Local rigidity of group actions of isometries on compact Riemannian manifolds

TL;DR

This paper studies the rigidity of group actions by isometries on compact Riemannian manifolds, showing that small perturbations under certain conditions are conjugate to original actions, extending classical rigidity results.

Contribution

It generalizes existing rigidity theorems to group actions on manifolds, establishing conditions under which perturbations are conjugate to isometries.

Findings

Rigidity holds for small perturbations satisfying Diophantine conditions.

Extends classical circle diffeomorphism rigidity to higher-dimensional manifolds.

Connects rigidity phenomena with Kazhdan's property (T).

Abstract

In this article, we consider perturbations of isometries on a compact Riemannian manifold $M$. We investigate the smooth (resp. analytic) rigidity phenomenon of groups of these isometries. As a particular case, we prove that if a finite family of smooth (resp. analytic) small enough perturbations is simultaneously conjugate to the family of isometries via a finitely smooth diffeomorphism, then it is simultaneously smoothly (resp. analytically) conjugate to it whenever the family of isometries satisfies a Diophantine condition. Our results generalize the rigidity theorems of Arnold, Herman, Yoccoz, Moser, etc. about circle diffeomorphisms which are small perturbations of rotations as well as Fisher-Margulis's theorem on group actions satisfying Kazhdan's property (T).