Proof that the first curvature vector P of a curve C is at right angles to C

First, some notation: write T,a for the (ordinary) partial derivative of the tensor T with respect to xa. This will avoid a lot of messy notation (as if there is not enough already!).

To show the result, we compute

Looking at the term on the right, we distribute to get 3 terms of the form

and we notice that the other two terms are identical to this (except for dummy variables). Thus, we are left with

as asserted.


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