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 $\{{\Omega }_k(g)\}$ 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.

arXiv comment27pages v2: an example added v3: the author's address was changed, the table of contents was changed, some typos were corrected. The final publication is available at Springer via http://dx.doi.org/10.1007/s00229-017-0969-9

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