When/how can you integrate tensors on manifolds and what does it mean?
I imagine that line integrals of tensors make sense when you have a connection,
since you can uniquely parallel transport all the tensors along the path to a common point on the paths and "sum" them there, but what about area, volume and higher dimensional integrals? What is required for that to make sense? Parallel transport does not seem sufficient anymore because there are no longer unique path to a common point and parallel transport is path dependent.
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