Parallel translation of curvature along geodesics

Author:
James J. Hebda

Journal:
Trans. Amer. Math. Soc. **299** (1987), 559-572

MSC:
Primary 53C20; Secondary 34A10

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869221-6

MathSciNet review:
869221

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Abstract: According to the Cartan-Ambrose-Hicks Theorem, two simply-connected, complete Riemannian manifolds are isometric if, given a certain correspondence between all the broken geodesics emanating from a point in one manifold, and all those emanating from a point in the other, the parallel translates of the curvature tensor agree along corresponding broken geodesics. For generic metrics on a surface, the hypothesis can be refined so that it is enough to compare curvature along corresponding unbroken geodesics in order to obtain the isometry.

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0869221-6

Article copyright:
© Copyright 1987
American Mathematical Society