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Conditions for the local existence of metric in a generic affine manifold

Published online by Cambridge University Press:  24 October 2008

Kuo-Shung Cheng  and
Wei-Tou Ni
Kuo-Shung Cheng
Affiliation:
Chiao Tung University and Tsing Hua University, Hsinchu, Taiwan, Republic of China
Wei-Tou Ni
Affiliation:
Chiao Tung University and Tsing Hua University, Hsinchu, Taiwan, Republic of China
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Abstract

For a manifold with a generic symmetric affine connection, explicit necessary and sufficient conditions for the local existence of metric compatible with the connection are obtained in terms of the Riemann tensor and its first-order covariant derivatives. If these conditions are satisfied, the solutions for metric are unique up to a constant scale factor and the absolute value of the signature is uniquely determined. Explicit formulae for the solutions are given in terms of integrals.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 87 , Issue 3 , May 1980 , pp. 527 - 534
Copyright
Copyright © Cambridge Philosophical Society 1980

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References

REFERENCES

(1)Cheng, K.-S. and Ni, W.-T.Necessary and sufficient conditions for the existence of metric in two-dimensional affine manifolds. Chinese J. Phys. 16 (1978), 228.Google Scholar
(2)Cheng, K.-S. and Ni, W.-T.Necessary and sufficient conditions for the existence of metric in three-dimensional affine manifolds. Chinese J. Phys. 17 (1979),Google Scholar