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Existence of Linear Connections with Respect to Which Given Tensor Fields are Parallel or Recurrent

Published online by Cambridge University Press:  22 January 2016

Yung-Chow Wong
Yung-Chow Wong*
Affiliation:
University of Hong Kong, Hong Kong
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Let M be an n-dimensional (n≥2) connected C∞-manifold with a linear connection. For simplicity, tensor fields on M will simply be called tensors on M. A tensor S on M is said to be parallel if its covariant derivative is everywhere zero in M, i.e., if ▽S = 0. S is said to be recurrent if its covariant derivative is equal to the tensor product of a covector and S itself, i.e., if ▽S = W⊗S, where W is called the recurrence covector.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 24 , June 1964 , pp. 67 - 108
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1964

References

[1] Chevalley, C., Theory of Lie groups I, Princeton University Press (1946).CrossRefGoogle Scholar
[2] Chrystal, G., Algebra II, London (1926).Google Scholar
[3] Eisenhart, L. P., Non-Riemannian geometry, American Math. Soc. (1927).Google Scholar
[4] Fukami, V., Affine connections in almost product manifolds with some structures, Tohoku Math. J., vol. 11 (1959), pp. 430–446.CrossRefGoogle Scholar
[5] Gantmacher, F. R., The theory of matrices I, New York, Chelsea (1959).Google Scholar
[6] Jacobson, N., Lectures in abstract algebra II, New York, Van Nostrand (1953).CrossRefGoogle Scholar
[7] Kurita, M., “Tensor fields and their parallelisim”, Nagoya Math. J., vol. 18 (1961), pp. 133–151.CrossRefGoogle Scholar
[8] MacDuffee, C. C., The theory of matrices, New York, Chelsea (1946).Google Scholar
[9] Nomizu, K., Lie groups and differential geometry, Tokyo, Japanese Math. Soc. (1956).Google Scholar
[10] Roth, W. E., “On direct product matrices”, Bulletin American Math. Soc, vol. 40 (1964), pp. 461–468.CrossRefGoogle Scholar
[11] Walker, A. G., “On parallel fields of partially null vector spaces”, Quarterly J. Math. (Oxford), vol. 20 (1949), pp. 135–145.Google Scholar
[12] Willmore, T. J., An introduction to differential geometry, Oxford University Press (1959).Google Scholar
[13] Wong, Yung-Chow, “Recurrent tensors on a linearly connected differentiate manifold”, Trans. American Math. Soc, vol. 99 (1961), pp. 325–341.CrossRefGoogle Scholar
[14] Wong, Yung-Chow, “Linear connections with zero torsion and recurrent curvature”, Trans. American Math. Soc, vol. 102 (1962), pp. 471–506.CrossRefGoogle Scholar