Metric of vector bundle of Riemann manifold.

Question

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 $\mathbb R^n$ ?

Answer 1

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.

Answer 2

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 $T_p M$.