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 $$(M,g)$$ be a Riemannian manifold, denote by $$\tau\colon\mathrm{T} M\to M$$ the tangent bundle over $$M$$. The Sasaki metric $$\hat g$$ on $$\mathrm{T} M$$ is uniquely defined by the following properties:
 * The map $$\tau\colon\mathrm{T} M\to M$$ is a Riemannian submersion.
 * The metric on each tangent space $$\mathrm{T}_p\subset \mathrm{T} M$$ is the Euclidean metric induced by $$g$$.
 * Assume $$\gamma(t)$$ is a curve in $$M$$ and $$v(t)\in\mathrm{T}_{\gamma(t)}$$ is a parallel vector field along $$\gamma$$. Note that $$v(t)$$ forms a curve in $$\mathrm{T} M$$. For the Sasaki metric, we have $$v'(t)\perp \mathrm{T}_{\gamma(t)}$$for any $$t$$; that is, the curve $$v(t)$$ normally crosses the tangent spaces $$\mathrm{T}_{\gamma(t)}\subset \mathrm{T} M$$.