Parallel translation of curvature along geodesics

Hebda, James J.

doi:10.1090/S0002-9947-1987-0869221-6

Parallel translation of curvature along geodesics
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by James J. Hebda PDF

Trans. Amer. Math. Soc. 299 (1987), 559-572 Request permission

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.

References

W. Ambrose, Parallel translation of Riemannian curvature, Ann. of Math. (2) 64 (1956), 337–363. MR 102841, DOI 10.2307/1969978

Differential geometry

Michael A. Buchner, Simplicial structure of the real analytic cut locus, Proc. Amer. Math. Soc. 64 (1977), no. 1, 118–121. MR 474133, DOI 10.1090/S0002-9939-1977-0474133-5
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
Marshall M. Cohen, A course in simple-homotopy theory, Graduate Texts in Mathematics, Vol. 10, Springer-Verlag, New York-Berlin, 1973. MR 0362320
Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York, Inc., New York, 1969. MR 0257325
Herman Gluck and David Singer, Scattering of geodesic fields. I, Ann. of Math. (2) 108 (1978), no. 2, 347–372. MR 506991, DOI 10.2307/1971170
James J. Hebda, Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem, Indiana Univ. Math. J. 31 (1982), no. 1, 17–26. MR 642612, DOI 10.1512/iumj.1982.31.31003
James J. Hebda, The local homology of cut loci in Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 1, 45–52. MR 695658, DOI 10.2748/tmj/1178229100
Shoshichi Kobayashi and Katsumi Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1963. MR 0152974
Sumner Byron Myers, Riemannian manifolds in the large, Duke Math. J. 1 (1935), no. 1, 39–49. MR 1545863, DOI 10.1215/S0012-7094-35-00105-3
Sumner Byron Myers, Connections between differential geometry and topology II. Closed surfaces, Duke Math. J. 2 (1936), no. 1, 95–102. MR 1545908, DOI 10.1215/S0012-7094-36-00208-9
John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989
V. Ozols, Cut loci in Riemannian manifolds, Tohoku Math. J. (2) 26 (1974), 219–227. MR 390967, DOI 10.2748/tmj/1178241180
C. T. C. Wall, Geometric properties of generic differentiable manifolds, Geometry and topology (Proc. III Latin Amer. School of Math., Inst. Mat. Pura Aplicada CNPq, Rio de Janeiro, 1976) Lecture Notes in Math., Vol. 597, Springer, Berlin, 1977, pp. 707–774. MR 0494233
Wolfgang Walter, Differential and integral inequalities, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. Translated from the German by Lisa Rosenblatt and Lawrence Shampine. MR 0271508
F. W. Warner, Conjugate loci of constant order, Ann. of Math. (2) 86 (1967), 192–212. MR 214005, DOI 10.2307/1970366