Rigidity of Riemannian manifolds with positive scalar curvature

Abstract

For the Bach-flat closed manifold with positive scalar curvature, we prove a rigidity theorem involving the Weyl curvature and the traceless Ricci curvature. Moreover, we provide a similar rigidity result with respect to the $L^{\frac{n}{2}}$-norm of the Weyl curvature, the traceless Ricci curvature, and the Yamabe invariant. In particular, we also obtain rigidity results in terms of the Euler–Poincaré characteristic.

Appendix: The proof of Corollaries 1.3 and 1.5

In this appendix section, we provide the details in proving Corollaries 1.3 and 1.5. For this, the following lemma by Catino [8, Lemma 4.1] is needed:

Lemma 4.1

Let $(M^4,g)$ be a closed manifold. Then

$$\begin{aligned} \begin{aligned} Y^2(M,[g])\ge \int _M(R^2-12|\mathring{R}_{ij}|^2)\,\mathrm{d}V_g, \end{aligned}\end{aligned}$$

(4.1)

with the inequality is strict unless $(M^4,g)$ is conformally Einstein.

By using (4.1), the pinching condition (1.9) can be written as

$$\begin{aligned} \begin{aligned}&\int _M(|W|^2+|\mathring{R}_{ij}|^2)\,\mathrm{d}V_g <\frac{1}{48}Y^2(M,[g]). \end{aligned}\end{aligned}$$

(4.2)

It follows that

$$\begin{aligned} \begin{aligned}&\int _M(|W|^2+|\mathring{R}_{ij}|^2)\,\mathrm{d}V_g-\frac{1}{48}Y^2(M,[g])\\&\quad <\int _M\Big (|W|^2+\frac{5}{4}|\mathring{R}_{ij}|^2-\frac{1}{48}R^2\Big )\,\mathrm{d}V_g, \end{aligned}\end{aligned}$$

(4.3)

from which Corollary 1.3 follows immediately.

On the other hand, (1.12) can be written as

$$\begin{aligned} \begin{aligned} \int _M(|W|^2+4|\mathring{R}_{ij}|^2)\,\mathrm{d}V_g<\frac{25}{486}Y^2(M,[g]) \end{aligned}\end{aligned}$$

(4.4)

which combined with (4.1) gives

$$\begin{aligned} \begin{aligned}&\int _M(|W|^2+4|\mathring{R}_{ij}|^2)\,\mathrm{d}V_g-\frac{25}{486}Y^2(M,[g])\\&\quad < \int _M\Big (|W|^2+\frac{374}{81}|\mathring{R}_{ij}|^2-\frac{25}{486}R^2\Big )\,\mathrm{d}V_g, \end{aligned}\end{aligned}$$

(4.5)

which completes the proof of Corollary 1.5.