Sasaki metric

The Sasaki metric is a natural choice of Riemannian metric on the tangent bundle of a Riemannian manifold.
Introduced by Shigeo Sasaki in 1958.

Construction

Let be a Riemannian manifold, denote by the tangent bundle over.
The Sasaki metric on is uniquely defined by the following properties:

The map is a Riemannian submersion.
The metric on each tangent space is the Euclidean metric induced by.
Assume is a curve in and is a parallel vector field along. Note that forms a curve in. For the Sasaki metric, we have for any ; that is, the curve normally crosses the tangent spaces.