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 $\mathrm {T} _{\gamma (t)}\subset \mathrm {T} M$.

References

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