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The Pestov identity on the frame bundle and associated homogeneous fibrations

Mihajlo Cekić, Thibault Lefeuvre, Andrei Moroianu, Uwe Semmelmann

Abstract

In this short note, we prove a global Pestov identity on the (orthonormal) frame bundle of a Riemannian manifold and deduce similar identities on associated homogeneous fibrations. As a particular example, this provides a concise proof of the Pestov identity on the unit tangent bundle of the manifold.

The Pestov identity on the frame bundle and associated homogeneous fibrations

Abstract

In this short note, we prove a global Pestov identity on the (orthonormal) frame bundle of a Riemannian manifold and deduce similar identities on associated homogeneous fibrations. As a particular example, this provides a concise proof of the Pestov identity on the unit tangent bundle of the manifold.

Paper Structure

This paper contains 14 sections, 10 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Content of this note
  4. Frame bundle geometry and structural equations
  5. Frame bundle geometry
  6. Unit tangent bundle geometry
  7. Geodesic and frame flows
  8. Structural equations
  9. Pestov identities
  10. Universal Pestov identity on the frame bundle
  11. Associated Pestov identity
  12. Homogeneous fibrations over spheres
  13. Associated Pestov identity
  14. Applications in negative curvature: frame flow ergodicity

Key Result

Lemma 2.1

For all $\xi,\xi' \in \Lambda^2 \mathbb{R}^n$, $\theta,\theta' \in \mathbb{R}^n$ and $w \in FM$ we have: where $\xi\theta$ is the vector obtained by applying $\xi$ (as an endomorphism of $\mathbb{R}^n$) to $\theta$.

Theorems & Definitions (23)

  • Lemma 2.1
  • proof
  • Lemma 2.2: Expression for the horizontal gradient
  • proof
  • Lemma 2.3: Commutation relation for the horizontal gradient
  • proof
  • Example 2.4
  • Lemma 2.5
  • proof
  • Theorem 3.1: Universal Pestov identity
  • ...and 13 more