TITLE: Metric on tangent bundle
QUESTION [8 upvotes]: Let $(M,g)$ be a compact Riemannian manifold, no boundary. 
I would like to know when is it possible to endow the tangent bundle $TM$ with an euclidean Riemannian metric. 
Do you know of a good reference?

REPLY [2 votes]: One of the ways of defining naturally a metric on the tangent bundle $TM$ is indeed (as Peter says above) a Sasaki metric defined in 1968. You can prove that $(TM, g_{\mathrm{Sas}})$ is flat if and only if $(M,g)$ is flat. The reference is for example the paper by Sigmundur Gudmundsson and Elias Kappos On the Geometry of Tangent Bundles. They give explicit calculations for Levi-Civita connection for this metric as well as curvature tensor.