On the Betti and Tachibana Numbers of Compact Einstein Manifolds

Abstract

Throughout the history of the study of Einstein manifolds, researchers have sought relationships between the curvature and topology of such manifolds. In this paper, first, we prove that a compact Einstein manifold $(M,g)$ with an Einstein constant $\alpha >0$ is a homological sphere when the minimum of its sectional curvatures $>\alpha /(n+2)$; in particular, $(M,g)$ is a spherical space form when the minimum of its sectional curvatures $>\alpha /n$. Second, we prove two propositions (similar to the above ones) for Tachibana numbers of a compact Einstein manifold with $\alpha <0$.

1. Introduction

The study of Einstein manifolds has a long history in Riemannian geometry. Throughout the history of the study of Einstein manifolds, researchers have sought relationships between curvature and topology of such manifolds. A. Besse [1] summarized the results. We present here some interesting facts related to the classification of all compact Einstein manifolds satisfying a suitable curvature inequality, which is one of the subjects of our research.

Recall that an n-dimensional $(n\ge 2)$ connected manifold M with a Riemannian metric g is said to be an Einstein manifold with Einstein constant $\alpha$ if its Ricci tensor satisfies $\mathrm{Ric}=\alpha g$; moreover, we have $\alpha =s/n$ for its scalar curvature s. Therefore, any Einstein manifold of dimensions two and three is a space form (i.e., has constant sectional curvature). The study of Einstein manifolds is more complicated in dimension four and higher (see [1] (p. 44)).

An important problem in differential geometry is to determine whether a smooth manifold admits an Einstein metric. When $\alpha >0$, the example are symmetric spaces, which include the sphere ${\mathbb{S}}^n\left(1\right)$ with $\alpha =n-1$ and the sectional curvature $sec=1$, the product of two spheres ${\mathbb{S}}^n\left(1\right)\times {\mathbb{S}}^n\left(1\right)$ with $\alpha =n-1$ and $0\le sec\le 1$, and the complex projective space $\mathbb{C}{P}^m={\mathbb{S}}^{2m+1}/{\mathbb{S}}^1$ with the Fubini–Study metric, $\alpha =2m+2$ and $1\le sec\le 4$ (see [2] (pp. 86, 118, 149–150)). Recall that if $(M,g)$ is a compact Einstein manifold with curvature bounds of the type $3n/(7n-4)<sec\le 1$, then $(M,g)$ is isometric to a spherical space form. This might be not the best estimate: for $n=4$ the sharp bound is $1/4$ (see [1] (p. 6)). In both these cases, the manifolds are real homology spheres (see [3] (p. XVI)). Therefore, any such manifold has the homology groups of an n-sphere; in particular, its Betti numbers are $b_1\left(M\right)=\dots =b_{n-1}\left(M\right)=0$.

One of the basic problems in Riemannian geometry was to classify Einstein four-manifolds with positive or nonnegative sectional curvature in the categories of either topology, diffeomorphism, or isometry (see, for example, [4,5,6,7]). It was conjectured that an Einstein four-manifold with $\alpha >0$ and non-negative sectional curvature must be either ${\mathbb{S}}^4,\mathbb{C}{\mathrm{P}}^2$, ${\mathbb{S}}^2\left(1\right)\times {\mathbb{S}}^2\left(1\right)$ or a quotient. For example, if the maximum of the sectional curvatures of a compact Einstein four-manifold is bounded above by $(2/3)\alpha$, or if $\alpha =1$ and the minimum of the sectional curvatures $\ge (1/6)(2-\sqrt{2})$, then the manifold is isometric to ${\mathbb{S}}^4,\mathbb{R}{\mathrm{P}}^4$ or $\mathbb{C}{\mathrm{P}}^2$ (see [6]). Classification of four-dimensional complete Einstein manifolds with $\alpha >0$ and pinched sectional curvature was obtained in [7].

Here, we consider this problem from another side. Given a Riemannian manifold $(M,g)$, the notion of symmetric curvature operator$\overline{R}$, acting on the space ${\Lambda }^{2}M$ of 2-forms, is an important invariant of a Riemannian metric (see [2] (p. 83); [8,9]). The Tachibana Theorem (see [10]) asserts that a compact Einstein manifold $(M,g)$ with $\overline{R}>0$ is a spherical space form. Later on, it was proved that compact manifolds with $\overline{R}>0$ are spherical space forms (see [11]).

Denote by $\stackrel{\circ }{R}$ the symmetric curvature operator of the second kind, acting on the space $S_0^{2}M$ of traceless symmetric two-tensors (see [1] (p. 52); [9,12]). Kashiwada (see [9]) proved that a compact Einstein manifold with $\stackrel{\circ }{R}>0$ is a spherical space form. This statement is an analogue of the theorem of Tachibana in [10]. In contrast, if a complete Riemannian manifold $(M,g)$ satisfies $sec\ge \delta >0$, then M is compact with $\mathrm{diam}(M,g)\le \pi /\sqrt{\delta }$ (see [2] (p. 251)).

Remark 1

(By [2] (Theorem 10.3.7)). There are manifolds with metrics of positive or nonnegative sectional curvature but not admitting any metric with $\overline{R}\ge 0$ (see also [2] (p. 352)). In particular, for three-dimensional manifolds the inequality $sec>0$ is equivalent to the inequality $\overline{R}>0$ (see [9]).

Using Kashiwada’s theorem from [9] we can prove the following.

Theorem 1.

Let $(M,g)$ be a compact Einstein manifold with Einstein constant $\alpha >0$, and let δ be the minimum of its positive sectional curvature. If $\delta >\alpha /n$, then $(M,g)$ is a spherical space form.

We can present a generalization of above result in the following form.

Theorem 2.

Let $(M,g)$ be a compact Einstein manifold with Einstein constant $\alpha >0$ and let δ be the minimum of its positive sectional curvature. If $\delta >\alpha /(n+2)$, then $(M,g)$ is a homological sphere.

Obviously, ${\mathbb{S}}^n\left(1\right)\times {\mathbb{S}}^n\left(1\right)$ is not an example for Theorem 1 because the minimum of its sectional curvature is zero and $\alpha =n-1$. On the other hand, the complex projective space $\mathbb{C}{\mathrm{P}}^m$ is an Einstein manifold with $\alpha =2m+2$ and sectional curvature bounded below by $\delta =1$. Then the inequality $\alpha <(n+2)\delta$ can be rewritten in the form $\delta >1$ because $n=2m$. Therefore, $\mathbb{C}{\mathrm{P}}^m$ is not an example for Theorem 1. Moreover, all even dimensional Riemannian manifolds with positive sectional curvature have vanishing odd-dimensional homology groups. Thus, Theorem 1 complements this statement (see [2] (p. 328)).

Let $(M,g)$ be an n-dimensional compact connected Riemannian manifold. Denote by ${\Delta }^{\left(p\right)}$ the Hodge Laplacian acting on differential p-forms on M for $p=1,\dots ,n-1$. The spectrum of ${\Delta }^{\left(p\right)}$ consists of an unbounded sequence of nonnegative eigenvalues which starts from zero if and only if the p-th Betti number $b_p\left(M\right)$ of $(M,g)$ does not vanish (see [13]). The sequence of positive eigenvalues of ${\Delta }^{\left(p\right)}$ is denoted by

$$0<{\lambda }_1^{\left(p\right)}<\dots <{\lambda }_m^{\left(p\right)}<\dots \to \infty .$$

In addition, if $F_p\left(\omega \right)\ge \sigma >0$ (see Equation (4) of $F_p$) at every point of M, then ${\lambda }_1^{\left(p\right)}\ge \sigma$ (see [13] (p. 342)). Using this and Theorem 1, we get the following.

Corollary 1.

Let $(M,g)$ be a compact Einstein manifold with positive Einstein constant α and sectional curvature bounded below by a constant $\delta >0$ such that $\delta >\alpha /(n+2)$. Then the first eigenvalue ${\lambda }_1^{\left(p\right)}$ of the Hodge Laplacian ${\Delta }^{\left(p\right)}$ satisfies the inequality ${\lambda }_1^{\left(p\right)}\ge (1/3)((n+2)\delta -\alpha )(n-p)$.

Remark 2.

In particular, if $(M,g)$ is a Riemannian manifold with curvature operator of the second kind bounded below by a positive constant $\rho >0$, then using the main theorem from [14], we conclude that ${\lambda }_1^{\left(p\right)}\ge \rho (n-p)$.

Conformal Killing p-forms ($p=1,\dots ,n-1$) were defined on Riemannian manifolds more than fifty years ago by S. Tachibana and T. Kashiwada (see [15,16]) as a natural generalization of conformal Killing vector fields.

The vector space of conformal Killing p-forms on a compact Riemannian manifold $(M,g)$ has finite dimension $t_p\left(M\right)$ named the Tachibana number (see e.g., [17,18,19]). Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ are conformal scalar invariants of $(M,g)$ satisfying the duality condition $t_p\left(M\right)=t_{n-p}\left(M\right)$. The condition is an analog of the Poincaré duality for Betti numbers. Moreover, Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ are equal to zero on a compact Riemannian manifold with negative curvature operator or negative curvature operator of the second kind (see [18,19]).

We obtain the following theorem, which is an analog of Theorem 1.

Theorem 3.

Let $(M,g)$ be an Einstein manifold with sectional curvature bounded above by a negative constant $-\delta$ such that $\delta >-\alpha /(n+2)$ for the Einstein constant α. Then Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ are zero.

2. Proof of Results

Let $(M,g)$ be an n-dimensional $(n\ge 2)$ Riemannian manifold and let $R_{ijkl}$ and $R_{ij}$ be, respectively, the components of the Riemannian curvature tensor and the Ricci tensor in orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{x}M$ at any point $x\in M$. We consider an arbitrary symmetric two-tensor $\phi$ on $(M,g)$. At any point $x\in M$, we can diagonalize $\phi$ with respect to g, using orthonormal basis $\{e_1,\dots ,e_n\}$ of $T_{x}M$. In this case, the components of $\phi$ have the form ${\phi }_{ij}={\lambda }_i{\delta }_{ij}$. Let $sec(e_i,e_j)$ be the sectional curvature of the plane of $T_{x}M$ generated by $e_i$ and $e_j$. We can express $sec(e_i,e_j)$ in the following form (see [1] (p. 436); [20]):

$$\frac{1}{2}{\sum }_{i\ne j}sec(e_i,e_j){({\lambda }_i-{\lambda }_j)}^2=R_{ijlk}{\phi }^{ik}{\phi }^{jl}+R_{ij}{\phi }^{ik}{\phi }_k^j$$

(1)

If $(M,g)$ is an Einstein manifold and its sectional curvature satisfies the inequality $sec\ge \delta$ for a positive constant $\delta$, then from Equation (1) we obtain the inequality

$$R_{ijlk}{\phi }^{ik}{\phi }^{jl}+\frac{s}{n}{\phi }^{ik}{\phi }_{ik}^{}\ge (\delta /2){\sum }_{i\ne j}({\lambda }_i-{\lambda }_j){}^2.$$

(2)

If ${\mathrm{trace}}_g\phi ={\sum }_i{\lambda }_i=0$, then the identity holds ${\sum }_i{\left({\lambda }_i\right)}^2=-2{\sum }_{i<j}{\lambda }_i{\lambda }_j$. In this case, the following identities are true:

$$\frac{1}{2}\sum _{i\ne j}{({\lambda }_i-{\lambda }_j)}^2=(n-1)\sum _i{\left({\lambda }_i\right)}^2-2\sum _{i<j}{\lambda }_i{\lambda }_j=n\sum _i{\left({\lambda }_i\right)}^2=n{\parallel \phi \parallel }^2.$$

Then the inequality in Equation (2) can be rewritten in the form

$$R_{ijlk}{\phi }^{ik}{\phi }^{jl}+\frac{s}{n}{\phi }^{ik}{\phi }_{ik}\ge n\delta {\parallel \phi \parallel }^2.$$

(3)

From Equation (3) we obtain the inequality

$$R_{ijlk}{\phi }^{ik}{\phi }^{jl}\ge (n\delta -\alpha ){\parallel \phi \parallel }^2.$$

Then $\stackrel{\circ }{R}>0$ for the case when $\alpha <n\delta$, where $\alpha =s/n$ is the Einstein constant of $(M,g)$. If $(M,g)$ is compact then it is a spherical space form (see [9]). Theorem 1 is proven.

Define the quadratic form

$$F_p\left(\omega \right)=R_{ij}{\omega }^{i{i}_2\dots i_p}{\omega }_{i_2\dots i_p}^j-\frac{p-1}{2}{R}_{ijkl}{\omega }^{ij{i}_3\dots i_p}{\omega }_{i_2\dots i_p}^{kl}$$

(4)

for the components ${\omega }_{i_1\dots i_p}=\omega (e_{i_1},\dots ,e_{i_p})$ of an arbitrary differential p-form $\omega$. If the quadratic form $F_p\left(\omega \right)$ is positive definite on a compact Riemannian manifold $(M,g)$, then the p-th Betti number of the manifold vanishes (see [21] (p. 61); [3] (p. 88)). At the same time, in [22] the following inequality

$$F_p\left(\omega \right)\ge p(n-p)\epsilon {\parallel \omega \parallel }^2>0$$

was proved for any nonzero p-form $\omega$ on a Riemannian manifold with $\overline{R}\ge \epsilon >0$. On the other hand, in [14] the inequality

$$F_p\left(\omega \right)\ge p(n-p)\delta {\parallel \omega \parallel }^2>0$$

was proved for any nonzero p-form $\omega$ on a Riemannian manifold with $\stackrel{\circ }{R}\ge \delta >0$. In these cases, $b_1\left(M\right),\dots ,b_{n-1}\left(M\right)$ are zero (see [21]). We can improve these results for the case of Einstein manifolds. First, we will prove the following.

Lemma 1.

Let $(M,g)$ be an Einstein manifold with Einstein constant α and sectional curvature bounded below by a constant $\delta >0$. If $\alpha <(n+2)\delta$ then

$$F_p\left(\omega \right)\ge (1/3)((n+2)\delta -\alpha )(n-p){\parallel \omega \parallel }^2>0$$

for any nonzero p-form ω and an arbitrary $1\le p\le n-1$.

Proof. 

Let $p\le [n/2]$, then we can define the symmetric traceless two-tensor ${\phi }^{\left(i_1{i}_2\dots i_p\right)}$ with components (see [14])

$${\phi }_{jk}^{\left(i_1{i}_2\dots i_p\right)}=\sum _{a=1}^p\left({\omega }_{i_1\dots i_{a-1}ji{}_{a+1}\dots i_p}{g}_{k{i}_a}+{\omega }_{i_1\dots i_{a-1}k{i}_{a+1}\dots i_p}{g}_{j{i}_a}\right)-\frac{2p}{n}{g}_{jk}{\omega }_{i_1\dots i_p}$$

for each set of values of indices $\left(i_1{i}_2\dots i_p\right)$ such that $1\le i_1<i_2<\dots <i_p\le n$. After long but simple calculations we obtain the identities (see also [14]),

$$\begin{array}{cc}\hfill & R_{ijkl}{\phi }^{il\left(i_1\dots i_p\right)}{\phi }_{\left(i_1\dots i_p\right)}^{jk}=p(\frac{2(n+4p)}{n}{R}_{ij}{\omega }^{i{i}_2\dots i_p}{\omega }_{i_2\dots i_p}^j\hfill \end{array}$$

$$\begin{array}{c}-3\left(p-1\right)R_{ijkl}{\omega }^{ij{i}_3\dots i_p}{\omega }_{i_3\dots i_p}^{kl}-\frac{4p}{n^2}s{\parallel \omega \parallel }^2);\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & {∥\overline{\phi }∥}^2=\frac{2p(n+2)(n-p)}{n}{\parallel \omega \parallel }^2,\hfill \end{array}$$

(6)

where

$$\begin{array}{c}\hfill {∥\overline{\phi }∥}^2=g^{ik}{g}^{jl}{g}_{i_1{j}_1}\dots g_{i_p{j}_p}{\phi }_{ij}^{{}_{\left(i_1\dots i_p\right)}}{\phi }_{kl}^{\left(j_1\dots j_p\right)},\\ \hfill {\parallel \omega \parallel }^2={\omega }^{i_1{i}_2\dots i_p}{\omega }_{i_1{i}_2\dots i_p}^{}=g^{i_1{j}_1}\dots g^{i_p{j}_p}{\omega }_{i_1\dots i_p}^{}{\omega }_{j_1\dots j_p}\end{array}$$

for $g^{ij}={\left(g^{-1}\right)}_{ij}$. If $(M,g)$ is an Einstein manifold, then Equations (4) and (5) can be rewritten in the form

$$F_p\left(\omega \right)=\frac{s}{n}{\parallel \omega \parallel }^2-\frac{p-1}{2}{R}_{ijkl}{\omega }^{ij{i}_3\dots i_p}{\omega }_{i_3\dots i_p}^{kl},$$

$$R_{ijkl}{\phi }^{il(i_1\dots i_p)}{\phi }_{\left(i_1\dots i_p\right)}^{jk}=p\left(\frac{2n+4p}{n^2}s{\parallel \omega \parallel }^2-3(p-1)R_{ijkl}{\omega }^{ij{i}_3\dots i_p}{\omega }_{i_3\dots i_p}^{kl}\right).$$

(7)

On the other hand, for a fixed set of values of indices $(i_1,i_2,\dots ,i_p)$ such that $1\le i_1<i_2<\dots <i_p\le n$, the equality in Equation (3) can be rewritten in the form

$$R_{ijkl}{\phi }^{il\left(i_1\dots i_p\right)}{\phi }_{}^{jk\left(i_1\dots i_p\right)}+\frac{s}{n}{\phi }^{ik\left(i_1\dots i_p\right)}{\phi }_{ik}^{\left(i_1\dots i_p\right)}\ge n\delta {\phi }^{kl\left(i_1\dots i_p\right)}{\phi }_{{}_{kl}}^{\left(i_1\dots i_p\right)}.$$

(8)

Then from Equation (8) we obtain the inequality

$$R_{ijkl}{\phi }^{il\left(i_1\dots i_p\right)}{\phi }_{\left(i_1\dots i_p\right)}^{jk}\ge \left(n\delta -\frac{s}{n}\right){∥\overline{\phi }∥}^2.$$

(9)

Using Equation (9) we deduce from Equation (7) the following inequality:

$$6p{F}_p\left(\omega \right)\ge \left(n\delta -\frac{s}{n+2}\right){∥\overline{\phi }∥}^2.$$

(10)

Thus, using Equation (6) we can rewrite Equation (10) in the following form:

$$F_p\left(\omega \right)\ge (1/3)((n+2)\delta -\alpha )(n-p){\parallel \omega \parallel }^2.$$

(11)

It is obvious that if the sectional curvature of an Einstein manifold $(M,g)$ satisfies the inequality $sec\ge \delta$ for a positive constant $\delta$, then the scalar curvature of $(M,g)$ satisfies the inequality $s\ge n(n-1)\delta >0$. In this case, if $(n-1)\delta \le \alpha <(n+2)\delta$, then from Equation (11) we deduce that the quadratic form $F_p\left(\omega \right)$ is positive definite for any $p\le [n/2]$. It is known [23] that $F_p\left(\omega \right)=F_{n-p}(\ast \omega )$ and ${\parallel \omega \parallel }^2={\parallel \ast \omega \parallel }^2$ for any p-form $\omega$ with $1\le p\le n-1$ and the Hodge star operator $\ast :{\Lambda }^{p}M\to {\Lambda }^{n-p}M$ acting on the space of p-forms ${\Lambda }^{p}M$. Therefore, the inequality in Equation (11) holds for any $p=1,\dots ,n-1$. □

Recall that if on an n-dimensional compact Riemannian manifold $(M,g)$ the quadratic form $F_p\left(\omega \right)$ is positive definite for any smooth p-form $\omega$ with $p=1,\dots ,n-1$, then the Betti numbers $b_1\left(M\right),\dots ,b_{n-1}\left(M\right)$ vanish (see [3] (p. 88); [13] (pp. 336–337)). In this case, Theorem 2 directly follows from Lemma 1.

If the curvature of an Einstein manifold $(M,g)$ satisfies $sec\le -\delta <0$ for a positive constant $\delta$, then the Einstein constant of $(M,g)$ satisfies the the obvious inequality $\alpha \le -(n-1)\delta <0$. On the other hand, from Equation (1) we deduce the inequality $R_{ijlk}{\phi }^{ik}{\phi }^{jl}\le -\left(n\delta +\alpha \right){∥\phi ∥}^2$. Therefore, if $\delta >-\alpha /n$, then $\stackrel{\circ }{R}<0$. In this case, the Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ are equal to zero (see [19]). We proved the following.

Proposition 1.

Let $(M^n,g)$ be an Einstein manifold with sectional curvature bounded above by a negative constant $-\delta$ such that $\delta >-\alpha /n$ for the Einstein constant α. Then the Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ are zero.

We can complete this result. If an Einstein manifold $(M^n,g)$ satisfies the curvature inequality $sec\le -\delta <0$ for a positive constant $\delta$, then from Equations (3) and (7) we deduce the inequality $F_p\left(\omega \right)\le -\frac{1}{3}((n+2)\delta +\alpha )(n-p){\parallel \omega \parallel }^2$ for any $p=1,\dots ,n-1$. Therefore, the Tachibana numbers $t_1\left(M\right),\dots ,t_{n-1}\left(M\right)$ of a compact Einstein manifold with sectional curvature bounded above by a negative constant $-\delta$ such that $\delta \ge -\alpha /(n+2)$ are zero.