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Local rigidity of actions of isometries on compact real analytic Riemannian manifolds

Laurent Stolovitch, Zhiyan Zhao

Abstract

In this article, we consider analytic perturbations of isometries of an analytic Riemannian manifold M. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on M to the same group of isometry it is a perturbation of. Our result relies on a "Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffemorphisms of the circle that are small perturbations of rotations.

Local rigidity of actions of isometries on compact real analytic Riemannian manifolds

Abstract

In this article, we consider analytic perturbations of isometries of an analytic Riemannian manifold M. We prove that, under some conditions, a finitely presented group of such small enough perturbations is analytically conjugate on M to the same group of isometry it is a perturbation of. Our result relies on a "Diophantine-like" condition, relating the actions of the isometry group and the eigenvalues of the Laplace-Beltrami operator. Our result generalizes Arnold-Herman's theorem about diffemorphisms of the circle that are small perturbations of rotations.
Paper Structure (22 sections, 31 theorems, 276 equations)

This paper contains 22 sections, 31 theorems, 276 equations.

Table of Contents

  1. Introduction and main result
  2. Real analytic Riemannian manifold and Grauert tube
  3. Exponential map and Grauert tube
  4. Moser's lemma for Riemannian geometry
  5. Spaces of sections of vector bundles
  6. Analyticity of eigenvectors of Laplacian ${\Delta}_{TM}$
  7. Action on $TM$
  8. Local rigidity of group action by isometries
  9. Group action and cohomology
  10. Precise statement of main theorems
  11. Examples of local analytic rigidity results
  12. Properties of box operator $\square$
  13. $d_0$ and $d_1$
  14. Kernel of $\square$
  15. Inverse of $\square$
  16. ...and 7 more sections

Key Result

Theorem 1.1

Let $M$ be a real analytic compact manifold with an analytic Riemannian metric. Let $G$ be a finitely presented group and let $\pi$ be a Diophantine $G$-group action by analytic isometries on $M$. We assume that $\dim\mathrm{Ker}\; \square <+\infty$. Let $\pi_0$ be an analytic $G$-group action by d

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Proposition 2.4
  • Remark 2.5
  • Definition 2.6
  • Theorem 2.7
  • ...and 35 more