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Given a riemannian manifold $(M,g)$, is there a canonical induced riemannian metric on the cotangent bundle?

Motivation: See this related question. Also, this looks like it is a canonical metric on $TM$, so would it make sense to map $T^*M$ to $TM$ using $g$ and then take the pullback metric on $T^*M$?

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Ashley
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    Your suggestion works - $g$ is a canonical bundle isomorphism (and thus certainly a diffeomorphism) between $TM$ and $T^*M$, so you can just transfer the metric from $TM$. – Anthony Carapetis Jan 07 '17 at 01:07

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