- #1

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*is*this method used, and I just haven’t read about it yet?

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- Thread starter Pencilvester
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- #1

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- #2

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Not unless the curvature is zero in that neighbourhood.I understand that this is the general case for any two points on a curved manifold, but what if we limit ourselves with the requirement that the two points be in the same local convex neighborhood?

If you only had two points, that would be natural. But you do not have only two points and typically you would like comparisons to imply an equivalence relation, i.e., if ##A_p \in V_p##, where ##V_p## is one of the tensor spaces at ##p##, then if ##A_p \sim A_q## and ##A_q \sim A_r##, then ##A_p \sim A_r##. This generally does not hold if you use the geodesic between ##p## and ##q## to define the relation between ##V_p## and ##V_q##, etc., simply because the geodesic from ##p## to ##r## does not need to be the composition of the geodesics from ##p## to ##q## and from ##q## to ##r##.If there’s only one unique geodesic between the two points, that seems like a pretty natural preferred path to me.

- #3

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Makes sense, thanks!if ##A_p \in V_p##, where ##V_p## is one of the tensor spaces at ##p##, then if ##A_p \sim A_q## and ##A_q \sim A_r##, then ##A_p \sim A_r##. This generally does not hold if you use the geodesic between ##p## and ##q## to define the relation between ##V_p## and ##V_q##, etc., simply because the geodesic from ##p## to ##r## does not need to be the composition of the geodesics from ##p## to ##q## and from ##q## to ##r##.

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