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I have the following question. Assume we have a Riemannian manifold $M$ with the induced metric given by $d$.

I am looking for a canonical way to compare two elements $v,w\in T^n M$, where $T$ denotes the tangent bundle of a manifold.

For instance, for $n=0$, we can compare two elements $v,w\in M$ by $d(v,w)$.

For $n=1$ we can compare two elements $v,w\in TM$ by parallel transporting the vector $w$ to the basepoint $\pi(v)$ of $v$ and setting $D(v,w):= d(\pi(v),\pi(w)) + \|v-\tilde w \|$, where $\tilde w$ denotes the result of parallel transport of the vector $w$.

This is how far I get, for $n=2$ I cannot think of a natural way to compare two points. Is there a natural notion of distance on $T^nM$ for $n>1$?

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If I'm not mistaken, there is a canonical Riemannian metric on the total space of the tangent bundle to a Riemannian manifold...I think the term to google is "Sasaki metric." –  Daniel Litt Aug 30 '11 at 12:06
    
I believe another good place to look into is the Cheeger-Gromoll metric –  Michael Kissner Aug 30 '11 at 12:11
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Dear Philipp, for any Riemannian manifold $(M,g)$, on the tangent bundle $TM$ there exists a natural Riemannian metric $g_T$, called the Sasaki metric on $TM$, such that the canonical projection $\tau_M:(TM,gT)\to (M,g)$ is a Riemannian submersion. –  Giuseppe Tortorella Aug 30 '11 at 12:11
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Sasaki T., On the differential geometry of tangent bundles of Riemannian manifolds, Tohoku Math. J. 10 1958, 338-354. –  Giuseppe Tortorella Aug 30 '11 at 12:17
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It is here: projecteuclid.org/DPubS/Repository/1.0/… –  Giuseppe Tortorella Aug 30 '11 at 12:20
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