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Whether there is a standard metric of vector bundle of Riemann manifold?

How is it defined ?

Using the exponent map ? Or just using the metric of $\mathbb R^n$ ?

Enhao Lan
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No, there is no standard metric on the tangent bundle of a Riemannian manifold. You may want, though, to take a look at the Sasaki metric and at the Cheeger-Gromoll metric, both of which are presented in this master's thesis.

Alex M.
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The Levi-Civita connection lets you choose a horizontal space in the tangent bundle of $TM$. The metric in the horizontal direction is then induced from the metric on $M$ itself, and the metric in the vertical direction stems from a canonical identification of the complementary vertical space with the fiber $T_p M$.

Mikhail Katz
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  • Isn't this the Sasaki metric? I wouldn't call it standard, maybe it is the simplest one, but why not choose Cheeger's metric instead? No, there is no standard metric on the tangent bundle of a Riemannian manifold. – Alex M. Dec 28 '15 at 13:04
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    @AlexM. you can try to clarify what your objectives are. The metric via a choice of a horizontal space is certainly a natural one, and Sasaki was at least a few decades before Cheeger, both of whom are great mathematicians :-) – Mikhail Katz Dec 28 '15 at 13:22