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 [math]\displaystyle{ (M,g) }[/math] be a Riemannian manifold, denote by [math]\displaystyle{ \tau\colon\mathrm{T} M\to M }[/math] the tangent bundle over [math]\displaystyle{ M }[/math]. The Sasaki metric [math]\displaystyle{ \hat g }[/math] on [math]\displaystyle{ \mathrm{T} M }[/math] is uniquely defined by the following properties:

• The map [math]\displaystyle{ \tau\colon\mathrm{T} M\to M }[/math] is a Riemannian submersion.
• The metric on each tangent space [math]\displaystyle{ \mathrm{T}_p\subset \mathrm{T} M }[/math] is the Euclidean metric induced by [math]\displaystyle{ g }[/math].
• Assume [math]\displaystyle{ \gamma(t) }[/math] is a curve in [math]\displaystyle{ M }[/math] and [math]\displaystyle{ v(t)\in\mathrm{T}_{\gamma(t)} }[/math] is a parallel vector field along [math]\displaystyle{ \gamma }[/math]. Note that [math]\displaystyle{ v(t) }[/math] forms a curve in [math]\displaystyle{ \mathrm{T} M }[/math]. For the Sasaki metric, we have [math]\displaystyle{ v'(t)\perp \mathrm{T}_{\gamma(t)} }[/math]for any [math]\displaystyle{ t }[/math]; that is, the curve [math]\displaystyle{ v(t) }[/math] normally crosses the tangent spaces [math]\displaystyle{ \mathrm{T}_{\gamma(t)}\subset \mathrm{T} M }[/math].

References

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