MathDB

15

Part of 2021 Alibaba Global Math Competition

Problems(1)

Positivity of integral on manifold with bound on Ricci tensor

Source: Alibaba Global Math Competition 2021, Problem 15

7/4/2021
Let (M,g)(M,g) be an nn-dimensional complete Riemannian manifold with n≥2n \ge 2. Suppose MM is connected and Ric≥(n−1)g\text{Ric} \ge (n-1)g, where Ric\text{Ric} is the Ricci tensor of (M,g)(M,g). Denote by dg\text{d}g the Riemannian measure of (M,g)(M,g) and by d(x,y)d(x,y) the geodesic distance between xx and yy. Prove that ∫M×Mcos⁡d(x,y)dg(x)dg(y)≥0.\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)(M,g) is isometric to the unit round sphere SnS^n.
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