots k_q}\varphi ^{jl}{}_{kk_3\dots k_q}, \end{array} \end{aligned}$$

(6)

where $\varphi ^{i_1,\dots ,i_q}= \varphi _{j_1,\dots ,j_q}g^{i_1j_1}\dots g^{i_qj_q}$ are local contravariant components of an arbitrary $\varphi \in S^qM$.

On the other hand, we recall that the point-wise symmetric curvature operator $\buildrel {\circ } \over {R}$: $S_0^2M\rightarrow S_0^2M$ defined on traceless symmetric two-tensor fields by equations

$$\begin{aligned} \buildrel {\circ } \over {R}(\varphi )_{ij}=R_{iklj}\varphi ^{kl} \end{aligned}$$

(7)

for the local contravariant components $\varphi ^{kl}=\varphi _{ij}g^{ik}g^{jl}$ of an arbitrary $\varphi \in S^2_0M$ is called as the curvature operator of the second kind (see [7]).

Remark In the monograph ([4], p. 52), the curvature operator of the second kind was defined by the formula $\buildrel {\circ } \over {R}(\varphi )_{ij}=R_{ikjl}\varphi ^{kl}$ since the local components $R_{jkli}$ of the Riemann curvature tensor were defined there by the identities $R_{ijkl}=g(R(e_i,e_j)e_k,e_l)$.

We say that $\buildrel {\circ } \over {R}\ge 0$ (resp., $\buildrel {\circ } \over {R}>0$) if the eigenvalues of $\buildrel {\circ } \over {R}$ as a bilinear form on $S_0^2M$ are non-negative (resp., strictly positive). At an arbitrary point $x\in M$, we choose orthogonal unit vectors $X,Y\in T_xM$ at an arbitrary point $x\in M$ and define the symmetric 2-tensor field $\theta =X\otimes Y+Y\otimes X$, then by direct calculation, we obtain

$$\begin{aligned} \begin{array}{c} g(\buildrel {\circ } \over {R}(\theta ),\theta )=R_{iklj}(X^iY^j+X^jY^i)(X^kY^l+X^lY^k)=\\ R_{iklj}X^iY^jX^lY^k+R_{iklj}X^jY^iX^kY^l=2\,sec(X,Y). \end{array} \end{aligned}$$

(8)

Therefore, if the operator $\buildrel {\circ } \over {R}$ is nonnegative (resp. strictly positive) defined on any section of the bundle $S_0^2M$, then the sectional curvature of (M, g) is everywhere non-negative (resp. positive) (see also ([7], p. 196)). In addition, if $X\in T_xM$ is a unit vector and we complete it to an orthonormal basis $\{X,e_2,\dots ,e_n\}$ for $T_xM$, then

$$\begin{aligned} Ric(X,X)=\sum \limits ^n_{i=2} sec(X,e_i). \end{aligned}$$

(9)

Therefore, the Ricci curvature of (M, g) is everywhere non-negative (resp. positive) if the operator $\buildrel {\circ } \over {R}$ is non-negative (resp. strictly positive), defined on any section of the bundle $S_0^2M$.

Remark We recall here that closed n-dimensional Riemannian manifolds, whose curvature operator of the second kind acting on symmetric traceless tensors is everywhere positive definite, are diffeomorphic to a spherical space form $\mathbb {S}/\Gamma$ (see [7]).

Next, if we suppose that $\buildrel {\circ } \over {R}\ge 0$ (resp., $\buildrel {\circ } \over {R}>0$), then $R_{ijkl}\varphi ^{ik}\varphi ^{jl}\le 0$ (resp., $R_{ijkl}\varphi ^{ik}\varphi ^{jl}<0$) for an arbitrary $\varphi \in S_0^2M$ with local components $\varphi _{ij}$ and, therefore, $R_{ijkl}\varphi ^{ik k_3\dots k_q}\varphi ^{jl}_{k_3\dots k_q}\le 0$ (resp., $R_{ijkl}\varphi ^{ik k_3\dots k_q}\varphi ^{jl}_{k_3\dots k_q}<0$) for an arbitrary $\varphi \in S_0^qM$ with local components $\varphi _{k_1\dots k_q}$ for $q\ge 2$ (see also [15]). Furthermore, $R_{ij}\varphi ^{i k_2\dots k_q}\varphi ^{j}_{k_2\dots k_q}\ge 0$ (resp., $R_{ij}\varphi ^{i k_2\dots k_q}\varphi ^{j}_{k_2\dots k_q}>0$) for an arbitrary $\varphi \in S^q_0M$ with local components $\varphi _{k_1\dots k_q}$ for $q\ge 2$.

As a result, from the above, we conclude that if (M, g) is a Riemannian manifold with positive semidefinite (resp. strictly positive) curvature operator of the second kind, then $Q(\varphi )=g(\mathfrak {R}(\varphi ),\varphi )\ge 0$ (resp. $Q(\varphi )=g(\mathfrak {R}(\varphi ),\varphi )>0$).

Let (M, g) be a closed Riemannian manifold and $\Delta _L\varphi =0$ for some $\varphi \in {C^{\infty }}(S^q_0M)$, then from (5), we deduce the following integral formulas

$$\begin{aligned} 0={\int _{M}}(\Vert \nabla \varphi \Vert ^2 + t\,Q(\varphi )){d\nu _{g}}\ge t{\int _{M}} Q(\varphi ){d\nu _{g}}. \end{aligned}$$

It is obvious that $\buildrel {\circ } \over {R}$ is positive-semidefinite on (M, g) and $\buildrel {\circ } \over {R}>0$ at some point at (M, g), then there are no non-zero $\Delta _L$-harmonic traceless symmetric q-tensor fields $(q\ge 2)$.

Theorem 3.1

Let (M, g) be a closed Riemannian manifold. If the curvature operator of the second kind $\buildrel {\circ } \over {R}$ is positive-semidefinite on (M, g), then $\dim _{\mathbb R} \Delta _L\le {n+q-1\atopwithdelims ()q}$. Moreover, if $\buildrel {\circ } \over {R}>0$ at every point of (M, g), then $\dim _{\mathbb R}\ker \Delta _L=0$.

Remark A form $\omega \in {C^{\infty }}(\Lambda ^qM)$ is harmonic if it is in the kernel of the Hodge Laplacian: $\Delta _H\omega =0$. The main motivation for considering this operator is that the dimension of the vector space of harmonic q-forms, denoted $\mathfrak { H}^q(M,\mathbb {R})$, is a topological invariant, and in fact equals to the q-th de Rham Betti number $b_q(M,\mathbb {R})$ of M. Recall that the curvature operator of the first kind is the self-adjoint endomorphism $\hat{R}$ of $\Lambda ^2M$, defined by: $g(\hat{R}(X\wedge Y),Z\wedge W)=R(X,Y,Z,W)$, where R is the Riemann tensor and X, Y, Z, W are tangent vectors (see ([14], p. 116)). Then, if $\hat{R}$, is positive-semidefinite on (M, g) then $dim_{\mathbb R}\Delta _H\le {n\atopwithdelims ()q}$. Moreover, if $\hat{R}>0$ at some point at (M, g), then there are no non-zero harmonic q-forms $(1\le q\le n-1)$. In this case $dim_{\mathbb R}\ker \Delta _H=b_q(M,\mathbb {R})=0$ (see ([14], p. 351)).

On the other hand, it is obvious that (4) is valid for any $q\ge 2$. In this case, from (4), we obtain the inequality

$$\begin{aligned} \Vert \varphi \Vert \Delta \Vert \varphi \Vert \ge t\,Q(\varphi ), \end{aligned}$$

(10)

for any $\varphi \in {C^{\infty }}(S^q_0M)\cap \ker \Delta _L$. If the curvature operator of the second kind of (M, g) is non-negative at an arbitrary point $x\in M$, then $Q(\varphi )=g(\mathfrak {R}(\varphi ),\varphi )\ge 0$. At the same time, if a Riemannian manifold has a non-negative curvature operator of the second kind, then it has non-negative sectional curvature (see, our formula (8)). On the other hand, if a complete non-compact Riemannian manifold has non-negative sectional curvature, then it must have infinite volume (see [20]). Therefore, we can reformulate our Theorem 2.2 in the following form.

Theorem 3.2

Let (M, g) be a connected noncompact and complete Riemannian manifold with non-negative curvature operator of the second kind of (M, g). Then, the vector space $L^q(\ker \Delta _L)$ is trivial for any $1<p<\infty$.

Remark This assertion is an analog of the vanishing theorem for q-forms of the space $L^2(\ker \Delta _H)$ on complete Riemannian manifolds (see [1]).

Let (M, g) be a connected closed Riemannian manifold and $\varphi$ be a $\Delta _L$- harmonic traceless symmetric q-tensor field $(q\ge 2)$ defined on (M, g), then proceeding from the formula (5) and using the strong maximum principle (see ([14], p. 75)), we conclude that the following corollary holds.

Corollary 3.1

Let (M, g) be a connected closed Riemannian manifold and $\varphi$ be a $\Delta _L$-harmonic traceless symmetric q-tensor field $(q\ge 2)$ defined on (M, g). If the curvature operator of the second kind of (M, g) is positive semi-define at every points, then $\Vert \varphi \Vert ^2$ is a constant function and $\varphi$ is invariant under parallel translation in (M, g).

3.2. We denote by $S^q_0M$ the subbundle of the bundle $S^qM$ on a Riemannian manifold defined by the condition $trace_g\varphi =\sum ^n_{i=1}\varphi (e_i,e_j,X_3,\dots ,X_q)=0$ for $\varphi \in S^q_0M$ and orthonormal basis $\{e_1\dots ,e_n\}$ of $T_xM$ at an arbitrary point $x\in M$ (see the details of the theory in our monograph [12]). The Laplacian $\Delta _L$ maps ${C^{\infty }} (S^qM)$ into itself. This property is a corollary of (3) for any $\varphi \in {C^{\infty }}( S^qM)$. In particular, the following equation holds

$$\begin{aligned} trace_g(\Delta _L\varphi )=\Delta _L(trace_g\varphi )=0 \end{aligned}$$

(11)

for an arbitrary $\varphi \in {C^{\infty }} (S^q_0M)$ and $q\ge 2$. Then, we can conclude that $\Delta _L$: $\varphi \in {C^{\infty }} (S^q_0M)\rightarrow \varphi \in {C^{\infty }} (S^q_0M)$ since $trace_g\varphi =0$. Furthermore, we also conclude that $trace_g\varphi \in \ker \Delta _L$ for an arbitrary $\varphi \in {C^{\infty }} (S^qM)$ such that $\varphi \in \ker \Delta _L$.

Lemma 3.1

Let $\Delta _L$: ${C^{\infty }} (S^qM)\rightarrow {C^{\infty }} (S^qM)$ be the Lichnerowicz Laplacian acting on $C^{\infty }$-sections of the bundle of covariant symmetric p-tensor fields $S^qM$ $(q\ge 2)$ on a Riemannian manifold (M, g). Then,

• (i) $trace_g\varphi \in \ker \Delta _L$ for an arbitrary $\varphi \in {C^{\infty }} (S^qM)$ such that $\varphi \in \ker \Delta _L$;

• (ii) $\Delta _L$ maps $S^q_0M$ into itself for the subbundle $S^q_0M$ of traceless covariant symmetric p-tensor fields on (M, g).

If $\varphi \in {C^{\infty }} (S^q_0M)$ and the sectional curvature of (M, g) is nonnegative at an arbitrary point $x\in M$, then $Q(\varphi )=g(\mathfrak {R}(\varphi ),\varphi )\ge 0$ (see [5]). In this case, for any $\varphi \in {C^{\infty }} (S^q_0M)\cap \ker \Delta _L$, we can reformulate our Theorem 2.2 in the following form.

Theorem 3.3

Let (M, g) be a connected noncompact and complete Riemannian manifold with non-negative sectional curvature. Then, there is no non-zero $\Delta _L$-harmonic traceless symmetric q-tensor field $(q\ge 2)$ such that it lies in $L^p$ for some $1<p<\infty$.

The following corollary is obvious (see also the proof of Theorem 2.2).

Corollary 3.2

Let (M, g) be a connected closed Riemannian manifold and $\varphi$ be a $\Delta _L$-harmonic traceless symmetric q-tensor field $(q\ge 2)$ defined on (M, g). If the section curvature of (M, g) is positive semi-define at every points, then $\Vert \varphi \Vert ^2$ is a constant function and $\varphi$ is invariant under parallel translation in (M, g).

In what follows, we will assume that (M, g) is a closed Riemannian manifold. In this case, the Lichnerowicz Laplacian $\Delta _L$: ${C^{\infty }} (S^qM)\rightarrow {C^{\infty }} (S^qM)$ has a discrete spectrum $Spec^{(q)}\Delta _L$ on (M, g) since it is an elliptic formally self-adjoint second-order differential operator. If $\lambda ^{(q)}_a \in Spec^{(q)}\Delta _L$ is an eigenvalue of $\Delta _L$ corresponding to an eigentensor $\varphi \in {C^{\infty }}(S^qM)$ for the case $q\ge 2$, then from (5), we obtain the integral inequality

$$\begin{aligned} \lambda ^{(q)}_a{\int _{M}}\Vert \varphi \Vert ^2{d\nu _{g}}\ge t{\int _{M}} Q(\varphi ){d\nu _{g}}. \end{aligned}$$

Let $\Delta _L$: ${C^{\infty }}(S^q_0M)\rightarrow {C^{\infty }}(S^q_0M)$ be the Lichnerowicz Laplacian acting on the bundle $S_0^q M$ of traceless symmetric q-tensors over (M, g). Then, if we suppose that the curvature operator of the second kind $\buildrel {\circ } \over {R}$: $S^2_0M\rightarrow S^2_0M$ of (M, g) is positive semidefinite (resp. strictly positive), then from the above formula, we conclude that $\lambda ^{(q)}_a\ge 0$ (resp. $\lambda ^{(q)}_a>0$). Namely, assume that the curvature operator of the second kind $\buildrel {\circ } \over {R}$: $S^2_0M\rightarrow S^2_0M$ of (M, g) satisfies the inequality $g(\buildrel {\circ } \over {R}(\varphi ),\varphi )\ge K\, \Vert \varphi \Vert ^2$ for certain positive number K and for any symmetric 2-tensor field $\varphi$. Taking two mutually orthogonal unite vector fields X and Y and putting $\varphi =X\otimes Y+Y\otimes X$ we find, from (8),

$$\begin{aligned} sec(X,Y)\ge K>0. \end{aligned}$$

(12)

Then, from (9) and (12), we obtain $Ric(X,X)\ge (n-1)\,K>0$. Thus,

$$\begin{aligned} \begin{array}{c} Q(\varphi )=\mathfrak {R}(\varphi )_{i_1\dots i_q}\varphi ^{i_1\dots i_q}= q R_{ij}\varphi ^{ik_2\dots k_q}\varphi ^j_{k_2\dots k_q} - q (q-1)R_{ijkl}\varphi ^{ikk_3\dots k_q}\varphi ^{jl}_{k_3\dots k_q} \\ \ge q(n-1) \,K\,\Vert \varphi \Vert ^2 + q(q-1) \,K\,\Vert \varphi \Vert ^2= q(n+q-2) \,K\,\Vert \varphi \Vert ^2 \end{array} \end{aligned}$$

which is positive for $n\ge 2$ and $q\ge 1$. In this case, from Theorem 2.3, we can conclude that the following corollary holds.

Corollary 3.3

Let (M, g) be an n-dimensional closed Riemannian manifold with the curvature operator of the second kind $\buildrel {\circ } \over {R}$: $S^2_0M\rightarrow S^2_0M$ satisfying the inequalities $\buildrel {\circ } \over {R}\ge 2K>0$ for certain positive number K. At the same time, let $\Delta _L$: ${C^{\infty }}(S^q_0M)\rightarrow {C^{\infty }}(S^q_0M)$ be the Lichnerowicz Laplacian that satisfies the Weitzenböck decomposition $\Delta _L=\bar{\Delta }+t\,\mathfrak {R}$ for a fixed positive number t. Then, for an arbitrary $a\ge 1$ the eigenvalues $\lambda ^{(q)}_a$ of $\Delta _L$ satisfy the inequality $\lambda ^{(q)}_a\ge q(n+q-2)\,t\,K$.

Remark Having discussed above the kernel of the Hodge Laplacian $\Delta _H$, we now turn our attention to its first positive eigenvalue, which we will denote by the symbol: $\mu _1^{(q)}$. Then, if (M, g) is an n-dimensional closed manifold with the curvature operator of the first kind $\hat{R}$ satisfying the inequalities $\hat{R}\ge K>0$ for certain positive number K, then $\mu _1^{(q)}\ge C(n,q)K$, where $C(n,q)=\min \{q(n-q+1); (q+1)(n-q)\}$ for all $q=1,\dots ,n-1$ (see [8]).

In particular, let (M, g) is a manifold with constant sectional curvature $K>0$ (see ([10], p. 203)), then its Riemann and Ricci curvature tensors are given by the identities $R_{ijkl}=K(g_{ik}g_{jl}-g_{il}g_{jk})$ and $R_{ij}=K(n-1)g_{ij}$, respectively. Then, using these equalities and the equality (7) for any symmetric form $\varphi \in S_0^q M$, we have

$$\begin{aligned} \begin{array}{c} g(\mathfrak {R}(\varphi ),\varphi )=qR_{ij}\varphi ^{ik_2\dots k_q}\varphi ^j_{k_2\dots k_q}- q(q-1)R_{ijkl}\varphi ^{ikk_2\dots k_q}\varphi ^{jl}_{k_2\dots k_q}= \\ (n-1)K\Vert \varphi \Vert ^2+q(q-1)K\Vert \varphi \Vert ^2=(n+q-2)K\Vert \varphi \Vert ^2\ge 0, \end{array} \end{aligned}$$

where equality is possible in the case $\varphi =0$. In this case, we deduce from (5) the integral inequality

$$\begin{aligned} (-\lambda ^{(q)}_a+q(n+q-2)tK){\int _{M}}\Vert \varphi \Vert ^2{d\nu _{g}}\le 0. \end{aligned}$$

Then, from the previous inequality, we conclude that $\lambda ^{(q)}_a\ge q(n+q-2)\,tK$ for the first non-zero eigenvalues $\lambda ^{(q)}_1$ of the Lichnerowicz Laplacian $\Delta _L$: ${C^{\infty }}(S^q_0M)\rightarrow {C^{\infty }}(S^q_0M)$ acting on the bundle $S^q_0M$ of traceless symmetric q-tensors on a Riemannian manifold with constant sectional curvature $K>0$. Thus, we have proved the following theorem.

Theorem 3.4

Let (M, g) be an n-dimensional closed Riemannian manifold with constant sectional curvature $K>0$ and let $\Delta _L$: ${C^{\infty }}(S^q_0M)\rightarrow {C^{\infty }}(S^q_0M)$ be the Lichnerowicz Laplacian acting on the bundle $S^q_0M$ of traceless symmetric q-tensors over (M, g). Then, the first non-zero eigenvalue $\lambda ^{(q)}_1$ of $\Delta _L$ satisfies the inequality $\lambda ^{(q)}_1\ge q(n+q-2)\,tK$.