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Covariant derivatives and parallelism for tensors
*November 3, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, covariant derivatives, parallelism, Riemannian metrics, tensors

3 comments

Tags: connections, covariant derivatives, parallelism, Riemannian metrics, tensors

3 comments

Time to continue the story for covariant derivatives and parallelism, and do what I promised yesterday on tensors.

Fix a smooth manifold with a connection . Then parallel translation along a curve beginning at and ending at leads to an isomorphism , which depends smoothly on . For any , we get isomorphisms depending smoothly on . (Of course, given an isomorphism of vector spaces, there is an isomorphism sending —the important thing is the inverse.) (more…)