Suppose \((M,g)\) denotes a \(d\)-dimensional Riemannian manifold and \(p\in M\). The mean sectional curvature at \(p\) is determined by the scalar curvature. To be more precise,
\[ \frac{1}{\Vol (\mathbb{S}^{d-1})} \int_{\mathbb{S}^{d-1}} m(v) \,d\sigma(w) = \frac{1}{d(d-1)} S(p). \]This follows by employing the geometric interpretation of the Ricci curvature.