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maddy
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I learned that there exists a difference between Lie transport and parallel transport and what that difference is in differential geometry, but I'm getting all confused again when I read the explanation given in the 'intro to differential forms' thread (below).
How do we do Lie transport of a vector without embedding the manifold in some higher dimensional manifold?
Thanks!
(posted by jeff)Intrinsic curvature is defined by using the fairly easy to understand idea of "parallel transport". Imagine some closed curve on a flat surface with the tail of a vector placed on a point of this curve. Now push the tail around the curve in such a way that in moving it between infinitessimally separated points on the curve, the vector is kept parallel to itself. When the tail returns to the starting point the vector will be pointing in the same direction as it was initially. However, in performing the same exercise on a curved surface, the final and initial orientations of the vector will in general differ.
How do we do Lie transport of a vector without embedding the manifold in some higher dimensional manifold?
Thanks!