A lemma of Élie Cartan
Author:
Robert Maltz
Journal:
Proc. Amer. Math. Soc. 53 (1975), 433-434
MSC:
Primary 53B05
DOI:
https://doi.org/10.1090/S0002-9939-1975-0410582-7
MathSciNet review:
0410582
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Abstract | References | Similar Articles | Additional Information
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.
- [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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Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1975-0410582-7
Keywords:
Affine connection,
parallel translation,
curvature,
torsion,
covariant differentiation,
tensor fields over mappings
Article copyright:
© Copyright 1975
American Mathematical Society
