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up vote 1 down vote favorite I know that if a map F between two surfaces preserves inner products of tangent vectors and is 1-1 and onto, then it must preserve intrinsic distances, but I’m not sure whether the inverse is true. Namely, is it true that any map F between two surfaces that is 1-1, onto, and preserves intrinsic distances will automatically preserve inner products of tangent vectors? Thanks a lot! geometry differential-geometry surfaces share | cite | improve this question asked Nov 25 at 7:05 David Petey Gao 43 4