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
Full-text PDF
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] Richard L. Bishop and Samuel I. Goldberg, Tensor analysis on manifolds, The Macmillan Co., New York; Collier-Macmillan Ltd., London, 1968. MR 0224010
- [2] D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Grossen, Lecture Notes in Mathematics, No. 55, Springer-Verlag, Berlin-New York, 1968 (German). MR 0229177
- [3] Robert Maltz, Isometric immersions into spaces of constant curvature, Illinois J. Math. 15 (1971), 490–502. MR 0282317
Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53B05
Retrieve articles in all journals with MSC: 53B05
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
