Sasaki metric explained

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 T M\to M the tangent bundle over M .The Sasaki metric \hat g on T M is uniquely defined by the following properties:

• The map

\tau\colon T M\to M is a Riemannian submersion.

• The metric on each tangent space

T _{p\subset} T M is the Euclidean metric induced by g .

• Assume

\gamma(t) is a curve in M and v(t)\in T _{\gamma(t)} is a parallel vector field along \gamma . Note that v(t) forms a curve in T M . For the Sasaki metric, we have v'(t)\perp T _{\gamma(t)} for any t ; that is, the curve v(t) normally crosses the tangent spaces T _{\gamma(t)} \subset T M .

References

• S. Sasaki, On the differential geometry of tangent bundle of Riemannian manifolds, Tôhoku Math. J.,10 (1958), 338–354.