题号:14626    题型:解答题    来源:2024年丘成桐大学生数学竞赛(几何与拓扑类)-无答案
Let $(M, g)$ be a closed oriented $n$-dimensional Riemannian manifold. Let $p \in M$ and $\operatorname{Ric}_p$ be the Ricci curvature tensor a $\mathrm{t} p, p$ be the scalar curvature at $p$ which is given by defined to be $S_p:=\frac{1}{n} \operatorname{Tr}_g\left(\operatorname{Ric}_p\right)$. Prove that the scalar curvature $S(p)$ at $p \in M$ is given by
$$
S_p=\frac{1}{\omega_{n-1}} \int_{S^{n-1}} \operatorname{Ric}_p(V, V) d S^{n-1}
$$
where $\omega_{n-1}$ is the area of the unit sphere $S^{n-1}$ in $T_p M$, $V \in S^{n-1}$ are unit vector fields, and $d S^{n-1}$ is the area eleme nt on $S^{n-1}$.
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