Pestov's Identity on frame bundles and applications

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 $P^{k}M$ 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 $M$.