Pestov's Identity on frame bundles and applications

TL;DR

This paper extends Pestov's Identity to frame bundles and explores its applications, including dynamical systems involving parallel transport on Grassmannians of oriented k-planes.

Contribution

It introduces a generalized Pestov's Identity on frame bundles and derives related integrated and restricted versions, with applications to dynamical systems.

Findings

Extended Pestov's Identity to k-tuple tangent bundles

Derived integrated and restricted versions on frame bundles

Applied results to parallel transport on Grassmannians

Abstract

In this article we lift Pestov's Identity on the tangent bundle of a Riemannian manifold $M$ to the bundle of $k$-tuples of tangent vectors. We also derive an integrated version and a restriction to the frame bundle $P^kM$ of $k$-frames. Finally, we discuss a dynamical application for the parallel transport on $\mathcal{G}_{or}^k(M)$, the Grassmannian of oriented $k$-planes of $M$.