Sectional Curvature, Isotropic Curvature, and Yau's Pinching Problem

TL;DR

This paper proves a new curvature condition implies a closed Riemannian manifold with finite fundamental group is homeomorphic to a spherical space form, extending sphere theorems and addressing Yau's pinching problem.

Contribution

It introduces a novel curvature inequality involving sectional curvatures that generalizes the sphere theorem under stronger pinching conditions.

Findings

Manifolds satisfying the curvature condition are homeomorphic to spherical space forms.

The result extends classical sphere theorems to a broader class of curvature conditions.

Provides partial resolution to Yau's 1990 pinching problem.

Abstract

We prove that if a closed Riemannian manifold $(M^n,g)$ has finite fundamental group and satisfies the curvature condition \begin{equation*} R_{1313} +R_{1414} +R_{2323} + R_{2424} > \tfrac{1}{2}\left(R_{1212} + R_{3434}\right) \end{equation*} for all orthonormal four-frame $\{e_1, e_2, e_3, e_4\} \subset T_pM$, then $M$ is homeomorphic to a spherical space form. This generalizes the famous sphere theorem under the stronger condition of $\frac{1}{4}$-pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990.

Equations

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\frac{1}{2}\left(R_{1212}+R_{3434}\right)$$

$$1<K(\sigma )\le 4$$

$$K({\sigma }_1)>\frac{1}{4}K({\sigma }_2)$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\frac{1}{2}\left(R_{1212}+R_{3434}\right)$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}-2R_{1234}>0$$

$$K_{\min }>\frac{n-1}{n+2}{S}_0$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\frac{4n(n-1)}{n^2-n+12}{S}_0$$

$$K_{\min }>\frac{n(n-1)}{n^2-n+12}{S}_0$$

$$6S_0=R_{1212}+R_{3434}+R_{1313}+R_{1414}+R_{2323}+R_{2424}.$$

$$R_{1212}+R_{3434}<2{\gamma }_n{S}_0,$$

$${\gamma }_n=\{\begin{array}{ll}\frac{n(n-1)}{n^2-n-6}& \text{if }n\ge 6\text{ or }n=4;\\ \frac{20}{17}& \text{if }n=5.\end{array}$$

$$K_{\max }<{\gamma }_n{S}_0$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}\ge \frac{1}{2}\left(R_{1212}+R_{3434}\right)$$

$$R(X,Y,Z,W)=-R(Y,X,Z,W)=-R(X,Y,W,Z)=R(Z,W,X,Y)$$

$$R(X,Y,Z,W)+R(X,Z,W,Y)+R(X,W,Y,Z)=0$$

$$K(\sigma )=R_{1212},$$

$$R_{ijkl}=R(e_i,e_j,e_k,e_l)$$

$$R_{ij}=\mathrm{Ric}(e_i,e_j)=\sum _{k=1}^{n}R(e_i,e_k,e_j,e_k)$$

$$S=\sum _{i=1}^n{R}_{ii}=2\sum _{1\le i<j\le n}{R}_{ijij}.$$

$$S_0=\frac{S}{n(n-1)}.$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\frac{1}{2}\left(R_{1212}+R_{3434}\right)$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}\ge 4K_{\min }>K_{\max }\ge \frac{1}{2}\left(R_{1212}+R_{3434}\right).$$

$$R_{1313}$$

$$R_{1414}$$

$$R_{2323}$$

$$R_{2424}$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\frac{1}{2}\left(R_{1212}+R_{3434}\right).$$

$$R_{1313}+R_{1414}+R_{2323}+R_{2424}>\gamma \left(R_{1212}+R_{3434}\right)$$

$$\sum _{1\le i\ne j\ne k\ne l\le n}\left(R_{ikik}+R_{ilil}+R_{jkjk}+R_{jljl}\right)$$

$$\sum _{1\le i\ne j\ne k\ne l\le n}\left(R_{ijij}+R_{klkl}\right)$$

$$\sum _{1\le i\ne j\ne k\ne l\le n}\left(R_{ikik}+R_{ilil}+R_{jkjk}+R_{jljl})\right)$$

Full text

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$, then $M$ is homeomorphic to a spherical space form. This generalizes the famous sphere theorem under the stronger condition of $\frac{1}{4}$-pinched sectional curvature. As an application, we provide a partial answer to a pinching problem proposed by Yau in 1990.

The author’s research is partially supported by NSF-DMS #2405257 and a start-up grant at Auburn University

1. Introduction

The celebrated sphere theorem due to Berger [Ber60] and Klingenberg [Kli61] asserts that if $(M^{n},g)$ is a complete, simply connected, $n$-dimensional Riemannian manifold whose sectional curvature $K$ satisfies

[TABLE]

for all two-planes $\sigma\subset T_{p}M$, then $M$ is homeomorphic to the $n$-sphere. In 2009, Brendle and Schoen [BS09] utilized the Ricci flow introduced by Hamilton [Ham82], together with the novel construction of invariant cones under Ricci flow by Böhm and Wilking [BW08], to upgrade the homeomorphism to diffeomorphism and weaken the pinching condition from global to pointwise, whereas the $n=4$ case was done much earlier by Chen [Che91] based on Hamilton’s convergence result for Ricci flow on four-manifolds [Ham86]. More precisely, it is proved that if a closed Riemannian manifold $(M^{n},g)$ has $\frac{1}{4}$-pinched sectional curvature in the sense that

[TABLE]

for all two-planes $\sigma_{1},\sigma_{2}\subset T_{p}M$, then $M$ is diffeomorphic to a spherical space form. Shortly after, the $\frac{1}{4}$-pinched sectional curvature condition was relaxed to $\frac{1}{4}$-pinched flag curvature, which means $K(\sigma_{1})>\frac{1}{4}K(\sigma_{2})$ for all two-planes $\sigma_{1},\sigma_{2}\subset T_{p}M$ that intersect in a line, by Andrews and Nguyen [AN09] for $n=4$ and Ni and Wilking [NW10] for $n\geq 5$. For more sphere theorems under various positivity conditions on curvature, we refer the reader to [MM88], [BW08], [Bre08], [CGT23], [Li24a, Li22, Li24b], [PW21], [NPW23], and the references therein.

The main purpose of this article is to prove the following sphere theorem under a much weaker pinching condition on the sectional curvature.

Theorem 1.1**.**

Let $(M^{n},g)$ be a closed Riemannian manifold of dimension $n\geq 4$. Suppose that $(M,g)$ has finite fundamental group and satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$, then $M$ is homeomorphic (diffeomorphic if $n=4$ or $n\geq 12$) to a spherical space form.

Clearly, condition (1.1) is implied by $\frac{1}{4}$-pinched sectional or flag curvature (see Lemma 2.1). Thus, Theorem 1.1 generalizes the above-mentioned sphere theorems of Berger [Ber60] and Klingenberg [Kli61], Brendle and Schoen [BS08], and Ni and Wilking [NW10]. We would like to point out that (1.1) is, in a certain sense, much weaker than $\frac{1}{4}$-pinched sectional or flag curvature, as it does not even imply positive Ricci curvature. This can be seen by considering the manifold $\mathbb{S}^{n-1}\times\mathbb{S}^{1}$, with the standard product metric, which satisfies (1.1) but does not have positive Ricci curvature.

A key step in the proofs of the $\frac{1}{4}$-pinched differentiable sphere theorems is to show that $\frac{1}{4}$-pinched sectional or flag curvature implies that $M\times\mathbb{R}^{2}$ has positive isotropic curvature, a condition that is equivalent to positive complex sectional curvature as discovered in [NW07]. Then a metric with positive complex sectional curvature on a closed manifold evolves under the normalized Ricci flow to a metric with constant sectional curvature, as shown by Brendle and Schoen [BS09]. On the contrary, condition (1.1) does not imply positive complex sectional curvature, as it does not even imply positive Ricci curvature. Instead, our proof of Theorem 1.1 replies on the key observation that (1.1) implies the positive isotropic curvature condition introduced by Micallef and Moore [MM88].

Theorem 1.2**.**

Let $R$ be an algebraic curvature tensor on an Euclidean vector space $V$ of dimension $n\geq 4$. If $R$ satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$, then $R$ has positive isotropic curvature, i.e.

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$.

The finite fundamental group assumption in Theorem 1.1 is only used to pass to the universal cover, to which one can apply the beautiful sphere theorem of Micallef and Moore [MM88] stating that a closed, simply connected Riemannian manifold with positive isotropic curvature is homeomorphic to the $n$-sphere. We also point out that the homeomorphism can be improved to diffeomorphism for $n=4$ using Hamilton’s classification of closed four-manifold with positive isotropic curvature [Ham97] (see also [CTZ12]) and for $n\geq 12$ with Brendle’s work [Bre19] (see also [Hua23, Corollary 1.3]). Dimensions $9\leq n\leq 11$ seem reachable due to a recent preprint of Chen [Che24], while the $5\leq n\leq 8$ cases remain completely open.

Theorem 1.2 establishes a novel connection between sectional curvature and isotropic curvature, and it has an application to Yau’s pinching problem that we explain now. In 1990, Yau asked in his “Open problems in geometry” [Yau93, Problem 12, page 4] (see also [SY94, page 369]): *“The famous pinching problem says that on a compact simply connected manifold if $K_{\min}>\frac{1}{4}K_{\max}>0$, then the manifold is homeomorphic to a sphere. If we replace $K_{\max}$ by normalized scalar curvature, can we deduce similar pinching theorems?” * Here, $K_{\min}$ and $K_{\max}$ denote the minimum and maximum of the sectional curvature, respectively, and the normalized scalar curvature $S_{0}$ is given by $S_{0}=\frac{1}{n(n-1)}S$, where $S$ denotes the scalar curvature. Yau’s pinching problem can be restated as follows (see [GX12, page 525]).

Problem 1.3** (Yau [Yau93]).**

Is a closed, simply connected Riemannian manifold $(M^{n},g)$ of dimension $n\geq 4$ satisfying

[TABLE]

homeomorphic to the $n$-sphere.

The best result up to date for Problem 1.3 is due to Gu and Xu [GX12], who gave an affirmative answer under the stronger condition $K_{\min}>\frac{n(n-1)}{n^{2}-n+6}S_{0}$. In addition, Problem 1.3 was answered affirmatively in dimension $4$ by Costa and Ribeiro [CR14] under a weaker condition, and for Einstein manifolds in all dimensions by Xu and Gu [XG14]. The constant $\frac{n-1}{n+2}$ is the best possible in view of the complex projective space $\mathbb{CP}^{m}$ with the Fubini-Study metric $g_{FS}$.

Our next result provides a partial answer to Yau’s pinching problem.

Theorem 1.4**.**

Let $(M^{n},g)$ be a closed Riemannian manifold of dimension $n\geq 4$. Suppose that $(M^{n},g)$ has finite fundamental group and satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$. Then $M$ is homeomorphic to a spherical space form. In particular, a closed, simply connected Riemannian manifold satisfying

[TABLE]

is homeomorphic to the $n$-sphere.

Theorem 1.4 improves the result of Gu and Xu [GX12] in two aspects: it improves the pinching constant in front of $S_{0}$ and relaxes $K_{\min}$ to the minimum of the sum of four sectional curvatures of the form $R_{1313}+R_{1414}+R_{2323}+R_{2424}$. In dimension four, Theorem 1.4 also recovers the result of Costa and Ribeiro [CR14] assuming $K^{\perp}_{\max}<2S_{0}$, where $K^{\perp}_{\max}$ denotes the maximum of biorthogonal (sectional) curvature. Their key observation is that $K^{\perp}_{\max}<2S_{0}$ implies positive isotropic curvature in dimension four. Indeed, the three conditions $K^{\perp}_{\max}<2S_{0}$, (1.2), and (1.1) are all equivalent when $n=4$, due to the identity

[TABLE]

It is also natural to replace $K_{\min}$, instead of $K_{\max}$, in the $\frac{1}{4}$-pinched sectional curvature condition by an appropriate multiple of $S_{0}$ and ask the same question as in Problem 1.3. Indeed, such a conjecture was formulated by Gu and Xu [GX12, Conjecture 2 on page 526], and they proved a partial result stating that a closed Riemannian manifold satisfying $K_{\max}<\frac{n(n-1)}{n(n-1)-\frac{12}{5}}S_{0}$ is diffeomorphic to a spherical space form. Here, we obtain an improvement of their result.

Theorem 1.5**.**

Let $(M^{n},g)$ be a closed Riemannian manifold of dimension $n\geq 4$. Suppose that $(M,g)$ has finite fundamental group and satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$, where

[TABLE]

Then $M$ is homeomorphic to a spherical space form. In particular, a closed, simply connected Riemannian manifold satisfying

[TABLE]

is homeomorphic to the $n$-sphere.

Theorems 1.4 and 1.5 will be derived as consequences of Theorem 1.1, in the sense that we obtain them by showing that (1.1) is implied by either (1.2) or (1.4). Same as in Theorem 1.1, the homeomorphism in Theorems 1.4 and 1.5 can be upgraded to diffeomorphism if $n=4$ or $n\geq 12$. For four-manifolds, topological and geometric rigidity results were obtained by Cao and Tran [CT22] under a variety of curvature conditions.

Sphere theorems often have corresponding rigidity results. Brendle and Schoen [BS08] proved that a closed Riemannian manifold with weakly $\frac{1}{4}$-pinched sectional curvature, in the sense that $K(\sigma_{1})\geq\frac{1}{4}K(\sigma_{2})\geq 0$ for all $\sigma_{1},\sigma_{2}\subset T_{p}M$, is either diffeomorphic to a spherical space form or isometric to a rank one compact symmetric space. Ni and Wilking [NW10] showed the same conclusion for manifolds with weakly $\frac{1}{4}$-pinched flag curvature. Here, we prove a rigidity result corresponding to Theorem 1.1.

Theorem 1.6**.**

Let $(M^{n},g)$ be a closed Riemannian manifold of dimension $n\geq 4$. Suppose that $(M^{n},g)$ has finite fundamental group and satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$. Then one of the following statements holds:

1. (1)

$(M,g)$* is flat;* 2. (2)

$M$* is homeomorphic to a spherical space form.* 3. (3)

$n=2m$* and $(M,g)$ is a Kähler manifold biholomorphic to $\mathbb{CP}^{m}$; * 4. (4)

the universal cover of $(M,g)$ is isometric to an irreducible compact symmetric space.

The key ingredients in the proof of Theorem 1.6 include Lemma 5.1 that prevents certain splitting, the structure theorem of reducible manifolds with nonnegative isotropic curvature in [MW93], and the classification of simply connected irreducible manifolds with nonnegative isotropic curvature in [Bre10, Theorem 9.30].

This article is organized as follows. In Section 2, we collect some preliminaries and establish some basic properties related to condition (1.1). In Section 3, we prove Theorem 1.2, namely (1.1) implies positive isotropic curvature, and then derive Theorem 1.1. In Section 3, we present the proofs of Theorems 1.4 and 1.5. In Section 4, we prove Theorem 1.6.

2. Curvature Conditions

In this section, we collect some preliminaries and prove several elementary properties and identities that will be used in subsequent sections. Throughout this paper, $(V,g)$ is a Euclidean vector space of dimension $n\geq 4$.

2.1. Preliminaries

Let $\wedge^{2}V$ denote the space of two-forms on $V$. Denote by $S^{2}_{B}(\wedge^{2}V)$ the space of algebraic curvature operators (or tensors) on $V$, that is to say, $S^{2}_{B}(\wedge^{2}V)$ consists of all symmetric operators $R:\wedge^{2}V\to\wedge^{2}V$ satisfying the symmetries

[TABLE]

and the first Bianchi identity

[TABLE]

for all $X,Y,Z,W\in V$.

Let $R\in S^{2}_{B}(\wedge^{2}V)$. The sectional curvature $K$ of $R$ of a two-plane $\sigma\subset V$ is defined as

[TABLE]

where $\{e_{1},e_{2}\}$ is any orthonormal basis of $\sigma$. Here and in the rest of this paper, we use the abbreviation

[TABLE]

when $\{e_{i}\}_{i=1}^{n}$ is an orthonormal basis of $V$. The Ricci curvature of $R$ is given by

[TABLE]

and the scalar curvature $S$ of $R$ is given by

[TABLE]

The normalized scalar curvature $S_{0}$ is given by

[TABLE]

2.2. Basic properties

Lemma 2.1**.**

Suppose $R\in S^{2}_{B}(\wedge^{2}V)$ has $\frac{1}{4}$-pinched sectional curvature or $\frac{1}{4}$-pinched flag curvature, then

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$.

Proof.

Let $\{e_{1},e_{2},e_{3},e_{4}\}$ be an arbitrary orthonormal four-frame in $V$. If $R$ has $\frac{1}{4}$-pinched sectional curvature, then

[TABLE]

If $R$ has $\frac{1}{4}$-pinched flag curvature, then we have $R_{ijij}>\frac{1}{4}R_{ikik}$, whenever $i,j,k$ are mutually distinct. In particular, we have

[TABLE]

Adding these four inequalities together produces

[TABLE]

The proof is complete. $\square$

Next, we show that positive scalar curvature is implied by a family of curvature conditions including (1.1).

Lemma 2.2**.**

Let $\gamma<2$ and $R\in S^{2}_{B}(\wedge^{2}V)$. Suppose that

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$. Then $R$ has positive scalar curvature. If equality is allowed in (2.1), then $R$ has nonnegative scalar curvature.

We point out that the $\gamma=0$ case covers a well-known result of Micallef and Wang [MW93, Page 659] stating that positive isotropic curvature implies positive scalar curvature.

To prove Lemma 2.2, we first establish two identities that will also be used in the proofs of Propositions 4.1 and 4.2 later.

Lemma 2.3**.**

Let $R\in S^{2}_{B}(\wedge^{2}V)$ and $\{e_{i}\}_{i=1}^{n}$ be an orthonormal basis of $V$. Then

[TABLE]

and

[TABLE]

Here, $i\neq j\neq k\neq l$ in the summation means $i,j,k,l$ are mutually distinct.

Proof.

To establish (2.3), we compute that

[TABLE]

Similarly, one verifies (2.3) as follows

[TABLE]

$\square$

We now prove Lemma 2.2.

Proof of Lemma 2.2.

Let $\{e_{i}\}_{i=1}^{n}$ be an orthonormal basis of $V$. Since $\{e_{i},e_{j},e_{k},e_{l}\}$ is an orthonormal four-frame in $V$ for $1\leq i,j,k,l\leq n$ mutually distinct, we have

[TABLE]

Using (2.3) and (2.3), one gets

[TABLE]

It follows that $S>0$ when $\gamma<2$. This finishes the proof. $\square$

2.3. Complex Projective Spaces

Let $R^{\mathbb{CP}^{m}}$ denote the Riemann curvature tensor of the complex projective space $\mathbb{CP}^{m}$ of complex dimension $m\geq 2$ with the Fubini-Study metric, normalized with constant holomorphic sectional curvature $4$. According to [BK78], $R^{\mathbb{CP}^{m}}$ is given by

[TABLE]

for $X,Y,Z,W\in T_{p}\mathbb{CP}^{m}$. This implies that, if $X$ and $Y$ are unit vectors with $g(X,Y)=0$, then

[TABLE]

In particular, the sectional curvatures of $R^{\mathbb{CP}^{m}}$ take values in the interval $[1,4]$. It follows from Lemma 2.1 that $R^{\mathbb{CP}^{m}}$ satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}$. Moreover, one easily sees that for an orthonormal four-frame of the form $\{e_{1},e_{2}=Je_{1},e_{3},e_{4}=Je_{3}\}$, we have the equality

[TABLE]

It is this example that inspired the author to formulate the curvature condition (1.1) and investigate it in this paper.

Similar properties are also satisfied by the quaternion-Kähler project space and the Cayley plane, with their canonical metrics.

3. Proof of Theorem 1.1

We recall the important notion of positive isotropic curvature introduced by Micallef and Moore [MM88].

Definition 3.1**.**

$R\in S^{2}_{B}(\wedge^{2}V)$ is said to have positive isotropic curvature if

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$. If $>$ is replaced with $\geq$, then $R$ is said to have nonnegative isotropic curvature.

We restate Theorem 1.2 below and present its proof.

Proposition 3.2**.**

Suppose that $R\in S^{2}_{B}(\wedge^{2}V)$ satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$. Then $R$ has positive isotropic curvature. If equality is allowed in (3.1), then $R$ has nonnegative isotropic curvature

Proof.

Let $\{e_{1},e_{2},e_{3},e_{4}\}$ be an arbitrary orthonormal four-frame in $V$. By assumption, we have

[TABLE]

Set

[TABLE]

One verifies that $\{e_{1}^{\prime},e_{2}^{\prime},e_{3}^{\prime},e_{4}^{\prime}\}$ is also an orthonormal four-frame in $V$. The assumption then implies

[TABLE]

where

[TABLE]

By direct calculations, we have

[TABLE]

[TABLE]

and

[TABLE]

Substituting the above identities into (3.3) produces

[TABLE]

We then replace $\{e_{1},e_{2},e_{3},e_{4}\}$ by $\{e_{1},e_{2},-e_{4},e_{3}\}$ in the above argument to obtain

[TABLE]

Finally, adding (3.2), (3), and (3) together yields

[TABLE]

where, in getting the last equality, we have used the first Bianchi identity

[TABLE]

Since the orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}$ is arbitrary, we conclude that $R$ has positive isotropic curvature. The statement for nonnegative isotropic curvature can be proved similarly. $\square$

Next, we prove Theorem 1.1.

Proof of Theorem 1.1.

Since $(M^{n},g)$ has finite fundamental group, its universal cover $(\widetilde{M},\tilde{g})$ is a closed and simply connected manifold. Moreover, $(\widetilde{M},\tilde{g})$ has positive isotropic curvature, in view of Proposition 3.2. Invoking the sphere theorem of Micallef and Moore [MM88], we conclude that $\widetilde{M}$ is homeomorphic to $\mathbb{S}^{n}$. Hence, $M$ is homeomorphic to a spherical space form.

The homeomorphism can be improved to diffeomorphism for $n=4$ using Hamilton’s classification of four-manifolds with positive isotropic curvature [Ham97] (see also [CTZ12]) and for $n\geq 12$ using Brendle’s recent work [Bre19] (see also [Hua23]). $\square$

4. Proofs of Theorems 1.4 and 1.5

Theorem 1.4 follows from Theorem 1.1 and the following proposition.

Proposition 4.1**.**

Suppose $R\in S^{2}_{B}(\wedge^{2}V)$ satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$. Then

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$.

Proof of Proposition 4.1.

We may assume $n\geq 5$, since the assumption and the conclusion coincide when $n=4$ due to the identity (1.3).

Let $\{e_{1},e_{2},e_{3},e_{4}\}$ be an arbitrary orthonormal four-frame in $V$ and we extend it to an orthonormal basis $\{e_{i}\}_{i=1}^{n}$ of $V$. For $1\leq i,j,k,l\leq n$ that are mutually distinct, $\{e_{i},e_{j},e_{k},e_{l}\}$ is an orthonormal four-frame in $V$, and the assumption implies

[TABLE]

where

[TABLE]

To prove Proposition 4.1, it suffices to establish the upper bound

[TABLE]

in view of

[TABLE]

To prove (4.2), we shall make use of the identity

[TABLE]

We begin with the $n=5$ case. Notice that

[TABLE]

where we have used in the inequality step that each bracket containing four terms is bigger than $4\eta_{n}S_{0}$ due to (4.1). It follows that

[TABLE]

as desired.

For $n=6$, we observe that

[TABLE]

Again, we have used in the last step that each bracket containing four terms is bigger than $4\eta_{n}S_{0}$ by (4.1). From this, we deduce

[TABLE]

and finish the $n=6$ case.

When $n=7$, one verifies (a bit tedious but elementary) that

[TABLE]

As before, each bracket containing four terms is bigger than $4\eta_{n}S_{0}$. Therefore, we have

[TABLE]

This completes the $n=7$ case.

Finally, we treat all $n\geq 8$. By (2.3) and (4.1), we have

[TABLE]

We also observe that, for any $n\geq 6$,

[TABLE]

Using the above two estimates, we obtain

[TABLE]

This establishes (4.2) for all $n\geq 8$.

We have completed the proof.

$\square$

Theorem 1.5 is a consequence of Theorem 1.1 and the following proposition.

Proposition 4.2**.**

Suppose $R\in S^{2}_{B}(\wedge^{2}V)$ satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$, where $\gamma_{n}$ is defined in (1.5). Then

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset V$.

Proof.

The $n=4$ case follows from the identity (1.3). We assume $n\geq 5$ below.

Let $\{e_{1},e_{2},e_{3},e_{4}\}$ be an arbitrary orthonormal four-frame in $V$ and we extend it to an orthonormal basis $\{e_{i}\}_{i=1}^{n}$ of $V$. The assumption implies that

[TABLE]

whenever $1\leq i,j,k,l\leq n$ are mutually distinct.

Using (2.3) and (4.3), we have, for any $n\geq 8$,

[TABLE]

One verifies that, for $n\geq 6$, we have

[TABLE]

This, together with (4.3), yields

[TABLE]

Therefore, we obtain, when $n\geq 8$, that

[TABLE]

In view of (1.5), this becomes

[TABLE]

as desired. This completes the proof for $n\geq 8$.

Next, we deal with the cases $5\leq n\leq 7$ separately. For $n=7$, we observe that

[TABLE]

where we have used (4.3) to obtain the inequality step.

In a similar fashion, the $n=6$ case follows from

[TABLE]

Finally, for $n=5$, we notice that

[TABLE]

Since $R_{1212}+R_{3434}<2\gamma_{5}S_{0}$, we arrive at

[TABLE]

The proof is complete.

$\square$

5. Rigidity

In this section, we present the proof of Theorem 1.6. We begin with the following lemma, which prevents non-flat manifolds satisfying (1.6) from certain splitting.

Lemma 5.1**.**

Suppose that $(M^{n},g)=(M_{1}^{k},g_{1})\times(M_{2}^{n-k},g_{2})$, $n\geq 4$ and $2\leq k\leq\frac{n}{2}$, satisfies

[TABLE]

for all orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{p}M$. If both $(M_{1},g_{1})$ and $(M_{2},g_{2})$ have nonnegative scalar curvature somewhere, then $(M^{n},g)$ is flat.

Proof.

For $i=1,2$, let $\sigma_{i}\subset T_{p_{i}}M_{i}$ be a two-plane that maximizes the sectional curvature on $(M_{i},g_{i})$. Let $\{e_{1},e_{2}\}$ be an orthonormal basis of $\sigma_{1}$ and $\{e_{3},e_{4}\}$ be an orthonormal basis of $\sigma_{2}$. For the orthonormal four-frame $\{e_{1},e_{2},e_{3},e_{4}\}\subset T_{(p_{1},p_{2})}M$, the curvature assumption implies

[TABLE]

Due to the product structure, we have $R_{1313}=R_{1414}=R_{2323}=R_{2424}=0$. It follows that $R_{1212}+R_{3434}\leq 0$. On the other hand, since both $(M_{1},g_{1})$ and $(M_{2},g_{2})$ have nonnegative scalar curvature somewhere, we have $R_{1212}\geq 0$ and $R_{3434}\geq 0$. Hence $R_{1212}=R_{3434}=0$. Since $R_{1212}$ and $R_{3434}$ maximize the sectional curvature of $(M_{1},g_{1})$ and $(M_{2},g_{2})$, respectively, we conclude that both $(M_{1},g_{1})$ and $(M_{2},g_{2})$ have nonpositive sectional curvature, and so does $(M,g)$. On the other hand, $(M,g)$ has nonnegative scalar curvature by Lemma 2.2. Hence, $(M,g)$ is flat.

$\square$

We are ready to prove Theorem 1.6.

Proof of Theorem 1.6.

Denote by $(\widetilde{M}^{n},\tilde{g})$ the Riemannian universal cover of $(M^{n},g)$. Since $M$ has finite fundamental group, $\widetilde{M}^{n}$ is also a closed manifold. By Proposition 3.2, $(\widetilde{M}^{n},\tilde{g})$ has nonnegative isotropic curvature.

We first consider the case that $(\widetilde{M}^{n},\tilde{g})$ is irreducible. By the classification of closed, simply connected, irreducible Riemannian manifolds with nonnegative isotropic curvature in [Bre10, Theorem 9.30], we conclude that $(\widetilde{M}^{n},\tilde{g})$ is either homeomorphic to $\mathbb{S}^{n}$, or Kähler and biholomorphic to $\mathbb{CP}^{\frac{n}{2}}$, or isometric to an irreducible compact symmetric space.

We then consider the case $(\widetilde{M}^{n},\tilde{g})$ is reducible and use Lemma 5.1 to prove that $(\widetilde{M}^{n},\tilde{g})$ must be flat in this case. Noticing the de Rham decomposition of $(\widetilde{M}^{n},\tilde{g})$ cannot have Euclidean factors, we have two possibilities by [MW93, Theorem 3.1]:

Case 1: $(\widetilde{M}^{n},\tilde{g})$ is isometric to $(N_{1}^{n_{1}},g_{1})\times\cdots\times(N_{r}^{n_{r}},g_{r})$, where $n=n_{1}+\cdots+n_{r}$, and for each $1\leq i\leq r$, either $n_{i}=2$, $N_{i}=\mathbb{S}^{2}$, and $g_{i}$ has nonnegative Gauss curvature, or $n_{i}\geq 3$, $N_{i}$ is compact and irreducible, and $g_{i}$ has nonnegative Ricci curvature. In view of Lemma 5.1, $(\widetilde{M}^{n},\tilde{g})$ must be flat.

Case 2: $(\widetilde{M}^{n},\tilde{g})$ is isometric to $(\Sigma^{2},g_{1})\times(N^{n-2},g_{2})$, where $\Sigma$ is a surface with Gauss curvature $\kappa$ negative somewhere, and $(N,g_{2})$ is a compact, irreducible Riemannian manifold with nonnegative complex sectional curvature. If $\kappa$ is nonnegative somewhere on $\Sigma$, then Lemma 5.1 implies that $(\tilde{M},\tilde{g})$ is flat. Therefore, $\Sigma$ must have negative Gauss curvature everywhere, and thus genus $g<0$. It follows that the universal cover of $\Sigma$, which is equal to $\Sigma$ itself, is diffeomorphic to $\mathbb{R}^{2}$, contradicting the fact that $\Sigma$ is a closed surface.

The proof is complete.

$\square$