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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A lemma of Élie Cartan

Author: Robert Maltz
Journal: Proc. Amer. Math. Soc. 53 (1975), 433-434
MSC: Primary 53B05
MathSciNet review: 0410582
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Abstract: We provide in this paper an alternative proof of the lemma of E. Cartan which states that parallel translation of curvature and torsion locally determines an affine connection. Our proof uses covariant differentiation of tensor fields over mappings in place of Cartan's exterior differential calculus.

References [Enhancements On Off] (What's this?)

  • [1] R. L. Bishop and S. I. Goldberg, Tensor analysis on manifolds, Macmillan, New York, 1968. MR 36 #7057. MR 0224010 (36:7057)
  • [2] D. Gromoll, W. Klingenberg and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Math., no. 55, Springer-Verlag, Berlin and New York, 1968. MR 37 #4751. MR 0229177 (37:4751)
  • [3] R. Maltz, Isometric immersions into spaces of constant curvature, Illinois J. Math. 15 (1971), 490-502. MR 43 #8029. MR 0282317 (43:8029)

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Keywords: Affine connection, parallel translation, curvature, torsion, covariant differentiation, tensor fields over mappings
Article copyright: © Copyright 1975 American Mathematical Society

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