- #1

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1. A tensor undergoes

*parallel transport*if, as it moves through a manifold, its covariant derivative is zero.

2.

*Covariant derivative*describes how a tensor changes as it moves through a manifold.

3. A tensor undergoes

*change*as it moves if it does not parallel transport.

So how do I get out of this loop? I have an intuitive sense of parallel transport, but I do not know how to describe it mathematically except by using the definition above (1). Perhaps the answer lies in the calculus of variations?