  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 $M$.
