**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* ≥ 2) connected manifold *M* with a Riemannian metric *g* is said to be an *Einstein manifold* with *Einstein constant α* if its Ricci tensor satisfies Ric = *α g*; moreover, we have *α* = *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 *α* > 0, the example are symmetric spaces, which include the sphere S*n*(1) with *α* = *n* − 1 and the sectional curvature sec = 1, the product of two spheres S*n*(1) × S*n*(1) with *α* = *n* − 1 and 0 ≤ sec ≤ 1, and the complex projective space C*Pm* = S2*m*<sup>+</sup>1/ S1 with the Fubini–Study metric, *α* = 2*m* + 2 and 1 ≤ sec ≤ 4 (see [2] (pp. 86, 118, 149–150)). Recall that if (*<sup>M</sup>*, *g*) is a compact Einstein manifold with curvature bounds of the type <sup>3</sup>*n*/(<sup>7</sup>*n* − 4) < sec ≤ 1, then (*<sup>M</sup>*, *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(*M*) = ... = *bn*−<sup>1</sup>(*M*) = 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, [4–7]). It was conjectured that an Einstein four-manifold with *α* > 0 and non-negative sectional curvature must be either S4, CP2, S<sup>2</sup>(1) × S<sup>2</sup>(1) or a quotient. For example, if the maximum of the sectional curvatures of a compact Einstein four-manifold is bounded above by (2/3) *α*, or if *α* = 1 and the minimum of the sectional curvatures ≥ (1/6)(2 − √2), then the manifold is isometric to S4, RP<sup>4</sup> or CP<sup>2</sup> (see [6]). Classification of four-dimensional complete Einstein manifolds with *α* > 0 and pinched sectional curvature was obtained in [7].

Here, we consider this problem from another side. Given a Riemannian manifold (*<sup>M</sup>*, *g*), the notion of symmetric *curvature operator R* ¯ , acting on the space Λ2*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 (*<sup>M</sup>*, *g*) with *R*¯ > 0 is a spherical space form. Later on, it was proved that compact manifolds with *R* ¯ > 0 are spherical space forms (see [11]).

Denote by *R* the symmetric *curvature operator of the second kind*, acting on the space *S*20*M* of traceless symmetric two-tensors (see [1] (p. 52); [9,12]). Kashiwada (see [9]) proved that a compact Einstein manifold with ◦ *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 (*<sup>M</sup>*, *g*) satisfies sec ≥ *δ* > 0, then *M* is compact with diam(*<sup>M</sup>*, *g*) ≤ *π*/√*<sup>δ</sup>* (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 R* ¯ ≥ 0 *(see also [2] (p. 352)). In particular, for three-dimensional manifolds the inequality* sec > 0 *is equivalent to the inequality R* ¯ > 0 *(see [9]).*

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

**Theorem 1.** *Let* (*<sup>M</sup>*, *g*) *be a compact Einstein manifold with Einstein constant α* > 0*, and let δ be the minimum of its positive sectional curvature. If δ* > *α*/*n, then* (*<sup>M</sup>*, *g*) *is a spherical space form.*

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

**Theorem 2.** *Let* (*<sup>M</sup>*, *g*) *be a compact Einstein manifold with Einstein constant α* > 0 *and let δ be the minimum of its positive sectional curvature. If δ* > *α*/(*n* + <sup>2</sup>)*, then* (*<sup>M</sup>*, *g*) *is a homological sphere.*

Obviously, S*n*(1) × S*n*(1) is not an example for Theorem 1 because the minimum of its sectional curvature is zero and *α* = *n* − 1. On the other hand, the complex projective space CP*m* is an Einstein manifold with *α* = 2*m* + 2 and sectional curvature bounded below by *δ* = 1. Then the inequality *α* < (*n* + 2) *δ* can be rewritten in the form *δ* > 1 because *n* = 2*m*. Therefore, CP*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 (*<sup>M</sup>*, *g*) be an *n*-dimensional compact connected Riemannian manifold. Denote by Δ(*p*) the *Hodge Laplacian* acting on differential *p*-forms on *M* for *p* = 1, ... , *n* − 1. The spectrum of Δ(*p*) consists of an unbounded sequence of nonnegative eigenvalues which starts from zero if and only if the *p*-th Betti number *bp*(*M*) of (*<sup>M</sup>*, *g*) does not vanish (see [13]). The sequence of positive eigenvalues of Δ(*p*) is denoted by

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

In addition, if *Fp*(*ω*) ≥ *σ* > 0 (see Equation (4) of *Fp*) at every point of *M*, then *λ*(*p*) 1 ≥ *σ* (see [13] (p. 342)). Using this and Theorem 1, we ge<sup>t</sup> the following.

**Corollary 1.** *Let* (*<sup>M</sup>*, *g*) *be a compact Einstein manifold with positive Einstein constant α and sectional curvature bounded below by a constant δ* > 0 *such that δ* > *α*/(*n* + <sup>2</sup>)*. Then the first eigenvalue λ*(*p*) 1 *of the Hodge Laplacian* Δ(*p*) *satisfies the inequality λ*(*p*) 1 ≥ (1/3) ((*n* + 2) *δ* − *α*) (*n* − *p*)*.*

**Remark 2.** *In particular, if* (*<sup>M</sup>*, *g*) *is a Riemannian manifold with curvature operator of the second kind bounded below by a positive constant ρ* > 0*, then using the main theorem from [14], we conclude that λ*(*p*) 1≥ *ρ* (*n* − *p*)*.*

*Conformal Killing p-forms* (*p* = 1, ... , *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 (*<sup>M</sup>*, *g*) has finite dimension *tp*(*M*) named the *Tachibana number* (see e.g., [17–19]). Tachibana numbers *<sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) are conformal scalar invariants of (*<sup>M</sup>*, *g*) satisfying the duality condition *tp*(*M*) = *tn*−*<sup>p</sup>*(*M*). The condition is an analog of the *Poincaré duality* for Betti numbers. Moreover, Tachibana numbers *<sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) 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* (*<sup>M</sup>*, *g*) *be an Einstein manifold with sectional curvature bounded above by a negative constant* −*δ such that δ* > −*<sup>α</sup>*/(*<sup>n</sup>* + 2) *for the Einstein constant α. Then Tachibana numbers <sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) *are zero.*

## **2. Proof of Results**

Let (*<sup>M</sup>*, *g*) be an *n*-dimensional (*n* ≥ 2) Riemannian manifold and let *Rijkl* and *Rij* be, respectively, the components of the Riemannian curvature tensor and the Ricci tensor in orthonormal basis {*<sup>e</sup>*1, ... ,*en*} of *TxM* at any point *x* ∈ *M*. We consider an arbitrary symmetric two-tensor *ϕ* on (*<sup>M</sup>*, *g*). At any point *x* ∈ *M*, we can diagonalize *ϕ* with respect to *g*, using orthonormal basis {*<sup>e</sup>*1, ... ,*en*} of *TxM*. In this case, the components of *ϕ* have the form *ϕij* = *λi <sup>δ</sup>ij*. Let sec (*ei*,*ej*) be the sectional curvature of the plane of *TxM* generated by *ei* and *ej*. We can express sec (*ei*, *ej*) in the following form (see [1] (p. 436); [20]):

$$\frac{1}{2} \sum\_{i \neq j} \sec\left(e\_i, e\_j\right) (\lambda\_i - \lambda\_j)^2 = R\_{ijlk} q^{ik} \varphi^{jl} + R\_{ij} q^{ik} \varphi^j\_k \tag{1}$$

If (*<sup>M</sup>*, *g*) is an Einstein manifold and its sectional curvature satisfies the inequality sec ≥ *δ* for a positive constant *δ*, then from Equation (1) we obtain the inequality

$$R\_{\bar{i}\bar{j}k}\boldsymbol{\varrho}^{\bar{i}k}\boldsymbol{\varrho}^{\bar{j}l} + \frac{s}{n}\boldsymbol{\varrho}^{\bar{i}k}\boldsymbol{\varrho}\_{\bar{i}k} \ge \left(\delta/2\right)\sum\_{i\neq j} \left(\lambda\_{\bar{i}} - \lambda\_{\bar{j}}\right)^{2}.\tag{2}$$

If trace*g ϕ* = ∑*i λi* = 0, then the identity holds ∑*i*(*<sup>λ</sup>i*)<sup>2</sup> = −2 ∑*<sup>i</sup>*<*<sup>j</sup> λi λj* . In this case, the following identities are true:

$$\frac{1}{2} \sum\_{i \neq j} (\lambda\_i - \lambda\_j)^2 = (n - 1) \sum\_i (\lambda\_i)^2 - 2 \sum\_{i < j} \lambda\_i \, \lambda\_j = n \sum\_i (\lambda\_i)^2 = n \| \, \oint \|^2.$$

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

$$R\_{ijlk}\boldsymbol{q}^{\vec{\text{ik}}}\boldsymbol{q}^{\vec{\text{ik}}} + \frac{s}{n}\boldsymbol{q}^{\vec{\text{ik}}}\boldsymbol{q}\_{ik} \ge n\,\delta \|\boldsymbol{q}\|^2. \tag{3}$$

From Equation (3) we obtain the inequality

$$\mathcal{R}\_{ijlk}\boldsymbol{q}^{ik}\boldsymbol{q}^{jl} \ge (n\delta - \boldsymbol{a})\left\|\boldsymbol{q}\right\|^2.$$

◦

Then *R* > 0 for the case when *α* < *n δ*, where *α* = *s*/*n* is the Einstein constant of (*<sup>M</sup>*, *g*). If (*<sup>M</sup>*, *g*) is compact then it is a spherical space form (see [9]). Theorem 1 is proven.

Define the quadratic form

$$F\_p(\omega) = R\_{i\bar{j}} \omega^{i\bar{i}\_2 \dots \bar{i}\_p} \omega^{\bar{j}}\_{i\_2 \dots i\_p} - \frac{p-1}{2} R\_{i\bar{j}kl} \omega^{i\bar{j}\bar{i}\_3 \dots \bar{i}\_p} \omega^{kl}\_{i\_2 \dots i\_p} \tag{4}$$

for the components *<sup>ω</sup>i*1...*ip* = *<sup>ω</sup>*(*ei*1 , ... ,*eip* ) of an arbitrary differential *p*-form *ω*. If the quadratic form *Fp*(*ω*) is positive definite on a compact Riemannian manifold (*<sup>M</sup>*, *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

$$F\_p(\omega) \ge p \left( n - p \right) \varepsilon \left\| \omega \right\|^2 > 0$$

was proved for any nonzero *p*-form *ω* on a Riemannian manifold with *R* ¯ ≥ *ε* > 0. On the other hand, in [14] the inequality

$$F\_p(\omega) \ge p(n-p)\delta \|\omega\|^2 > 0$$

was proved for any nonzero *p*-form *ω* on a Riemannian manifold with ◦ *R* ≥ *δ* > 0. In these cases, *b*1(*M*), ... , *bn*−<sup>1</sup>(*M*) are zero (see [21]). We can improve these results for the case of Einstein manifolds. First, we will prove the following.

**Lemma 1.** *Let* (*<sup>M</sup>*, *g*) *be an Einstein manifold with Einstein constant α and sectional curvature bounded below by a constant δ* > 0*. If α* < (*n* + 2)*δ then*

$$F\_p(\omega) \ge (1/3)((n+2)\delta - a)(n-p) \|\omega\|^2 > 0$$

*for any nonzero p-form ω and an arbitrary* 1 ≤ *p* ≤ *n* − 1*.*

**Proof.** Let *p* ≤ [*n*/2], then we can define the symmetric traceless two-tensor *ϕ*(*<sup>i</sup>*1*i*2...*ip*) with components (see [14])

$$\log\_{jk}^{\left(i\_{1}i\_{2}\ldots i\_{p}\right)} = \sum\_{a=1}^{p} \left(\omega\_{i\_{1}\ldots i\_{a-1}ji\_{a+1}\ldots i\_{p}} g\_{ki\_{a}} + \omega\_{i\_{1}\ldots i\_{a-1}ki\_{a+1}\ldots i\_{p}} g\_{ji\_{a}}\right) - \frac{2p}{n} g\_{jk} \omega\_{i\_{1}\ldots i\_{p}}$$

for each set of values of indices *i*1 *i*2 ... *ip* such that 1 ≤ *i*1 < *i*2 < ... < *ip* ≤ *n*. After long but simple calculations we obtain the identities (see also [14]),

$$\begin{split} \mathcal{R}\_{ijkl} \, \mathop{\rho}^{il} \begin{split} \boldsymbol{\varrho}^{il} \left( {}^{i\_{1}...i\_{p}} \right) \boldsymbol{\varrho}^{jk} \Big|\_{ (i\_{1}...i\_{p}) } &= \boldsymbol{p} \left( \frac{2(n+4p)}{n} \boldsymbol{R}\_{ij} \, \boldsymbol{\omega}^{j} {}^{i\_{2}...i\_{p}} \boldsymbol{\omega}^{j} {}\_{i\_{2}...i\_{p}} \right. \\ -\cline{3-4} \, (p-1) \, \mathop{\mathcal{R}}\_{ijkl} \boldsymbol{\omega}^{ij} {}^{j} {}^{j} \boldsymbol{\omega}^{kl} \, \mathop{\boldsymbol{\omega}}^{kl} \boldsymbol{\omega}^{j} {}\_{i\_{3}...i\_{p}} &- \frac{4p}{n^{2}} \boldsymbol{s} \, \|\boldsymbol{\omega}\|^{2} \right); \\ \boldsymbol{\varrho} \quad \boldsymbol{2n}(\boldsymbol{u}+\boldsymbol{\Im})(\boldsymbol{u}-\boldsymbol{n}) &= \boldsymbol{\varrho}. \end{split} \tag{5}$$

$$\|\|\phi\|\|^2 = \frac{2p(n+2)(n-p)}{n} \|\|\omega\|\|^2,\tag{6}$$

where

$$\begin{aligned} \|\|\boldsymbol{\varrho}\|\|^2 &= \mathcal{g}^{ik}\mathcal{g}^{jl}\mathcal{g}\_{i\_1j\_1}\dots\mathcal{g}\_{i\_pj\_p}\mathcal{q}\_{i\_j}^{(i\_1\dots i\_p)}\mathcal{q}\_{kl}^{(j\_1\dots j\_p)},\\ \|\|\boldsymbol{\omega}\|\|^2 &= \omega^{i\_1i\_2\dots i\_p}\omega\,\_{i\_1i\_2\dots i\_p} = \mathcal{g}^{i\_1j\_1}\dots\mathcal{g}^{i\_pj\_p}\omega\,\_{i\_1\dots i\_p}\mathcal{w}\,\_{j\_1\dots j\_p}.\end{aligned}$$

for *gij* = (*g* <sup>−</sup><sup>1</sup>)*ij*. If (*<sup>M</sup>*, *g*) is an Einstein manifold, then Equations (4) and (5) can be rewritten in the form

$$F\_p(\omega) = \frac{s}{n} \|\omega\|^2 - \frac{p-1}{2} \operatorname{R}\_{ijkl} \omega^{ij\,i\_3...i\_p} \omega^{kl}\_{\,i\_3...i\_{p'}} $$

$$R\_{i\bar{j}\bar{k}l} \, q^{jl(i\_1...i\_p)} \, q^{j\bar{k}}\_{\left(i\_1...i\_p\right)} = p \left( \frac{2n+4p}{n^2} s \, \|\, \omega \|\|^2 - 3(p-1)R\_{i\bar{j}\bar{k}l} \, \omega^{j\bar{j}i\_3...i\_p} \, \omega^{k\bar{l}}\_{i\_3...i\_p} \right). \tag{7}$$

On the other hand, for a fixed set of values of indices (*<sup>i</sup>*1, *i*2, ... , *ip*) such that 1 ≤ *i*1 < *i*2 < ... < *ip* ≤ *n*, the equality in Equation (3) can be rewritten in the form

$$R\_{ijkl} \, \boldsymbol{\varrho}^{\rm il} \left( {}^{i\_1 \ldots i\_p} \right) \boldsymbol{\varrho}^{\rm jk} \left( {}^{i\_1 \ldots i\_p} \right) + \frac{s}{n} \, \boldsymbol{\varrho}^{\rm ik} \left( {}^{i\_1 \ldots i\_p} \right) \boldsymbol{\varrho}\_{\rm ik}^{\left( {}^{i\_1 \ldots i\_p} \right)} \geq n \, \boldsymbol{\delta} \, \boldsymbol{\varrho}^{\rm kl} \left( {}^{i\_1 \ldots i\_p} \right) \boldsymbol{\varrho}\_{\rm kl}^{\left( {}^{i\_1 \ldots i\_p} \right)} \, . \tag{8}$$

Then from Equation (8) we obtain the inequality

$$\left(R\_{ijkl}\,\,\,\dot{\boldsymbol{q}}^{jl}\left(^{i\_1\dots i\_p}\right)\,\,\dot{\boldsymbol{q}}^{jk}\right.\tag{9}$$

$$\left(^{i\_1\dots i\_p}\right)\_{\left(^{i\_1}\dots i\_p\right)} \geq \left(\,^{n\delta}-\frac{s}{n}\right)\,\,\left\|\,\,\boldsymbol{\phi}\right\|^2\,.\tag{9}$$

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

$$
\left\|\Phi\right\| F\_{\mathcal{P}}(\omega) \ge \left(n\,\delta - \frac{s}{n+2}\right) \left\|\left\|\bar{\varphi}\right\|\right\|^2. \tag{10}
$$

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

$$F\_p(\omega) \ge (1/3)((n+2)\,\delta - \mathfrak{a})\,(n-p)\|\omega\|\,^2. \tag{11}$$

It is obvious that if the sectional curvature of an Einstein manifold (*<sup>M</sup>*, *g*) satisfies the inequality sec ≥ *δ* for a positive constant *δ*, then the scalar curvature of (*<sup>M</sup>*, *g*) satisfies the inequality *s* ≥ *n*(*n* − 1) *δ* > 0. In this case, if (*n* − 1) *δ* ≤ *α* < (*n* + 2) *δ*, then from Equation (11) we deduce that the quadratic form *Fp*(*ω*) is positive definite for any *p* ≤ [*n*/2]. It is known [23] that *Fp*(*ω*) = *Fn*−*<sup>p</sup>*(<sup>∗</sup> *ω*) and *ω*-2 = - ∗ *ω*-2 for any *p*-form *ω* with 1 ≤ *p* ≤ *n* − 1 and the Hodge star operator ∗ : Λ*pM* → Λ*<sup>n</sup>*−*pM* acting on the space of *p*-forms Λ*pM*. Therefore, the inequality in Equation (11) holds for any *p* = 1, . . . , *n* − 1.

Recall that if on an *n*-dimensional compact Riemannian manifold (*<sup>M</sup>*, *g*) the quadratic form *Fp*(*ω*) is positive definite for any smooth *p*-form *ω* with *p* = 1, ... , *n* − 1, then the Betti numbers *b*1(*M*), ... , *bn*−<sup>1</sup>(*M*) 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 (*<sup>M</sup>*, *g*) satisfies sec ≤ −*δ* < 0 for a positive constant *δ*, then the Einstein constant of (*<sup>M</sup>*, *g*) satisfies the the obvious inequality *α* ≤ −(*<sup>n</sup>* − 1) *δ* < 0. On the other hand, from Equation (1) we deduce the inequality *Rijlkϕikϕjl* ≤ − (*n δ* + *α*) *ϕ* -2. Therefore, if *δ* > <sup>−</sup>*<sup>α</sup>*/*<sup>n</sup>*, then ◦ *R* < 0. In this case, the Tachibana numbers *<sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) are equal to zero (see [19]). We proved the following.

**Proposition 1.** *Let* (*Mn*, *g*) *be an Einstein manifold with sectional curvature bounded above by a negative constant* −*δ such that δ* > −*<sup>α</sup>*/*<sup>n</sup> for the Einstein constant α. Then the Tachibana numbers <sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) *are zero.*

We can complete this result. If an Einstein manifold (*Mn*, *g*) satisfies the curvature inequality sec ≤ −*δ* < 0 for a positive constant *δ*, then from Equations (3) and (7) we deduce the inequality *Fp*(*ω*) ≤ −13 ((*n* + 2) *δ* + *α*)(*n* − *p*)*ω*-2 for any *p* = 1, ... , *n* − 1. Therefore, the Tachibana numbers *<sup>t</sup>*1(*M*), ... , *tn*−<sup>1</sup>(*M*) of a compact Einstein manifold with sectional curvature bounded above by a negative constant −*δ* such that *δ* ≥ −*<sup>α</sup>*/(*<sup>n</sup>* + 2) are zero.

**Author Contributions:** Investigation, V.R., S.S. and I.T.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The authors declare no conflict of interest.
