Parallel translation of curvature along geodesics

Author:
James J. Hebda

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

MSC:
Primary 53C20; Secondary 34A10

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.

**[1]**W. Ambrose,*Parallel translation of Riemannian curvature*, Ann. of Math. (2)**64**(1956), 337–363. MR**0102841****[2]**W. Blaschke,*Differential geometry*. I, Chelsea, New York, 1967.**[3]**Michael A. Buchner,*Simplicial structure of the real analytic cut locus*, Proc. Amer. Math. Soc.**64**(1977), no. 1, 118–121. MR**0474133**, 10.1090/S0002-9939-1977-0474133-5**[4]**Jeff Cheeger and David G. Ebin,*Comparison theorems in Riemannian geometry*, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. MR**0458335****[5]**Marshall M. Cohen,*A course in simple-homotopy theory*, Springer-Verlag, New York-Berlin, 1973. Graduate Texts in Mathematics, Vol. 10. MR**0362320****[6]**Herbert Federer,*Geometric measure theory*, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969. MR**0257325****[7]**Herman Gluck and David Singer,*Scattering of geodesic fields. I*, Ann. of Math. (2)**108**(1978), no. 2, 347–372. MR**506991**, 10.2307/1971170**[8]**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**, 10.1512/iumj.1982.31.31003**[9]**James J. Hebda,*The local homology of cut loci in Riemannian manifolds*, Tôhoku Math. J. (2)**35**(1983), no. 1, 45–52. MR**695658**, 10.2748/tmj/1178229100**[10]**Shoshichi Kobayashi and Katsumi Nomizu,*Foundations of differential geometry. Vol I*, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR**0152974****[11]**Sumner Byron Myers,*Riemannian manifolds in the large*, Duke Math. J.**1**(1935), no. 1, 39–49. MR**1545863**, 10.1215/S0012-7094-35-00105-3**[12]**Sumner Byron Myers,*Connections between differential geometry and topology II. Closed surfaces*, Duke Math. J.**2**(1936), no. 1, 95–102. MR**1545908**, 10.1215/S0012-7094-36-00208-9**[13]**John Nash,*The imbedding problem for Riemannian manifolds*, Ann. of Math. (2)**63**(1956), 20–63. MR**0075639****[14]**V. Ozols,*Cut loci in Riemannian manifolds*, Tôhoku Math. J. (2)**26**(1974), 219–227. MR**0390967****[15]**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) Springer, Berlin, 1977, pp. 707–774. Lecture Notes in Math., Vol. 597. MR**0494233****[16]**Wolfgang Walter,*Differential and integral inequalities*, Translated from the German by Lisa Rosenblatt and Lawrence Shampine. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 55, Springer-Verlag, New York-Berlin, 1970. MR**0271508****[17]**F. W. Warner,*Conjugate loci of constant order*, Ann. of Math. (2)**86**(1967), 192–212. MR**0214005**

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

Article copyright:
© Copyright 1987
American Mathematical Society