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 $\{\Omega_k(g)\}$ for closed Riemannian manifolds. The value of $\Omega_k(g)$ delicately reflects the spherical part of the manifold. Indeed, $\Omega_1(g)$ and $\Omega_2(g)$ characterize the standard sphere. 
Masayuki Aino
math.DG