The Lichnerowicz-Type Laplacians: Vanishing Theorems for Their Kernels and Estimate Theorems for Their Smallest Eigenvalues

Abstract

In the present paper, we prove several vanishing theorems for the kernel of the Lichnerowicz-type Laplacian and provide estimates for its lowest eigenvalue on closed Riemannian manifolds. As an example of the Lichnerowicz-type Laplacian, we consider the Hodge–de Rham Laplacian acting on forms and ordinary Lichnerowicz Laplacian acting on symmetric tensors. Additionally, we prove vanishing theorems for the null spaces of these Laplacians and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. Specifically, we will prove our version of the famous differential sphere theorem, which we will apply to the aforementioned problems concerning the ordinary Lichnerowicz Laplacian.

1. Introduction

1.1. Definitions and Notations

This paper is concerned with researching geometric analyses, and the tools of elliptic partial differential equations are used to obtain new results in differential geometry.The main theme of our paper is centered around Laplace’s equation and the eigenvalues of the Laplacian. To achieve this, we consider a Riemannian vector bundle $E\to M$ of rank r over a Riemannian manifold $(M,g)$ associated with the principal frame bundle by a representation $\rho$ of $SO\left(n\right)$.We define the Laplace-type operator ${\Delta }_E={\nabla }^{*}\nabla +t\mathfrak{R}$ on $C^{\infty }$-sections of a Riemannian vector bundle $E\to M$ where $\nabla$ is the covariant derivative of the connection induced on the vector bundle by the Levi-Civita connection of g, $\mathfrak{R}$ is a curvature term of ∇, and t is a real number (see, for example, refs. [1,2]). The operator ${\Delta }_E$ is self-adjoint and elliptic, and we will list its eigenvalues as the following:

$${{Spec}^{\left(r\right)}{\Delta }_E:\lambda }_1\le {\lambda }_2\le \cdots \le {\lambda }_k\le {\lambda }_{k+1}\le \cdots .$$

We prove several vanishing theorems for its null space and find estimates for its lowest eigenvalue on a closed Riemannian manifold. This operator is referred to as the Lichnerowicz-type Laplacian or, in other words, the Lichnerowicz-like Laplacian in several papers and monographs (see [1,2,3,4,5] and etc.). Despite its significant importance in mathematics and mathematical physics through Laplace-type operators (such as the Dirac Laplacian, Hodge–de Rham Laplacian, Schrödinger operators, Lichnerowicz Laplacian, etc.) and the associated Bochner–Weitzenböck formulas (cf. [3], pp. 54–58), only a limited amount of general information is known about this Laplacian. We will study this Laplacian in more detail. This will be achieved by using the Bochner technique (see [6], pp. 333–364; refs. [5,7]), an important tool for geometric analysis (see, for example, refs. [4,8]). At the same time, we recall here that, in the well-known monograph [3], 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 Weitzenbock decomposition and further of estimating its lowest eigenvalue”. This will serve as our guide in the study of Laplacians in the present paper. In this paper, we will prove vanishing theorems for the null spaces of Laplacian kernels and find estimates for their lowest eigenvalues on closed Riemannian manifolds with suitably bounded curvature operators of the first kind, sectional and Ricci curvatures. For example, in the next section, we will prove several vanishing theorems for the null space of the Lichnerowicz-type Laplacian, and we will estimate the minimum eigenvalue section $\phi$ of E with respect to the Lichnerowicz-type Laplacian. To accomplish this, we will use a Bochner–Lichnerowicz–Weitzenböck type formula.

In the Section 2 and Section 3, we will consider two important examples of the Lichnerowicz-type Laplacian: the Hodge–de Rham Laplacian acting on the bundle of differential forms ${\Lambda }^{q}M\to M$ and the Lichnerowicz Laplacian acting on the bundle of symmetric tensors $S^{p}M\to M$. In both cases, $\mathfrak{R}$ is the curvature term in the Bochner–Weitzenböck formula and $t=1$. In addition, we note that these Laplacians are examples of the Lichnerowicz-type Laplacian, which is a Laplace operator admitting a Weitzenbock decomposition(see [1,2]; ref. [3], p. 53; ref. [4], p. 104).

The main theorems of these sections will be devoted to linear theory, i.e., the Laplace equation and eigenvalues for the Laplacian, and the methods of dealing with these problems are quite often useful in the study of nonlinear partial differential equations that arise in geometry. Namely, as an application of the aforementioned results, we will prove several vanishing theorems for the null spaces of the Lichnerowicz and Hodge–de Rham Laplacians and find estimates of their lowest eigenvalues on n-dimensional closed Riemannian manifolds. In particular, in the Section 3 we will complement the classical results of Gallo-Meyer in regard to vanishing the kernel and estimating the eigenvalues of the Hodge–de Rham Laplacian (see [9,10,11]).

1.2. Preliminary Results

The four section of our paper is dedicated to proving a new version of the differentiable sphere theorem and exploring its connection with results on null spaces and the smallest eigenvalues of the ordinary Lichnerowicz Laplacian. This operator is of fundamental importance in mathematical physics and appears in many problems of Riemannian geometry, including the theories of infinitesimal Einstein deformations and the stability of Einstein manifolds (see [3]).

We shall delve deeper into the differentiable sphere theorem and its implications for the large-scale geometry of Riemannian manifolds. Here, and throughout this work, we will also present our own version of this theorem and its application to the geometry of the Lichnerowicz Laplacian. Specifically, the topological sphere theorem provides a sufficient condition for an n-dimensional Riemannian manifold $(M,g)$ to be homeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$.

In 1942, Hopf conjectured that a simply connected manifold with a pinched sectional curvature is a sphere (see [12]). Klingenberg gave an affirmative answer to this question, proving the following statement: Let $(M,g)$ be a compact, simply connected n-dimensional Riemannian manifold which is weakly 1/4-pinched in the global sense. Then, $(M,g)$ is homeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$ or isometric to a compact symmetric space of rank one (see [13]). We recall that a Riemannian manifold $(M,g)$ is said to be weakly α-pinched in the global sense if the sectional curvature of $(M,g)$ satisfies $\alpha \le sec\le 1$. The pinching constant is defined as the following. Let $K_{min}\left(x\right):={\inf }_{{\sigma }_x\subset T_{x}M}sec\left({\sigma }_x\right)$ and $K_{max}\left(x\right):={\sup }_{{\sigma }_x\subset T_{x}M}sec\left({\sigma }_x\right),$ then $\alpha =K_{min}/K_{max}$ for $K_{\min }=\inf \left\{K_{\min }\left(x\right):x\in M\right\}$ and $K_{\max }=\sup \left\{K_{\max }\left(x\right):x\in M\right\}.$ We say that $(M,g)$ is strictly$\alpha$-pinched in the global sense if the strict inequality $\alpha <sec<1$ holds. Around 1960, M. Berger and W. Klingenberg gave their affirmative answer to Hopf’s question (see [13]): Let $(M,g)$ be a closed, simply connected Riemannian manifold whose sectional curvatures lie in the interval (1,4]. Then, $(M,g)$ is homeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$. More generally, Berger (also see [13]) proved that a closed, simply connected Riemannian manifold with a weakly $1/4$-pinched sectional curvature is either homeomorphic to ${\mathbb{S}}^n$ or isometric to a compact symmetric space of rank one. In 1978, the following was proven (see [14], p. 79): the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ acting on 2-forms is positive as soon as the sectional curvature is weakly$\alpha$-pinched with $\alpha =1-3/(2\left[n/2\right]+1).$ At the same time, it is obvious that $\alpha >1/4$ for $n\ge 2.$ Therefore, in this case $(M,g)$ is homeomorphic to ${\mathbb{S}}^n$. On the other hand, the following celebrated differentiable sphere theorem holds: A closed Riemannian manifold with a two-positive curvature operator of the first kind is diffeomorphic to a spherical space form. We call the operator of the first kind two-positive if the sum of the smallest two eigenvalues is positive. This 2008 theorem is due to Böhm and Wilking [15].

We recall here that an n-dimensional closed Riemannian manifold $(M,g)$ with a positive curvature operator of the second kind $\stackrel{°}{R}$: $S_0^{2}M\to S_0^{2}M$ acting on traceless symmetric 2-tensors is diffeomorphic to a spherical space form ${\mathbb{S}}^n/\Gamma$ (see [16]). In particular, if $(M,g)$ is simply connected, then $(M,g)$ is diffeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$.

In turn, in 1969 the following was proven [17], p. 390: if the sectional curvature of $(M,g)$ is $\alpha$-pinched in the following form

$${\displaystyle \frac{p-1}{p}}<\alpha \le 1\mathrm{for}n=2p\mathrm{or}{\displaystyle \frac{2p^2-1}{2p(p+1)}}<\alpha \le 1\mathrm{for}n=2p+1,$$

then $\stackrel{°}{R}$ is positive-definite on $S_0^{2}M$. However, we note that $({\displaystyle \frac{p-1}{p}},1]$ and $({\displaystyle \frac{2p^2-1}{2p(p+1)}},1]$ are subsets of $(1/4,1]$ for $p\ge 2$. Therefore, in this case, we are unable to derive a new sphere theorem that is connected with the curvature operator of the second kind.

Moreover, let us give an example which is known well as the differentiable sphere theorem (also see [18,19]). First, recall the necessary definitions and notations. Since the unit sphere in $T_{x}M$ at an arbitrary point $x\in M$ is a closed set, then the sectional curvature satisfies the double inequality $K_{min}\left(x\right)\le sec\left({\sigma }_x\right)\le K_{max}\left(x\right)$.

In contrast, the Riemannian manifold $(M,g)$ is weakly pointwise $1/4$-pinched if its sectional curvature satisfies the inequality $K_{max}\left(x\right)=4K_{min}\left(x\right)$ at each point $x\in M.$ Then, if n-dimensional $(n\ge 4)$ is closed and a weakly pointwise weakly $1/4$-pinched Riemannian manifold $(M,g)$ is not locally symmetric, then it is diffeomorphic to a spherical space form ${\mathbb{S}}^n/\Gamma$ (see [13]). In particular, if $(M,g)$ is simply connected, then $(M,g)$ is diffeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$.

In turn, Xu and Gu proved in [20] another version of the differentiable sphere theorem. However, first, let us recall that the Ricci tensor is defined by the formula $Ric\left(X\right)={\sum }_{a=2}^{n}K(X,e_a)$ for given any unit vector $X\in T_{x}M$ and an orthonormal basis $\{e_1,\cdots ,e_n\}$ for $T_{x}M$, such that $X=e_1$ at an arbitrary point $x\in M$. In this case, we have the double inequality $(n-1)K_{\min }\left(x\right)\le Ric\left(X\right)\le {(n-1)K}_{\max }\left(x\right)$. In addition, define ${Ric}_{\min }\left(x\right):={\inf }_{X\in T_{x}M}Ric\left(X\right)$ for any unit vector $X\in T_{x}M$. Then, the “differentiable sphere theorem for manifolds with positive Ricci curvature” of Xu and Gu is as follows: an n-dimensional $(n\ge 2)$ closed Riemannian manifold $(M,g)$ whose Ricci curvature and sectional curvatures satisfy the double inequality

$${(n-1)K}_{\max }\left(x\right)-{6/5K}_{\max }\left(x\right)<{Ric}_{\min }\left(x\right)\le {(n-1)K}_{\max }\left(x\right)$$

at each point when $x\in M$ is diffeomorphic to a spherical space form ${\mathbb{S}}^n/\Gamma$. In particular, if $(M,g)$ is simply connected, then $(M,g)$ is diffeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$.

One of the main results of this section is our version of the differentiable sphere theorem for manifolds with positive sectional curvature: an n-dimensional $(n\ge 2)$ closed Riemannian manifold $(M,g)$ is diffeomorphic to a spherical space form ${\mathbb{S}}^n/\Gamma$ if the Ricci curvature $Ric$ of $(M,g)$ satisfies the double inequality

$${(n-1)K_{min}\left(x\right)\le Ric}_{\max }\left(x\right)<(n-1)K_{min}\left(x\right)+K_{min}\left(x\right),$$

where ${Ric}_{\max }\left(x\right):={\sup }_{X\in T_{x}M}Ric\left(X\right)$ for any unit vector $X\in T_{x}M$ at each point $x\in M$. In particular, if $(M,g)$ is simply connected, then $(M,g)$ is diffeomorphic to the Euclidean n-sphere ${\mathbb{S}}^n$. In addition, we recall that the simplest examples of spherical space forms are the Euclidean sphere ${\mathbb{S}}^n$ or a real projective space $\mathbb{R}{\mathbb{P}}^n$. Furthermore, when n is even, these are the only examples.

We will use our version of the famous differential sphere theorem and apply it to the above problems concerning the ordinary Lichnerowicz Laplacian.

1.3. Final Remark

This paper continues the authors’ research on the geometry of Lichnerowicz-type Laplacians, which was initiated in their other two articles [21,22].

2. The Lichnerovich-Type Laplacian: A Brief Overview of Its Theory and New Results

2.1. The Lichnerowicz Type Laplacian as a Generalization of the Classical Lichnerowicz Laplacian

Let $(M,g)$ be a connected Riemannian manifold of dimension $n\ge 2$ with the Levi-Civita connection ∇. We shall assume all our manifolds are smooth (i.e., $C^{\infty })$, complete and connected. Further, let $E\to M$ be a vector bundle of rank $r$ over $(M,g)$ associated with the principal frame bundle by a representation $\rho$ of $SO\left(n\right)$. The Riemannian metric g and the Levi-Civita connection ∇ induce a scalar product and a compatible connection on $E\to M$, respectively (see [1], p. 378). In addition, the scalar products and compatible connections of $E\to M$, as well as of M, will be denoted by the same symbols g and ∇. Moreover, we can define the $L^2$-global scalar product on $C^{\infty }$-sections of E by the formula $⟨\theta ,\vartheta ⟩={\int }_{M}g(\theta ,\vartheta )d{v}_g$ for $\theta ,\vartheta \in C^{\infty }\left(E\right)$ and introduce the associated Hilbert space $L^2\left(E\right)$. Then, using the $L^2$-structures on $C^{\infty }\left(E\right)$, we define the rough Laplacian or the connection Laplacian on $C^{\infty }$-sections of a Riemannian vector bundle $E\to M$ by the formula ${\overline{\Delta }=\nabla }^{*}\circ \nabla$, where ${\nabla }^{*}$ is the formal adjoint operator with respect to the $L^2$-product of the compatible connection ∇: $C^{\infty }\left(E\right)\to C^{\infty }(T^{*}M\otimes E)$ (see also [3], pp. 34, 52; ref. [1], p. 378; ref. [23]). Due to the above definitions and notations, we can consider the Lichnerowicz-type Laplacian ${\Delta }_E$: $C^{\infty }\left(E\right)\to C^{\infty }\left(E\right)$ defined on $C^{\infty }$-sections of a Riemannian vector bundle $E\to M$, which is a second-order elliptic linear differential operator symmetric with respect to the $L^2\left(E\right)$-inner product and satisfies the (see [1,2]; ref. [4], p. 104; ref. [3], p. 53)

$${\Delta }_E=\overline{\Delta }+t\mathfrak{R}$$

(1)

where $t\in \mathbb{R}$ is a suitable constant and $\mathfrak{R}$ is the Weitzenböck curvature operator. The curvature term $\mathfrak{R}$ depends linearly on the curvature of the covariant derivative ∇ on the bundle E and is defined as a symmetric endomorphism ${\mathfrak{R}}_x:E_x\to E_x$ at each point x in M (see [1], p. 379).

Let $(M,g)$ be a closed manifold. Since ${\Delta }_E$ is elliptic, the kernel $Ker{\Delta }_E$ is finite-dimensional and the following $L^2$-orthogonal decomposition is valid (see [3], p. 464):

$$C^{\infty }\left(E\right)=Ker{\Delta }_E⨁Im{\Delta }_E.$$

A smooth section $\phi \in C^{\infty }\left(E\right)$ is called ${\Delta }_E$-harmonic if $\phi \in Ker{\Delta }_E$ (see [4], p. 104). We note that ${\Delta }_E$-harmonic sections satisfy the (strong) unique continuation property. Moreover, in local coordinates, the condition ${\Delta }_E=\overline{\Delta }+t\mathfrak{R}=0$ becomes a system of $r$ elliptic differential equations satisfying the structural assumptions of Aronszajn–Cordes (see Appendix in [4]).

We define the vector space $\mathcal{H}\left(E\right)=\left\{\phi \in C^{\infty }\left(E\right):{\Delta }_E\phi =0\right\}.$ Then, from the above, we conclude that the dimension of the vector space $\mathcal{H}\left(E\right)$ over a closed Riemannian manifold $(M,g)$ is finite. Next, we define the conditions for the triviality of this vector space over a closed Riemannian manifold. The value of Weitzenböck expansions lies mainly in the disappearance theorems (see [24,25]). These theorems will be the topic of the next paragraph.

2.2. New Vanishing Theorems for Harmonic Forms

Multiplying the Weitzenböck decomposition Formula (1) by $\phi$, we can deduce the Bochner-Weitzenböck formula (also see [1], p. 389; ref. [4], p. 105; ref. [3], p. 53)

$${\displaystyle \frac{1}{2}}\Delta {∥\phi ∥}^2=-g({\Delta }_E\phi ,\phi )+tg(\mathfrak{R}\left(\phi \right),\phi )+{∥\nabla \phi ∥}^2$$

(2)

for $\Delta :={\mathrm{trace}}_g\nabla \circ \nabla$ and any smooth section $\phi \in C^{\infty }\left(E\right)$. Let $(M,g)$ be a closed manifold and $\phi \in \mathcal{H}\left(E\right)$; then, integrating over $(M,g)$, gives

$$0={\int }_M\left(tg(\mathfrak{R}\left(\phi \right),\phi )+{∥\nabla \phi ∥}^2\right)d{v}_g\ge t{\int }_{M}g\left(\mathfrak{R}\right(\phi ),\phi )d{v}_g,$$

(3)

where $d{v}_g$ is the volume form of $(M,g)$. Since $(M,g)$ is closed, we can define the number (see [1], p. 379)

$${\mathfrak{R}}_{min}=\inf \left\{{\mathfrak{R}}_{min}\left(x\right):x\in M\right\}$$

for ${\mathfrak{R}}_{min}\left(x\right)=\inf \left\{{{g(\mathfrak{R}\phi ,\phi )}_x:{\phi }_x\in E}_x,{g(\phi ,\phi )}_x=1\right\}$. In this case, we deduce from (3) the inequality

$$0\ge \left(t{\mathfrak{R}}_{min}\right){\int }_M{∥\phi ∥}^{2}d{v}_g.$$

(4)

Then, from (4) we conclude that $\phi =0$, if $t>0$ and ${\mathfrak{R}}_{min}>0,$ or if $t<0$ and ${\mathfrak{R}}_{min}<0$.

As a result, we can formulate a theorem.

Theorem 1.

Let ${\Delta }_E$ be the Laplacian defined on $C^{\infty }$-sections of a Riemannian fibre bundle $E\to M$ of rank $r$ over a closed Riemannian manifold $(M,g)$, satisfying (1). Then, the vector space $\mathcal{H}\left(E\right)$ of ${\Delta }_E$-harmonic forms is trivial if $t>0$ and ${\mathfrak{R}}_{min}>0,$ or if $t<0$ and ${\mathfrak{R}}_{min}<0.$

Next, we recall that the Riemann curvature tensor R of $(M,g)$ at each of its point lies in ${\mathrm{Sym}}^2\left(\mathfrak{s}o\left(n\right)\right)$ for the Lie algebra $\mathfrak{s}o\left(n\right)\cong {\Lambda }^2{\mathbb{R}}^n$. In this case, thanks to the representation $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E,$ we obtain the Weitzenböck curvature operator $\mathfrak{R}$ in ${\mathrm{Sym}}^2\left(\mathrm{End}E\right),$ and the composition of endomorphisms gives a self-adjoint endomorphism of E. More details: the Weitzenböck curvature operator $\mathfrak{R}$ is defined as (see [2]; ref. [26], p. 1214)

$$\mathfrak{R}=-\sum _{a,b}{R}_{ab}\mathrm{d}\rho \left(E_a\right)\mathrm{d}\rho \left(E_b\right),$$

where $\left\{E_a\right\}$ is an orthonormal basis of the Lie algebra $\mathfrak{s}o\left(n\right)\cong {\Lambda }^2{\mathbb{R}}^n,$ and the curvature operator of the first kind $\widehat{R}$: ${\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ is identified with $\widehat{R}={\sum }_{a,b}{R}_{ab}{E}_a⨂E_b$ (see [1], p. 345; ref. [2,26]). Note that $\mathfrak{R}$ is self-adjoint since $\widehat{R}$ is self-adjoint, and $\mathrm{d}\rho \left(X\right)$ are skew-adjoint, as $\rho$ is orthogonal. Moreover, as observed by [26], p. 1215, if $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and $\widehat{R}$ is positive definite at each point $x\in M$, then $\mathfrak{R}$ is also positive definite at each point $x\in M$. In addition, if $\widehat{R}$ is positive semi-definite at each point $x\in M$, then so is $\mathfrak{R}$. Due to the above definitions and notations, we can formulate a theorem.

Theorem 2.

Let ${\Delta }_E$ be the Laplacian on $C^{\infty }$-sections of a Riemannian fibre bundle $E\to M$ of rank $r$ over a closed Riemannian manifold $(M,g)$, satisfying (1) for $t>0$. If the representation $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ is positive semi-definite at each point $x\in M$, then $\nabla \phi =0$ for an arbitrary $\phi \in \ker {\Delta }_E$ and ${\dim }_{\mathbb{R}}ker{\Delta }_E\le r$. In particular, if $\widehat{R}$ is positive definite at each point $x\in M$, ${dim}_{\mathbb{R}}\ker {\Delta }_E=0.$

Proof. 

Using the above notation and results, we can conclude that the proof of our vanishing theorem is a verbatim reproduction of that of Theorem V from [1], p. 379: Let $E\to M$ be a Riemannian vector bundle of rank r over the closed Riemannian manifold $(M,g)$. If the symmetric endomorphism $\mathfrak{R}$ (Formula (1)) is positive semi-definite, then ${\dim }_{\mathbb{R}}ker{\Delta }_E\le r$. In particular, if $\mathfrak{R}>0$ at each point $x\in M$, then ${dim}_{\mathbb{R}}\ker {\Delta }_E=0$.

If $(M,g)$ is a connected and complete Riemannian manifold and $E\to M$ is a Riemannian vector bundle of rank $r$ over $M,$ then we can define the vector space (compare with [4], p. 104)

$$L^k\left(Ker{\Delta }_E\right)=\left\{\phi \in \mathcal{H}\left(E\right):∥\phi ∥\in L^k\left(M\right)\right\},$$

where ${∥\phi ∥}^2=g(\phi ,\phi )$. Next, we will define the conditions for the triviality of this vector space over a complete non-compact Riemannian manifold.

Using (1) and the second Kato inequality (see [1], p. 380)

$$∥\phi ∥·\Delta ∥\phi ∥\ge -g(\overline{\Delta }\phi ,\phi )$$

we obtain the following inequality:

$$∥\phi ∥·\Delta ∥\phi ∥\ge -g({\Delta }_L\phi ,\phi )+tg(\mathfrak{R}\left(\phi \right),\phi ).$$

(5)

In particular, for a harmonic section $\phi \in C^{\infty }\left(E\right)$, we have

$$∥\phi ∥·\Delta ∥\phi ∥\ge tg(\mathfrak{R}\left(\phi \right),\phi ).$$

At the same time, we recall that if $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and $\widehat{R}$ is positive definite, then $\mathfrak{R}$ is also positive definite. If, in addition, $t>0,$ then $∥\phi ∥·\Delta ∥\phi ∥\ge 0.$ In this case, by the S.-T. Yau theorem the above inequality we conclude that for the any positive number $k>1,$ either ${\int }_M{∥\phi ∥}^{k}d{v}_g=\infty$ or $∥\phi ∥=C$ for some constant $C\ge 0$ (see [27]). In particular, if $∥\phi ∥\in L^k\left(M\right)$ for some the positive number $k>1$ and the volume of $(M,g)$ is infinite, then the constant C is zero. At the same time, we recall that, if a complete non-closed Riemannian manifold $(M,g)$ with a non-negative curvature operator of the second kind has a non-negative sectional curvature (see [16]), then its volume is infinite (see [27]). □

For this case, we have the following

Theorem 3.

Let ${\Delta }_E$ be the Laplacian on $C^{\infty }$-sections of a Riemannian fibre bundle $E\to M$ of rank r over a complete and non-compact Riemannian manifold $(M,g)$, satisfying (1) for $t>0$. If the representation ρ: $\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ is positive semi-definite at each point $x\in M$, then $L^k\left(Ker{\Delta }_L\right)$ is trivial for any $k>1$.

2.3. Eigenvalues of the Lichnerowicz-Type Laplacian

An eigensection of the Lichnerowicz-type Laplacian ${\Delta }_E$ is $\phi \in C^{\infty }\left(E\right)$, satisfying the conditions ${\Delta }_E{\phi }_x=\lambda {\phi }_x$ and $\phi \ne 0$ at each point $x\in M,$ where $\lambda$ is a constant called the eigenvalue of ${\Delta }_E$ corresponding to $\phi .$ Since ${\Delta }_E$ is an elliptic operator, we can conclude that, on a closed manifold $(M,g),$ the space $E\left(\lambda \right)$ of the eigensection of ${\Delta }_E$ associated with the eigenvalue $\lambda$ has a finite dimension. An estimate for the dimension of this space can be found in [1], p. 389. Moreover, the set of all eigenvalues of the Lichnerowicz-type Laplacian ${\Delta }_E$ is discrete. Following article [1], we denote by ${Spec}^{\left(r\right)}{\Delta }_E={\left\{{\lambda }_a\right\}}_{a\ge 1}$ (resp. ${{Spec}^{\left(0\right)}\overline{\Delta }=\left\{{\overline{\lambda }}_a\right\}}_{a\ge 1}$) the non-decreasing sequence of the eigenvalues of ${\Delta }_E$ (resp. $\overline{\Delta }$) counted with multiplicities. Then, for the case $t=1$, the variational characterization of the eigenvalues of these spectra will be as follows (see [1], p. 393):

$${\lambda }_a\ge {\overline{\lambda }}_a+{\mathfrak{R}}_{min}$$

(6)

for all $a\ge 1$. Here, we recall that the rough Laplacian $\overline{\Delta }$ acting on $C^{\infty }\left(E\right)$ is an order 2 elliptic operator and that its spectrum on a closed $(M,g)$ is an unbounded sequence of real numbers ${Spec}^{\left(0\right)}\overline{\Delta }={\left\{{\overline{\lambda }}_a\right\}}_{a\in \mathbb{N}}$ which can be increasingly ordered (see [23])

$${{0=\overline{\lambda }}_0\le \overline{\lambda }}_1\le {\overline{\lambda }}_2\le \cdots \le {\overline{\lambda }}_k\le {\overline{\lambda }}_{k+1}\le \cdots$$

with the following convention: ${\overline{\lambda }}_0$ is the zero eigenvalue with multiplicity $\dim (Ker\nabla ).$ In cases where there is no parallel section i.e., $\dim (Ker\nabla )=0$, the spectrum starts with the positive eigenvalue ${\overline{\lambda }}_1$. In addition, each eigenvalue ${\overline{\lambda }}_a$ has finite multiplicity, and the eigenspaces $E_{{\overline{\lambda }}_a}$ corresponding to distinct eigenvalues are $L^2$-orthogonal. Next, let $\overline{\Delta }\phi ={\overline{\lambda }}_a\phi$ for a nonzero parallel section $\phi \in C^{\infty }\left(E\right);$ then the second Kato inequality (see [1], p. 380) gives

$$∥\phi ∥·\Delta ∥\phi ∥\ge -g(\overline{\Delta }\phi ,\phi )=-{\overline{\lambda }}_a{∥\phi ∥}^2$$

for any $\phi \in C^{\infty }\left(E\right)$. As a result, we obtain the inequality

$$\Delta ∥\phi ∥\ge -{\overline{\lambda }}_a∥\phi ∥.$$

Then, after integrating, we deduce

$$0\ge -{\overline{\lambda }}_a{\int }_M∥\phi ∥d{v}_g.$$

From this inequality, we conclude that ${\overline{\lambda }}_a>0$ for any $a\ge 1$. In particular (see [26], p. 1215), if we suppose that $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and $\widehat{R}$ is positive semi-definite, then $\mathfrak{R}$ is also positive semi-definite. In this case, if $t=1,$ then, from Theorem 2 and Equation (6), we conclude ${\lambda }_a\ge {\overline{\lambda }}_a+{\mathfrak{R}}_{min}\ge 0$ for all $a\ge 1.$

If ${\Delta }_E{\phi }_x={\lambda }_a{\phi }_x$ for some ${\phi }_x\ne 0$ at each point $x\in M$, then, after integrating (2), we deduce

$$0={\int }_M\left(-{\lambda }_a{\parallel \phi \parallel }^2+tg(\mathfrak{R}\left(\phi \right),\phi )+{\parallel \nabla \phi \parallel }^2\right)d{v}_g\ge$$

$$\ge (-{\lambda }_a+t{\mathfrak{R}}_{min}){\int }_M{∥\phi ∥}^{2}d{v}_g.$$

(7)

for an arbitrary $t>0$. In turn, from (7) we can conclude that ${\lambda }_a\ge t{\mathfrak{R}}_{min}$. On other hand, if we define the number ${\mathfrak{R}}_{max}=\sup \{{\mathfrak{R}}_{max}\left(x\right):x\in M\}$ for

$${\mathfrak{R}}_{\max }\left(x\right)=\sup \{g{(\mathfrak{R}\phi ,\phi )}_x:{\phi }_x\in E_x,g{(\phi ,\phi )}_x=1\},$$

we then can deduce the inequality

$$0\ge (-{\lambda }_a+t{\mathfrak{R}}_{\max }){\int }_M{∥\phi ∥}^{2}d{v}_g$$

for an arbitrary $t<0.$ Then, from the integral inequality above, we obtain that ${\lambda }_a\ge t{\mathfrak{R}}_{max}$ for all $a\ge 1.$ As a result, we formulate the following statement.

Theorem 4.

Let ${\Delta }_E:C^{\infty }\left(E\right)\to C^{\infty }\left(E\right)$ be the Lichnerowicz-type Laplacian defined on $C^{\infty }$-sections of a Riemannian vector bundle $E\to M$ over a closed Riemannian manifold $(M,g).$ Let ${\Delta }_E$ satisfy the Weitzenböck decomposition formula ${\Delta }_E=\overline{\Delta }+t\mathfrak{R}$ where $t\in \mathbb{R}$, $t\ne 0$, and $\mathfrak{R}$ is the Weitzenböck curvature operator, defining a symmetric endomorphism $\mathfrak{R}:E_x\to E_x$ at each $x\in M.$ Suppose that ${\left\{{\lambda }_a\right\}}_{a\ge 1}$ is a non-decreasing sequence of nonzero eigenvalues of ${\Delta }_E$, counted with multiplicities. Then, for a fixed positive number t, we have ${\lambda }_a\ge t{\mathfrak{R}}_{min}$, and, for a fixed negative number t, ${\lambda }_a\ge t{\mathfrak{R}}_{max}$.

In particular (see [26], p. 1215), if we suppose that $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and $\widehat{R}$ is positive definite, then $\mathfrak{R}$ is also positive definite. In this case, if $t>0$, then, from Theorem 1, we obtain that $\lambda \ge t{\mathfrak{R}}_{min}>0$ for all $a\ge 1.$ As a result, we formulate the following statement.

Corollary 1.

Let ${\Delta }_E:C^{\infty }\left(E\right)\to C^{\infty }\left(E\right)$ be the Lichnerowicz-type Laplacian defined on $C^{\infty }$-sections of a Riemannian vector bundle over a closed Riemannian manifold $(M,g)$. Let ${\Delta }_E$ satisfy the Weitzenböck formula ${\Delta }_E=\overline{\Delta }+t\mathfrak{R}$, where $t>0$. Suppose that ${\left\{{\lambda }_a\right\}}_{a\ge 1}$ is a non-decreasing sequence of ${\Delta }_E$ nonzero eigenvalues counted with multiplicities. If the representation ρ: $\mathfrak{s}o\left(n\right)\to EndE$ has no trivial factors and the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ is positive definite, then ${\lambda }_a>0$ for all $a\ge 1.$

The following consequences are obvious.

Corollary 2.

Let ${\Delta }_E$: $C^{\infty }\left(E\right)\to C^{\infty }\left(E\right)$ be the Lichnerowicz-type Laplacian defined on $C^{\infty }$-sections of a Riemannian vector bundle over a closed Riemannian manifold $(M,g)$. Let ${\Delta }_E$ satisfies the Weitzenböck formula ${\Delta }_E=\overline{\Delta }+t\mathfrak{R}$, where $t>0$. If the representation $\rho :\mathfrak{s}o\left(n\right)\to EndE$ has no trivial factors and the sectional curvature of $(M,g)$ is weakly α-pinched with $\alpha =1-(3/(2\left[n/2\right]+1))$, then ${dim}_{\mathbb{R}}ker{\Delta }_E=0.$

Corollary 3.

Let ${\Delta }_E$: $C^{\infty }\left(E\right)\to C^{\infty }\left(E\right)$ be the Lichnerowicz-type Laplacian defined on $C^{\infty }$-sections of a Riemannian vector bundle over a closed Riemannian manifold $(M,g)$. Let ${\Delta }_E$ satisfy the Weitzenböck decomposition formula ${\Delta }_E=\overline{\Delta }+t\mathfrak{R},$ where $t>0.$ Suppose that ${\left\{{\lambda }_a\right\}}_{a\ge 1}$ is a non-decreasing sequence of ${\Delta }_E$ nonzero eigenvalues counted with multiplicities. If the representation $\rho :\mathfrak{s}o\left(n\right)\to \mathrm{End}E$ has no trivial factors and the sectional curvature of $(M,g)$ is weakly α-pinched with $\alpha =1-(3/(2\left[n/2\right]+1))$, then ${\lambda }_a>0$ for all $a\ge 1.$

3. The Hodge–De Rham Laplacian: Theorems on the Vanishing of Its Kernel and an Estimated Theorem for Its Smallest Eigenvalue

3.1. The Hodge–De Rham Laplacian as an Example of Lichnerowicz-Type Laplacian

In this section, we will consider the Hodge–de Rham Laplacian as an example of Lichnerowicz-type Laplacian. For this, we set $t=1$ and $E={\Lambda }^{q}M$ for the vector bundle of exterior differential q-forms $(1\le q\le n-1)$ of rank $r=\dim {\Lambda }^q(T_{x}M)=({\displaystyle \genfrac{}{}{0pt}{}{n}{q}})$ over an n-dimensional Riemannian manifold $(M,g).$ In this case, ${\Delta }_E$ from (1) is the well-known Hodge–de Rham Laplacian${\Delta }_H:C^{\infty }\left({\Lambda }^{q}M\right)\to C^{\infty }\left({\Lambda }^{q}M\right)$ where the Weitzenböck curvature operator ${\mathfrak{R}:=\mathfrak{R}}_q$ restricted to p-forms (see details in [6]).

Next, let $(M,g)$ be covered by a system of coordinate neighborhoods $\left\{U,x^1,\dots ,x^n\right\}$ where U denotes a neighborhood and $x^1,\dots ,x^n$ denote local coordinates in U. Then, we can define the natural frame $\left\{X_1=\partial /\partial x^1,\dots ,X_n=\partial /\partial x^n\right\}$ in an arbitrary coordinate neighborhood $\left\{U,x^1,\dots ,x^n\right\}$. In this case, $g_{ij}=g(X_i,X_j)$ are local components of the metric tensor $g$ with the indices $i,j,k,l,\cdots \in \{1,2,\dots ,n\}$. Next, we denote by $R_{ik}$ and $R_{ikjl}$ the local components of the Ricci $Ric$ and curvature R tensors, respectively. Then, the Hodge–de Rham Laplacian ${\Delta }_H:C^{\infty }\left({\Lambda }^{q}M\right)\to C^{\infty }\left({\Lambda }^{q}M\right)$ with respect to local coordinates $x^1,\dots ,x^n$ has the form

$${\Delta }_H{\omega }_{i_1\dots i_q}=\overline{\Delta }{\omega }_{i_1\dots i_q}+{{\mathfrak{R}}_p\left(\omega \right)}_{i_1\dots i_q},$$

where (see, for example, ref. [28])

$${{\mathfrak{R}}_q\left(\omega \right)}_{i_1\dots i_p}=\sum _{a=1}^q{g}^{jk}{R}_{i_{a}j}{\omega }_{i_1\dots i_{a-1}k{i}_{a+1}\dots i_p}-\sum _{\begin{array}{c}a,b=1\\ a\ne b\end{array}}^q{g}^{jk}{g}^{lm}{R}_{i_a{i}_{b}jl}{\omega }_{i_1\dots i_{a-1}k{i}_{a+1}\dots i_{b-1}l{i}_{b-1}\dots i_p}$$

for $\omega \in C^{\infty }\left({\Lambda }^{q}M\right)$. In particular, ${\mathfrak{R}}_1=Ric$. In turn, (2) can be rewritten in the form

$${\displaystyle \frac{1}{2}}\Delta {∥\omega ∥}^2=-g({\Delta }_H\omega ,\omega )+g({\mathfrak{R}}_q\left(\omega \right),\omega )+{∥\nabla \omega ∥}^2,$$

where (see, for example, ref. [29])

$$g({\mathfrak{R}}_q\left(\omega \right),\omega )=q\left(g^{ij}{g}^{i_2{j}_2}\dots g^{i_q{j}_q}{R}_{ij}{\omega }_{{ii}_2\dots i_q}{\omega }_{{jj}_2\dots j_q}\right.-$$

$$\left.{{\displaystyle \frac{q-1}{2}}g}^{ij}{g}^{kl}{g}^{i_3{j}_3}\dots g^{i_q{j}_q}{R}_{i_a{i}_{b}jl}{\omega }_{{iki}_3\dots i_q}{\omega }_{{jlj}_3\dots j_q}\right)=$$

$$q\left(R_{ij}{\omega }^{{ii}_2\dots i_q}{\omega }_{j_2\dots j_q}^j-{\displaystyle \frac{q-1}{2}}{R}_{ikjl}{\omega }^{{iki}_3\dots i_q}{\omega }_{j_3\dots j_q}^{jl}\right)$$

for $\left(g^{ij}\right)={\left(g_{ij}\right)}^{-1}$. In particular, we have

$$\Delta {∥\omega ∥}^2\ge 2g({\mathfrak{R}}_q\left(\omega \right),\omega )$$

for an arbitrary $\omega \in C^{\infty }\left({\Lambda }^{q}M\right)\cap \ker {\Delta }_H$. We recall that, on a closed Riemannian manifold, by the Hodge’s theorem the dimension of the kernel of ${\Delta }_H:C^{\infty }\left({\Lambda }^{q}M\right)\to C^{\infty }\left({\Lambda }^{q}M\right)$ equals the $q^{th}$ Betti number ${\mathfrak{b}}_q\left(M\right)$, and so the Laplacians determine the Euler characteristic $\chi \left(M\right)$. Therefore, we can formulate a vanishing theorem for the kernel of ${\Delta }_H$, which is a corollary of Theorem 2 (also see the classical theorem on the kernel of ${\Delta }_H$ in [6], p. 351; ref. [5], p. 334; ref. [9], pp. 336–337).

Corollary 4.

Let ${\Delta }_H$ be the Hodge–de Rham Laplacian defined on $C^{\infty }$-sections of the fibre bundle of exterior differential q-forms $(1\le q\le n-1)$ over a closed n dimensional Riemannian manifold $(M,g)$. If the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ is positive semi-definite, then $\nabla \phi =0$ for an arbitrary $\phi \in \ker {\Delta }_H$ and ${\dim }_{\mathbb{R}}ker{\Delta }_H={\mathfrak{b}}_q\left(M\right)\le ({\displaystyle \genfrac{}{}{0pt}{}{n}{q}})$. In particular, if $\widehat{R}$ is positive definite at each point $x\in M$, then ${dim}_{\mathbb{R}}\ker {\Delta }_H={\mathfrak{b}}_q\left(M\right)=0.$

Remark 1.

We recall that Böhm and Wilking showed by Ricci-flow techniques that positive curvature operator implies that a closed manifold $(M,g)$ is diffeomorphic (not isometric) to a spherical space form (see [15]).

For the case of a complete and non-compact Riemannian manifold, we obtain the following corollary from Theorem 3.

Corollary 5.

Let ${\Delta }_H$ be the Hodge–de Rham Laplacian defined on $C^{\infty }$-sections of the fibre bundle of exterior differential q-forms over a complete and non-compact n-dimensional Riemannian manifold $(M,g)$ for $(1\le q\le n-1)$. If the curvature operator of the first kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ of $(M,g)$ is positive semi-definite, then $L^k\left(Ker{\Delta }_H\right)$ is trivial for any number $k>1$.

Remark 2.

Our statement above generalizes the following, now classic result from [30,31]: If ${\mathfrak{R}}_q$ is positive semi-define at every point of a complete Riemannian manifold $(M,g),$ then $L^2$-harmonic q-form is parallel. In particular, if either exists a point $x\in M$ such that ${\mathfrak{R}}_q$ is strictly positive at $x$ or the volume of $(M,g)$ is infinite, then every $L^2$-harmonic q-form is identically zero. In addition, in our paper (see [32]), we gave a classification of connected complete, locally irreducible Riemannian manifolds with a non-positive curvature operator that admits a nonzero closed or closed conformal $L^2$-Killing form (see [28]). Furthermore, we proved vanishing theorems for these forms on some complete Riemannian manifolds. These statements can serve as examples of our above results and methods.

3.2. New Theorems on the Kernel of the Hodge Laplacian and Its Eigenvalues

Having discussed the kernel of the Hodge Laplacian ${\Delta }_H$, we now turn our attention to its first positive eigenvalue on a closed Riemannian manifold, which we will denote by ${\lambda }_1^{\left[q\right]}$. Note here that the superscript $\left[q\right]$ refers to the degree of the involved eigenform. We also recall that the spectrum ${Spec}^{\left(q\right)}{\Delta }_H$ of the Hodge Laplacian consists only of non-negative eigenvalues with finite multiplicity. We also denote its positive eigenvalues counted with multiplicity by

$$0={\lambda }_0^{\left[q\right]}<{\lambda }_1^{\left[q\right]}\le {\lambda }_2^{\left[q\right]}\le \cdots \le {\lambda }_k^{\left[q\right]}\le {\lambda }_{k+1}^{\left[q\right]}\le \cdots ,$$

where the multiplicity of the eigenvalue 0 is equal to the q-th Betti number ${\mathfrak{b}}_q\left(M\right)$ of $(M,g)$, by the Hodge–de Rham theory (see, for example, [6]; ref. [9], p. 339). The case $q=0$ corresponds to the Laplace–Beltrami $\overline{\Delta }=\delta \nabla$ operator acting on $C^{\infty }$-functions. At the same time, we know from [14], p. 78 that if all eigenvalues of $\widehat{R}$ lie in $\left[{\widehat{r}}_{min},{\widehat{r}}_{max}\right],$ then the sectional curvature $sec$ satisfies $1/2{\widehat{r}}_{min}\le sec\le 1/2{\widehat{r}}_{max}$. Therefore, if the inequality $\widehat{R}\ge C>0$ holds, then, from the above, we have $sec\ge 1/2C$. In this case, $Ric\ge 1/2(n-1)C,$ and, as Lichnerowicz has already proven, ${\lambda }_q^{\left[0\right]}\ge 1/2nC$ (see, for example, ref. [9], p. 82). A similar result can be formulated about the eigenvalues of the Laplacians $\overline{\Delta }$ and ${\Delta }_H$.

Theorem 5.

Let $(M,g)$ be an n-dimensional closed Riemannian manifold. Let $\overline{\Delta }$ and ${\Delta }_H$ be the rough and Hodge–de Rham Laplacians acting defined on $C^{\infty }$-sections of the fibre bundle ${\Lambda }^{q}M$ of differential q-forms, $1\le q\le n-1$. If the curvature operator of the second kind $\widehat{R}:{\Lambda }^{2}M\to {\Lambda }^{2}M$ satisfies the inequality $\widehat{R}\ge C>0$ and $(M,g)$ is not isometric to the Euclidean n-sphere ${\mathbb{S}}^n$ with its standard metric, then ${\overline{\lambda }}_a^{\left[q\right]}>1/2nC$ and ${\lambda }_q^{\left[q\right]}>1/2nC+q(n-q)C$ for any eigenvalues ${\overline{\lambda }}_a^{\left[q\right]}$ and ${\lambda }_q^{\left[q\right]}$ of ${Spec}^{\left(q\right)}\overline{\Delta }$ and ${Spec}^{\left(q\right)}{\Delta }_H,$ respectively.

Proof. 

Let $(M,g)$ be an n-dimensional closed Riemannian manifold. We recall that, if there exists a positive constant C on $(M,g)$ such that the inequality $g(\widehat{R}\omega ,\omega )\ge C{∥\omega ∥}^2$ holds for any two-form $\omega \in {\Lambda }^{2}M$, then the inequality $g({\mathfrak{R}}_q\left(\omega \right),\omega )\ge q(n-q)C{∥\omega ∥}^2$ holds for any $\omega \in {\Lambda }^{q}M$ and $1\le q\le n-1$ (see [10]). In addition, equality holds for a Riemannian manifold isometric to the Euclidean n-sphere ${\mathbb{S}}^n$ with its standard metric. In other words, C serves here as a lower bound for the eigenvalues of the curvature operator of the first kind $\widehat{R}$ of $(M,g)$, and, in turn, ${\mathfrak{R}}_{min}=q(n-q)C$ serves here as a lower bound for the eigenvalues of the Weitzenböck curvature operator ${\mathfrak{R}}_q$ of $(M,g),$ respectively. In this case, the variational characterization of the eigenvalues (6) is as follows

$${\lambda }_a^{\left[q\right]}\ge {\overline{\lambda }}_a^{\left[q\right]}+q(n-q)C$$

(8)

for any ${\lambda }_a^{\left[q\right]}{\in Spec}^{\left(q\right)}{\Delta }_H$ and ${\overline{\lambda }}_a^{\left[q\right]}\in {Spec}^{\left(q\right)}\overline{\Delta }$. In contract with the Hodge Laplacian ${\Delta }_H$, the kernel of the rough Laplacian $\overline{\Delta }$ acting on q-forms consists of a parallel q-form whose dimension is not a topological invariant.Therefore, if ${\overline{\lambda }}_a^{\left[q\right]}=0$, then the associated eigenspace consists of parallel q-forms. At the same time, it is well-known that there are no parallel q-forms $(1\le q\le n-1)$ on a closed Riemannian manifold with a positive curvature operator of the first kind $\widehat{R}$ (see [6], p. 351). Therefore, in our case, ${\overline{\lambda }}_a^{\left[q\right]}\ne 0,$ i.e., for the metric $g$ with $\widehat{R}\ge C>0$, all eigenvalues of the rough and Hodge Laplacians acting on q-forms, $1\le q\le n-1$, are nonzero.

At the same time, for the metric $g$ with $\widehat{R}\ge C>0$ and every q, $1\le q\le 1/2n,$ inequality (8) can be rewritten as the first Gallot-Meyer inequality (see, for example, inequality (3.4) from [33])

$${\lambda }_a^{\left[q\right]}\ge qC+q(n-q)C.$$

In turn, for the metric $g$ with $\widehat{R}\ge C>0$ and every q, $1/2n\le q\le n-1$, inequality (8) can be rewritten as the second Gallot-Meyer inequality (see, for example, inequality (3.3) from [33])

$${\lambda }_a^{\left[q\right]}\ge (n-q)C+q(n-q)C,$$

because $1\le (n-q)\le 1/2n.$ Moreover, two lower bounds of ${\lambda }_a^{\left[q\right]}$ are optimal because they can be achieved for a Riemannian manifold $(M,g)$ isometric to the Euclidean n-sphere ${\mathbb{S}}^n$ with its standard metric; in other words, for this variety, the equalities are valid in both cases (see also [9], p. 342). Therefore, if a Riemannian manifold $(M,g)$ isometric to the Euclidean n-sphere ${\mathbb{S}}^n,$ then the first Gallot-Meyer inequality can be rewritten as the equality ${\lambda }_a^{\left[q\right]}=qC+q(n-q)C$ for every q, $1\le q\le 1/2n$. In this case, from (8) we deduce ${\overline{\lambda }}_a^{\left[q\right]}\le 1/2nC$ for all $a\ge 1.$ A similar conclusion can be drawn for the case when $1/2n\le q\le n-1$. Therefore, ${\overline{\lambda }}_a^{\left[q\right]}>1/2nC$ for an n-dimensional closed Riemannian manifold with $\widehat{R}\ge C>0$ and not isometric to the Euclidean n-sphere ${\mathbb{S}}^n$. Then, from (8), we deduce that ${\lambda }_q^{\left[q\right]}>1/2nC+q(n-q)C$. □

Then, our corollary holds.

Reamrk. 3.

If ${\overline{\lambda }}_a^{\left[q\right]}>1/2nC$, then both strictly Gallo-Meyer inequalities will automatically follow from (8) for an n-dimensional closed Riemannian manifold with $\widehat{R}\ge C>0$ and not be isometric to the Euclidean n-sphere ${\mathbb{S}}^n$.

4. The Lichnerowicz Laplacian: From a New Differentiable Sphere Theorem to a Vanishing Theorem for Its Kernel and an Estimated Theorem for Its Smallest Eigenvalue

4.1. Lichnerowicz Laplacian Defined on Symmetric Tensors

We let $t=1$ and $E=S^{p}M$ is the bundle of symmetric covariant tensor fields over a Riemannian manifold $(M,g)$. For this case, we have the pointwise identity $\dim S^p\left(T_{x}M\right)=({\displaystyle \genfrac{}{}{0pt}{}{n+p-1}{p}})$ at an arbitrary point $x\in M$. In this case, ${\Delta }_E$ from (1) is the well-known Lichnerowicz Laplacian ${\Delta }_L$: $C^{\infty }\left(S^{p}M\right)\to C^{\infty }\left(S^{p}M\right)$ acting on the space of covariant symmetric p-tensors. The Weitzenböck decomposition Formula (1) can be expressed in the form ${\Delta }_L=\overline{\Delta }+{\mathfrak{R}}_p$, where ${\mathfrak{R}}_p$ is the Weitzenböck curvature operator. It is now an algebraic operator, representing the restriction of the Weitzenböck curvature operator $\mathfrak{R}$ to symmetric p-tensors. This differential operator, initially introduced by Lichnerowicz in [18], is self-adjoint, elliptic, and preserves the symmetries of tensor fields. Furthermore, the Weitzenböck curvature operator ${\mathfrak{R}}_p$: $S^{p}M\to S^{p}M$ satisfies the following identities (see [18], p. 315):

$$g(\mathfrak{R}\left(\phi \right),{\phi }^{\prime })=g(\phi ,\mathfrak{R}\left({\phi }^{\prime }\right)),{trace}_g\mathfrak{R}\left(\phi \right)=\mathfrak{R}\left({trace}_g\phi \right)$$

(9)

for any $\phi ,{\phi }^{\prime }\in S^{p}M$. If ${\Delta }_L\phi =0$, the tensor field $\phi \in C^{\infty }\left(S^{p}M\right)$ is referred as ${\Delta }_L$-harmonic. We will continue to denote the vector space of these tensors as $\mathcal{H}\left(S^{p}M\right).$

The Lichnerowicz Laplacian ${\Delta }_L:C^{\infty }\left(S^{p}M\right)\to C^{\infty }\left(S^{p}M\right)$ has the form

$${\Delta }_L{\phi }_{i_1\dots i_p}=\overline{\Delta }{\phi }_{i_1\dots i_p}+{{\mathfrak{R}}_p\left(\phi \right)}_{i_1\dots i_p},$$

where (see [3], p. 54; ref. [18], p. 315)

$${{\mathfrak{R}}_p\left(\phi \right)}_{i_1\dots i_p}=\sum _{a=1}^p{g}^{jk}{R}_{i_{a}j}{\phi }_{i_1\dots i_{a-1}k{i}_{a+1}\dots i_p}-\sum _{\begin{array}{c}a,b=1\\ a\ne b\end{array}}^p{g}^{jk}{g}^{lm}{R}_{i_{a}j{i}_{b}l}{\phi }_{i_1\dots i_{a-1}k{i}_{a+1}\dots i_{b-1}l{i}_{b-1}\dots i_p}$$

for the local components ${\phi }_{i_1\cdots i_p}=\phi (X_{i_1},\cdots ,X_{i_p})$ of an arbitrary $\phi \in C^{\infty }\left(S^{p}M\right).$

For the case when $p=2,$ as described in the papers [18], p. 316 and [17], p. 388, the Lichnerowicz Laplacian is defined by the formula

$${\Delta }_L{\phi }_{ij}=\overline{\Delta }{\phi }_{ij}+(R_{ik}{\phi }_j^k+R_{jk}{\phi }_i^k-2R_{ikjl}{\phi }^{kl}).$$

(10)

We recall here that the symmetric operator $\stackrel{°}{R}$: $S_0^{2}M\to S_0^{2}M$ determined by the equations $\stackrel{°}{R}{\left(\phi \right)}_{ij}=R_{iklj}{\phi }^{kl}$ is called as the curvature operator of the second kind (see [16]). We can also say that $\stackrel{°}{R}>0$ if the eigenvalues of $\stackrel{°}{R}$ as a bilinear form on $S_0^{2}M$ are positive definite everywhere on $(M,g)$. In addition, the well-known (see, for example, ref. [16]) that if $\stackrel{°}{R}>0$ (resp., $\stackrel{°}{R}\ge 0$), then the sectional curvature is positive (resp., non-negative). Moreover, easy calculations give that $(M,g)$ is Einstein (i.e., $Ric=s/n·g$) if, and only if, $\stackrel{°}{R}$ maps $S_0^{2}M$ into itself at each point $x\in M$. In particular, $\stackrel{°}{R}$ is identical with $s/n(n-1)\times \left(theidentitymap\right)$ on $S_0^2{\mathbb{S}}^n$ for the Euclidean sphere ${\mathbb{S}}^n$ with the standard metric $g_0$. Therefore, the curvature operator $\stackrel{°}{R}$ is positive on the Euclidean sphere ${\mathbb{S}}^n$. Using the above, we can rewrite (10) in the following form

$${\Delta }_L{\phi }_{ij}=\overline{\Delta }{\phi }_{ij}+(R_{ik}{\phi }_j^k+R_{jk}{\phi }_i^k+2{\stackrel{°}{R}(\phi )}_{ij}).$$

We showed above that the curvature operator of the second kind arises naturally as a term in the Lichnerowicz Laplacian, including the curvature tensor.

4.2. A New Variant of Differential Sphere Theorem

Let us consider the quadratic form ${\mathcal{Q}}_p\left(\phi \right):S^{p}M\otimes S^{p}M\to \mathbb{R}$ defined by the equality

$${\mathcal{Q}}_p\left(\phi \right)=g({\mathfrak{R}}_p\left(\phi \right),\phi )={\mathfrak{R}\left(\phi \right)}_{i_1\cdots i_p}{\phi }^{i_1\cdots i_p}=$$

$$=p(R_{ij}{{\phi }^{ik{k}_3\dots k_p}\phi }_{{kk}_3\dots k_p}^j-(p-1)R_{ijkl}{\phi }^{ik{k}_3\dots k_p}{\phi }_{k_3\dots k_p}^{jl})$$

(11)

where ${\phi }^{i_1\dots i_p}=g^{i_1{j}_1}\dots g^{i_p{j}_p}{\phi }_{j_1\dots j_p}$ are local contravariant components of an arbitrary $\phi \in S^{p}M$. In particular, from (10) by direct calculation we obtain that the quadratic form ${\mathcal{Q}}_2\left(\phi \right)$ has the form

$${\mathcal{Q}}_2\left(\phi \right)=g({\mathfrak{R}}_2\left(\phi \right),\phi )=\mathfrak{R}\left(\phi \right){\phi }_{ij}{\phi }^{ij}$$

$$=2(R_{ij}{{\phi }^{ik}\phi }_k^j-R_{ijkl}{\phi }^{ik}{\phi }^{jl}).$$

(12)

In the last case, there exists an orthonormal basis $e_1,\dots ,e_n$ of $T_{x}M$ at any point $x\in M$ such that ${\phi }_x(e_i,e_j)={\epsilon }_i{\delta }_{ij}$ for the Kronecker delta ${\delta }_{ij}$. Then, the quadratic form ${\mathcal{Q}}_2\left(\phi \right)$ can be rewritten in the form

$$g({\mathfrak{R}}_2\left({\phi }_x\right),{\phi }_x)=\sum _{i<j}sec(e_i,e_j){({\epsilon }_i-{\epsilon }_j)}^2$$

(13)

where $sec(e_i,e_j)=R(e_i,e_j,e_i,e_j)$ is the sectional curvature $sec\left({\sigma }_x\right)$ of $(M,g)$ in direction to ${\sigma }_x=span\left\{e_i,e_j\right\}\subset T_{x}M$ at an arbitrary point $x\in M$ (see [17], pp. 387–388). In this case, we rewrite (13) in the form

$$R_{ij}{{\phi }^{ik}\phi }_k^j-R_{ijkl}{\phi }^{ik}{\phi }^{jl}=\sum _{i<j}sec(e_i,e_j){({\epsilon }_i-{\epsilon }_j)}^2.$$

Next, we will consider a smooth section of a sub-bundle $S_0^{p}M\subset S^{p}M$ of covariant symmetric p-tensor fields which are totally traceless; that is, traceless on any pair of indices, i.e., ${\sum }_{i=1}^n\phi (e_i,e_i,X_3\cdots ,X_n)=0$ for an orthonormal basis $\left\{e_1,e_2,\cdots ,e_n\right\}$ for $T_{x}M$ at an arbitrary point $x\in M.$ It is well-known that $\dim S_0^p\left(T_{x}M\right)=({\displaystyle \genfrac{}{}{0pt}{}{n+p-1}{p}})-({\displaystyle \genfrac{}{}{0pt}{}{n+p-3}{n-1}})$.

Since the unit sphere in $T_{x}M$ at an arbitrary point $x\in M$ is a closed set, then there exists the number $K\left(x\right)$ such that, for all 2-planes, ${\sigma }_x\subset T_{x}M$, the inequality $K\left(x\right)\le sec\left({\sigma }_x\right)$ is satisfied. In other words, $K\left(x\right)=K_{\min }\left(x\right)$, where $K_{\min }\left(x\right)$ is the minimum of the sectional curvature at $x\in M$. Then, the following equalities hold:

$$\sum _{i<j}\sec (e_i,e_j){({\epsilon }_i-{\epsilon }_j)}^2\ge K_{\min }\left(x\right)\sum _{i<j}{({\epsilon }_i-{\epsilon }_j)}^2$$

$$=K_{\min }\left(x\right)\left((n-1)\sum _i{\left({\epsilon }_i\right)}^2-2\sum _{i<j}{\epsilon }_i{\epsilon }_j\right)$$

$$=K_{min}\left(x\right)\left(n\sum _i{({\epsilon }_i)}^2-{(\sum _i{\epsilon }_i)}^2\right)=n{K}_{\min }\left(x\right){∥\phi ∥}^2$$

for ${\sigma }_x=span\left\{e_i,e_j\right\}$ and $\left({trace}_g\phi \right)\left(x\right)={\epsilon }_1+\cdots +{\epsilon }_n=0.$. Then, from the above, we conclude that at each point $x\in M$ the following inequality is satisfied

$$R_{ij}{{\phi }^{ik}\phi }_k^j-R_{ijkl}{\phi }^{ik}{\phi }^{jl}\ge n{K}_{\min }\left(x\right){∥\phi ∥}^2$$

for the local contravariant components ${{\phi }^{kl}=g^{ki}{g}^{lj}\phi }_{ij}$ of an arbitrary $\phi \in S_0^2\left(T_{x}M\right)$ at each point $x\in M.$ The last inequality we can rewrite in the form

$$g(\stackrel{°}{R}(\phi ),\phi )\ge n{K}_{\min }\left(x\right){∥\phi ∥}^2-R_{ij}{{\phi }^{ik}\phi }_k^j.$$

Using the above, we can state that, if the inequality $R_{ij}{{\phi }^{ik}\phi }_k^j<n{K}_{\min }\left(x\right){∥\phi ∥}^2$ holds for any $\phi \in S_0^2\left(T_{x}M\right)$ at each point, $x\in M,$ then the condition $\stackrel{°}{R}>0$ is satisfied. In conclusion, we note that, if the inequality $R_{ij}{X}^i{X}^j<n{K}_{\min }\left(x\right){∥X∥}^2$ holds for any $X\in T_{x}M$, then the inequality $R_{ij}{{\phi }^{ik}\phi }_k^j<n{K}_{\min }\left(x\right){∥\phi ∥}^2$ also holds for any nonzero $\phi \in S_0^2\left(T_{x}M\right)$ at each point $x\in M$ (also see [20], p. 82). Therefore, from (12) and the above comments, we can conclude the following statement.

Lemma 1.

The curvature operator of the second kind $\stackrel{°}{R}$ is positive everywhere on an n-dimensional $(n\ge 2)$ Riemannian manifold $(M,g)$ if the sectional curvature and Ricci curvature $Ric$ of $(M,g)$ satisfy the inequality $Ric\left(X\right)<n{K}_{\min }\left(x\right)$ for any $X\in T_{x}M$ at each point $x\in M.$

Next, if the Ricci curvature $Ric$ of $(M,g)$ satisfies the strict inequality $Ric\left(X\right)<$$n{K}_{\min }\left(x\right)$ for any unit vector $X\in T_{x}M$ and the minimum $K_{\min }\left(x\right)$ of the sectional curvature at an arbitrary point $x\in M$, then the section curvature of $(M,g)$ is positive for all 2-planes ${\sigma }_x\subset T_{x}M$.

Since the unit sphere in $T_{x}M$ at an arbitrary point $x\in M$ is a closed set, there exists the number ${Ric}_{\max }\left(x\right):={\sup }_{X\in T_{x}M}Ric\left(X\right)$. Then, with the above condition, the Ricci curvature $Ric$ of $(M,g)$ satisfies the strict inequality $Ric\left(X\right)<n{K}_{\min }\left(x\right)$ for any unit vector $X\in T_{x}M$ and the minimum $K_{\min }\left(x\right)$ of the sectional curvature of $(M,g)$ at an arbitrary its point $x\in M$ we can replace with the following more-closed condition: ${Ric}_{\max }\left(x\right)<{nK}_{\min }\left(x\right)$ at each point $x\in M$. At the same time, we recall that if $(M,g)$ is a closed Riemannian manifold such that $\stackrel{°}{R}$ is positive, then M is diffeomorphic to a spherical space form (see [16]). Therefore, can formulate the following corollary (compare with [20]).

Corollary 6.

Let $(M,g)$ be an n-dimensional $(n\ge 3)$ closed Riemannian manifold and $Ric$ be its Ricci tensor satisfying the inequality

$${Ric}_{\max }\left(x\right)<{nK}_{\min }\left(x\right)$$

(14)

for ${Ric}_{\max }\left(x\right):={\sup }_{X\in T_{x}M}Ric\left(X\right)$ and the minimum $K_{\min }\left(x\right)$ of the sectional curvature of $(M,g)$ at each point $x\in M$. Then, $(M,g)$ is diffeomorphic to a spherical space form ${\mathbb{S}}^n/\Gamma$. In particular, if $(M,g)$ is simply connected, then $(M,g)$ is diffeomorphic to the Euclidean sphere ${\mathbb{S}}^n.$

We note that the simplest examples of spherical space forms are the sphere ${\mathbb{S}}^n$ and the real projective space $\mathbb{R}{\mathbb{P}}^n$. Furthermore, when $n$ is even, these are the only examples. In particular, for $n=2$ the “differentiable sphere theorem” can be rewritten as follows: a closed surface of positive Gaussian curvature is diffeomorphic to ${\mathbb{S}}^2$ or $\mathbb{R}{\mathbb{P}}^2$. Therefore, we can formulate the following corollary.

Corollary 7.

Let $(M,g)$ be even dimensionclosed Riemannian manifold theRicci curvature satisfying the inequality ${Ric}_{\max }\left(x\right)<{nK}_{\min }\left(x\right)$ for ${Ric}_{\max }\left(x\right):={\sup }_{X\in T_{x}M}Ric\left(X\right)$ and the minimum $K_{\min }\left(x\right)$ of the sectional curvature of $(M,g)$ at each point $x\in M.$ Then, it is the Euclidean sphere or real projective space.

Remark 4.

Ogiue and Tachibana proved in [34] that an n-dimensional closed Riemannian manifold $(M,g)$ with the positive curvature operator of the second kind is a rational homology spheres. We can reformulate this statement in the following form: an n-dimensional $(n\ge 2)$ closed Riemannian manifold $(M,g)$ is a real homology sphere if its Ricci and sectional curvatures satisfy the inequality ${Ric}_{\max }\left(x\right)<{nK}_{\min }\left(x\right)$ at each of its point $x\in M$.

4.3. Vanishing Theorems for Harmonic Symmetric Tensors

The value of Weitzenböck expansions lies mainly in the disappearance theorems (see [24]). These theorems will be the topic of this section.

Since the curvature operator of the second kind $\stackrel{°}{R}$ defined on traceless symmetric two-tensors on a closed Riemannian manifold $(M,g)$ is positive, there exists a positive number C such that $R_{ijkl}{\phi }^{il}{\phi }^{jk}\ge C{∥\phi ∥}^2>0$ for an arbitrary $\phi \in S_0^{2}M$ such that $∥\phi ∥\ne 0$(see [35]). In other words, C is the smallest eigenvalue of the symmetric operator $\stackrel{°}{R}$: $S_0^{2}M\to S_0^{2}M$ defined on a closed manifold $(M,g)$. Let $(M,g)$ be a closed manifold. In this case, the inequalities $R_{ijkl}{\phi }^{il}{\phi }^{jk}\ge C{∥\phi ∥}^2\ge K_{\min }\left(x\right){∥\phi ∥}^2\ge 0$ are also satisfied for an arbitrary $\phi \in S_0^{2}M$. As a result, we have the following inequalities (see [5], pp. 82, 85 and [35])

$$R_{ijkl}{\phi }^{il{i}_3\dots .i_p}{\phi }_{i_3\dots .i_p}^{jk}\ge C{\phi }^{i_1\dots .i_p}{\phi }_{i_1\dots .i_p}\ge K_{\min }\left(x\right){\phi }^{i_1\dots .i_p}{\phi }_{i_1\dots .i_p},$$

$$R_{ij}{\phi }^{i{k}_2\dots k_p}{\phi }_{k_2\dots k_p}^j\ge (n-1)K_{\min }\left(x\right){∥\phi ∥}^2.$$

Therefore, the quadratic form ${\mathcal{Q}}_p\left(\phi \right)$ must satisfy the following conditions:

$$g({\mathfrak{R}}_p\left(\phi \right),\phi )={\mathfrak{R}\left(\phi \right)}_{i_1\dots i_p}{\phi }^{i_1\dots i_p}=$$

$$={p·R}_{ij}{{\phi }^{ik{k}_3\dots k_p}\phi }_{{kk}_3\dots k_p}^j-p(p-1)R_{ijkl}{\phi }^{ik{k}_3\dots k_p}{\phi }_{k_3\dots k_p}^{jl}$$

$$={p·R}_{ij}{{\phi }^{ik{k}_3\dots k_p}\phi }_{{kk}_3\dots k_p}^j+p(p-1)R_{ijkl}{\phi }^{jk{k}_3\dots k_p}{\phi }_{k_3\dots k_p}^{il}\ge$$

$$\ge p(n+p-2)K_{\min }\left(x\right){∥\phi ∥}^2\ge 0.$$

(15)

Equality in (15) is possible only if $\phi =0$.

Next, from (9), we conclude that ${\Delta }_L:C^{\infty }\left(S_0^{p}M\right)\to C^{\infty }\left(S_0^{p}M\right)$. Then, by direct calculations we obtain from (1) the following formula:

$${\displaystyle \frac{1}{2}}\Delta {∥\phi ∥}^2=-g({\Delta }_L\phi ,\phi )+{∥\nabla \phi ∥}^2+g({\mathfrak{R}}_p\left(\phi \right),\phi ).$$

(16)

Let $(M,g)$ be a closed Riemannian manifold and $K_{\min }\left(x\right)$ be the minimum of the sectional curvature at its point x$\in M$. If ${\Delta }_L\phi =0$ for some $\phi \in C^{\infty }\left(S_0^{p}M\right),$ then, from (16), we deduce the integral inequality

$$0={\int }_M({∥\nabla \phi ∥}^2+g({\mathfrak{R}}_p\left(\phi \right),\phi ))d{v}_g\ge$$

$$\ge p(n+p-2){\int }_M{K_{\min }\left(x\right)∥\phi ∥}^{2}d{v}_g$$

(17)

From (17), we can conclude that there are no nonzero ${\Delta }_L$-harmonic traceless symmetric p-tensor $(p\ge 2)$ fields on closed manifolds $(M,g)$ with a positive sectional curvature. Threfore, n the following theorem holds.

Theorem 6.

If the Ricci curvature $Ric$ of an n-dimensional $(n\ge 2)$ closed Riemannian manifold $(M,g)$ satisfies the following inequality ${Ric}_{\max }\left(x\right)<{nK}_{\min }\left(x\right)$ for ${Ric}_{\max }\left(x\right):={\sup }_{X\in T_{x}M}Ric\left(X\right)$ and the minimum $K_{\min }\left(x\right)$ of the sectional curvature at each point $x\in M$, then the vector space $\mathcal{H}\left(S^{p}M\right)$ of harmonic p-forms is trivial for any $p\ge 2.$

Next, if we suppose that $sec$ is positive and ${Ric}_{max}\left(x\right)<n{K}_{min}\left(x\right)$ at each point $x\in M$, then, from the above, we have the inequality $R_{ijkl}{\phi }^{jk}{\phi }^{il}\ge 0$ for an arbitrary $\phi \in S_0^{2}M$ with local components ${\phi }_{ij}$. Therefore, the inequality $R_{ijkl}{\phi }^{j{kk}_3\dots k_p}{\phi }_{k_3\dots k_p}^{il}\ge 0$ is satisfied for an arbitrary $\phi \in S_0^{p}M$ with local components ${\phi }_{i_1\dots i_p}$ for $p\ge 2$(also see [34]). At the same time, we have the double inequality

$$(n-1)K_{\min }{∥\phi ∥}^2\le R_{ij}{\phi }^{i{k}_2\dots k_p}{\phi }_{k_2\dots k_p}^j\le n{K}_{\min }{∥\phi ∥}^2$$

for an arbitrary $\phi \in S_0^{p}M$ with local components ${\phi }_{i_1\dots i_p}$ for $p\ge 2$ (also see [11], p. 82). As a result, from (11), we conclude that the quadratic form

$$g({\mathfrak{R}}_p\left(\phi \right),\phi )=p(R_{ij}{{\phi }^{ik{k}_3\dots k_p}\phi }_{{kk}_3\dots k_p}^j+(p-1)R_{ijkl}{\phi }^{jk{k}_3\dots k_p}{\phi }_{k_3\dots k_p}^{il})\ge$$

$$\ge p(n-1)K_{\min }\left(x\right){∥\phi ∥}^2\ge 0.$$

(18)

Equality in (18) is possible only if $\phi =0$.

Next, consider the vector space $Ker{\Delta }_L$ on a closed Riemannian manifold $(M,g).$ If ${\Delta }_L\phi =0$ for $\phi \in C^{\infty }\left(S_0^{p}M\right)$, then, from (18), we deduce the integral formula

$$0={\int }_M({∥\nabla \phi ∥}^2+g({\mathfrak{R}}_p\left(\phi \right),\phi ))d{v}_g\ge p(n-1)K_{\min }{\int }_M{∥\phi ∥}^{2}d{v}_g.$$

From the above inequality, we can conclude that there are no nonzero ${\Delta }_L$-harmonic traceless symmetric p-tensor $(p\ge 2)$ fields on closed manifolds $(M,g)$ with a positive sectional curvature. Therefore, the following corollary holds.

Corollary 8.

If $(M,g)$ is an n-dimensional $(n\ge 2)$ closed Riemannian manifold satisfies the condition ${Ric}_{\max }\le n{K}_{\min }$, then ${dim}_{\mathbb{R}}Ker{\Delta }_L=0$ for the Lichnerowicz Laplacian ${\Delta }_L:C^{\infty }\left(S_0^{p}M\right)$→$C^{\infty }\left(S_0^{p}M\right)$ and for any $p\ge 2.$

For the case of complete non-compact Riemannian manifolds we have the following

Theorem 7.

Let $(M,g)$ be a connected complete non-compact Riemannian manifold and ${\Delta }_L:C^{\infty }\left(S^{p}M\right)$→$C^{\infty }\left(S^{p}M\right)$ be the Lichnerowicz Laplacian. If the sectional curvature is a non-negative at each point of $(M,g)$, then $L^r\left(Ker{\Delta }_L\right)$ is trivial for any number $r>1$.

Proof. 

Let $(M,g)$ be a connected complete non-compact Riemannian manifold and ${\Delta }_L:C^{\infty }\left(S^{p}M\right)\to C^{\infty }\left(S^{p}M\right)$ be the Lichnerowicz Laplacian. By direct calculation, we deduce from (13) the following inequality (also see (5)):

$$∥\phi ∥·\Delta ∥\phi ∥\ge -g({\Delta }_L\phi ,\phi )+g({\mathfrak{R}}_p\left(\phi \right),\phi ).$$

In this case, if $\phi \in C^{\infty }\left(S^{p}M\right)$ is a ${\Delta }_L$-harmonic section and the quadratic form $g({\mathfrak{R}}_p\left(\phi \right),\phi )$ is non-negative at each point of M, then the right-hand side of the above inequality is also non-negative. At the same time, the fact that $\sec \ge 0$ implies negative-semi-definiteness of $g({\mathfrak{R}}_p\left(\phi \right),\phi )$ for all $\phi \in C^{\infty }\left(S^{p}M\right)$, and $p\ge 2$ was previously obtained in [22], p. 1218. At the same time, a complete non-compact Riemannian manifold with a non-negative sectional curvature has an infinite volume [7]. To complete the proof, we need to refer to Theorem 2. □

4.4. A Lower Bound for the Eigenvalues of the Lichnerowicz Laplacian Defined on Symmetric Tensors

The Lichnerowicz Laplacian ${\Delta }_L$: $C^{\infty }\left(S_0^{p}M\right)\to C^{\infty }\left(S_0^{p}M\right)$ has a discrete spectrum ${Spec}^{\left(p\right)}{\Delta }_L={\left\{{\lambda }_a^{\left[p\right]}\right\}}_{a\ge 1}$ on $(M,g)$, since it is an elliptic and formally self-adjoint second-order differential operator on a closed $(M,g)$. If ${\lambda }_a^{\left[p\right]}\in {Spec}^{\left(p\right)}{\Delta }_L$ is a nonzero eigenvalue of ${\Delta }_L$ corresponding to a nonzero eigentensor $\phi \in C^{\infty }\left(S_0^{p}M\right)$ for the case $p\ge 2$, then, from (17), we obtain the integral inequality

$$({\lambda }_a^{\left[p\right]}-p(n+p-2)K_{\min }){\int }_M{\parallel \phi \parallel }^{2}d{v}_g\ge 0.$$

In this case, we can conclude that the following theorem holds.

Theorem 8.

Let $(M,g)$ be an n-dimensional $(n\ge 2)$ closed Riemannian manifold $(M,g)$ with a strictly positive sectional curvature and the Ricci curvature satisfying the inequality ${Ric}_{\max }<n{K}_{\min }$ for the minimum $K_{min}$ of the sectional curvature; then, a nonzero eigenvalue ${\lambda }_a^{\left[p\right]}$ of the Lichnerowicz Laplacian ${\Delta }_L$: $C^{\infty }\left(S_0^{p}M\right)\to C^{\infty }\left(S_0^{p}M\right)$ satisfies the inequality ${\lambda }_a^{\left[p\right]}\ge p(n+p-2)K_{\min }$.

4.5. On the Stability of the Einstein Metrics

Finally, we will consider the Lichnerowitz Laplacian ${\Delta }_L$ acting on the vector bundle $S_0^{2}M$ of symmetric traceless 2-tensor fields, which can be regarded as infinitesimal Einstein deformations of the metric g and, therefore, arises in the analysis of the stability of the Einstein metrics (see details in [3], chapter 12).

Namely, let g be an Einstein metric on a closed manifold M, i.e., $Ric={\displaystyle \frac{s}{n}}g$ for the scalar curvatures of $(M,g).$ A symmetric 2-tensor field $\phi$ is an infinitesimal Einstein deformation of g if, and only if, it satisfies the following equation (see [3], p. 347):

$${\Delta }_L\phi =2{\displaystyle \frac{s}{n}}\phi ;\delta \phi =0;{trace}_g\phi =0.$$

(19)

If $\phi \in {trace}_g^{-1}\left(0\right)\cap {\Delta }^{-1}\left(0\right)$, then it is called a transverse traceless tensor or $TT$-tensor. Therefore, if $\phi$ is an infinitesimal Einstein deformation of g, then it is a $TT$-tensor and an eigenform of the Lichnerowicz Laplacian ${\Delta }_L$ and $2{\displaystyle \frac{s}{n}}$ is its eigenvalue. On the other hand, if $2{\displaystyle \frac{s}{n}}$ is not an eigenvalue of ${\Delta }_L$, then g is not deformable, i.e., Einstein deformations do not exist. We also recall that, if an Einstein metric g does not have infinitesimal Einstein deformations, then it is called rigid(see [3], p. 347).

Equation (10) of the Lichnerowicz Laplacian can be rewritten in the form

$${\Delta }_L{\phi }_{ij}=\overline{\Delta }{\phi }_{ij}+R_{ik}{\phi }_j^k+R_{jk}{\phi }_i^k+2\stackrel{°}{R}{\left(\phi \right)}_{ij}$$

(20)

for the curvature operator of the second kind $\stackrel{°}{R}:S^{2}M\to S^{2}M$. Then, using (19) and (20), we can rewrite Equation (16) in the form

$${\displaystyle \frac{1}{2}}\Delta {∥\phi ∥}^2={∥\nabla \phi ∥}^2+2g(\stackrel{°}{R}\left(\phi \right),\phi ).$$

(21)

Next, from (21), we deduce the integral formula

$${\int }_M({∥\nabla \phi ∥}^2+2g(\stackrel{°}{R}\left(\phi \right),\phi ))d{v}_g=0.$$

It is obvious that the inequality $\stackrel{°}{R}>0$ conflicts with the above integral equality. In addition, the inequality ${Ric}_{\max }<n{K}_{\min }$ can be rewritten in the form $s<n^2{K}_{\min }.$ Therefore, the following statement holds.

Corollary 9.

Let $(M,g)$ be a closed Einstein manifold of dimensional $n\ge 2$. If its scalar curvature s satisfies the inequality $s<n^2{K}_{\min },$ where $K_{\min }$ is the minimum of the sectional curvature of $(M,g)$, then its metric is rigid.

Remark 5.

Firstly, we recall here that there are only three known Einstein four-manifolds with positive sectional curvature; namely, the four-sphere ${\mathbb{S}}^4$ and the four-dimensional real projective space $\mathbb{R}{\mathbb{P}}^4$ with a positive constant sectional curvature and the complex projective space $\mathbb{C}{\mathbb{P}}^2$ with the Fubini Study metric. Therefore, we can conclude that Einstein four-manifolds with positive sectional curvatures and $\alpha <n{K}_{min}$ are Euclidean spheres or real projective spaces. Secondary, we can also recall that an n-dimensional ($n>2$) Riemannian manifold has a positive constant sectional curvature and is not an infinitesimal Einstein deformable (see [3], p. 132).

5. Summary

The Bochner technique has remained one of the main analytical methods in regard to Riemannian geometry for more than half a century (see, for instance, the monographs [1]; ref. [6], pp. 333–364; ref. [4], among others). It is an important part of geometric analysis (see, for instance, ref. [8]). In a famous monograph by Besse about the Bochner technique, the following is written: the Bochner technique is a method of proving vanishing theorems for the null space of a Laplace operator admitting a Weitzenböck decomposition and further of estimating its lowest nonzero eigenvalue (see [3], p. 53). In this article, we have considered a generalized form of the famous Lichnerowicz Laplacian (see [18]) and shown how the Bochner technique works for this operator while giving two important examples of it, namely the Hodge–de Rham Laplacian acting on forms and the usual Lichnerowicz Laplacian acting on symmetric tensors. Note that other examples of the generalized Lichnerowicz Laplacian can be found in the literature (see, for instance, ref. [3], sections 1.143–1.155, refs. [21,22]).