Sasaki metric

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Sasaki metric a natural choice of Riemannian metric on the tangent bundle of Riemannian manifold. Introduced by Shigeo Sasaki in 1958.

Construction[edit]

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 .

References[edit]

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