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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

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