Riemannian Invariants that Characterize Rotational Symmetries of the Standard Sphere

TL;DR

This paper introduces new Riemannian invariants that precisely identify the rotational symmetries of the standard sphere, extending classical spectral characterizations.

Contribution

The authors define a novel family of invariants for closed Riemannian manifolds that uniquely characterize the standard sphere among all manifolds.

Findings

 and  invariants characterize the standard sphere

The invariants reflect the spherical component of the manifold

New invariants extend spectral characterizations of spheres

Abstract

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.