Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

Abstract

Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants {Ω k (g)} for closed Riemannian manifolds. The value of Ω k (g) sharply reflects the spherical part of the manifold. Indeed, Ω 1 (g) and Ω 2 (g) characterize the standard sphere. Contents 16 4. Computation and examples 18 4.1. The product of Einstein manifolds 18 4.2. The case of Heisenberg manifolds 19 Appendix A. The proof of Lemma 2.5 24 References 26