Spectral Characterization of Besse and Zoll Properties for Symmetric Convex Contact Spheres

Abstract

A closed contact manifold is called Besse when all its Reeb orbits are closed, and Zoll when they have the same minimal period. Motivated by the study of symmetric closed characteristics on symmetric compact convex hypersurfaces of Wang [22], we firstly introduce some variant concepts for symmetric convex contact spheres, i.e., a symmetric convex contact sphere is called Symmetric-Besse when all its Reeb orbits are symmetric, and Symmetric-Zoll when they have the same minimal period, then we provide a characterization for Symmetric-Besse and Symmetric-Zoll convex contact spheres in terms of the S¹-equivariant spectral invariants of symmetric Reeb orbits.

Keywords

Symmetric Reeb orbits / Symmetric-Besse / Symmetric-Zoll / spectral characterization / 37J46 / 58E05

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