Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants $\{{\mathrm{\Omega }}_k(g)\}$ for closed Riemannian manifolds. The value of ${\mathrm{\Omega }}_k(g)$ delicately reflects the spherical part of the manifold. Indeed, ${\mathrm{\Omega }}_1(g)$ and ${\mathrm{\Omega }}_2(g)$ characterize the standard sphere.