Sasaki metric

Short description: Type of Riemannian 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 $\tau :\mathrm{T}M\to M$ the tangent bundle over $M$. The Sasaki metric $\hat{g}$ on $\mathrm{T}M$ is uniquely defined by the following properties:

• The map $\tau :\mathrm{T}M\to M$ 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 $v^{\prime }(t)\perp {\mathrm{T}}_{\gamma (t)}$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.