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In this paper 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
$\cal{G}_{or}^{k} (M)$
, the Grassmannian of oriented k-planes of M.
