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…)