ALIBABA 2021 Problem 15

Let $(M,g)$ be an $n$ -dimensional complete Riemannian manifold with $n \ge 2$ . Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$ , where $\text{Ric}$ is the Ricci tensor of $(M,g)$ . Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$ . Prove that $$\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.$$ Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$ .