Let $(M,g)$ be a closed oriented $n$-dimensional Riemannian manifold. Let $p\in M$ and ${\mathrm{Ric}}_p$ be the Ricci curvature tensor a $\mathrm{t}p,p$ be the scalar curvature at $p$ which is given by defined to be $S_p:=\frac{1}{n}{\mathrm{Tr}}_g\left({\mathrm{Ric}}_p\right)$. Prove that the scalar curvature $S(p)$ at $p\in M$ is given by
$$S_p=\frac{1}{{\omega }_{n-1}}{\int }_{S^{n-1}}{\mathrm{Ric}}_p(V,V)d{S}^{n-1}$$
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 $S^{n-1}$.