Sasaki metric
Sasaki metric a natural choice of Riemannian metric on the tangent bundle of Riemannian manifold. Introduced by Shigeo Sasaki in 1958.
Construction
Let be a Riemannian manifold, denote by the tangent bundle over . The Sasaki metric on is uniquely defined by the following properties:
- The map is a Riemannian submersion.
- The metric on each tangent space is the Euclidean metric induced by .
- Assume is a curve in and is a parallel vector field along . Note that forms a curve in . For the Sasaki metric, we have for any ; that is, the curve normally crosses the tangent spaces .
gollark: Do so.
gollark: ... did mgollark freeze? <:bees:724389994663247974>.
gollark: I was trying to have mgollark generate a response to that starting with "Actually,", but it's being very slow.
gollark: What's a loess?
gollark: It fits the data so well!
References
- S. Sasaki, On the differential geometry of tangent bundle of Riemannian manifolds, Tôhoku Math. J.,10 (1958), 338–354.
This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.