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Positivity of integral on manifold with bound on Ricci tensor

Source: Alibaba Global Math Competition 2021, Problem 15

July 4, 2021

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Problem Statement

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

.

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