Levi-Civita transport

From Diffgeom
Revision as of 19:48, 18 May 2008 by Vipul (talk | contribs) (1 revision)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

Definition

Let M be a differential manifold and g a Riemannian metric on M (thus (M,g) is a Riemannian manifold). Parallel transport or Levi-Civita transport on the tangent bundle is the rule that associates, to any smooth curve, the transport along that curve as per the Levi-Civita connection.

We can also define Levi-Civita transport on tensor powers of the tangent bundle, tensor powers of the cotangent bundle, and tensor products of these, because we can define the Levi-Civita connection on each of these.