Hostname: page-component-745bb68f8f-g4j75 Total loading time: 0 Render date: 2025-01-11T04:01:46.218Z Has data issue: false hasContentIssue false
  • English
  • Français

Recurrent tensors and holonomy group

Published online by Cambridge University Press:  17 April 2009

D.K. Datta
D.K. Datta
Affiliation:
Department of Mathematics, University of Rhode Island, Kingston, USA and Department of Mathematics, Institute of Advanced Studies, Australian National University, Canberra, ACT.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let M be a connected C∞. A method is being introduced here to study the action of the holonomy group and the restricted holonomy group of Γ on a recurrent tensor. The main result of this paper is that if the recurrence covector W of a recurrent tensor S on M is an exact form then the tensor S is invariant under the holonomy group of Γ and if W is a closed form then S is invariant under the restricted holonomy group of Γ. In the last section, this result is applied to some particular cases including the case of a riemannian manifold with recurrent curvature.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 10 , Issue 1 , February 1974 , pp. 71 - 77
Copyright
Copyright © Australian Mathematical Society 1974

References

[1]Ludden, Gerald D., “Perfect tensors on a manifold”, J. Differential Geometry 2 (1968), 41–53.Google Scholar
[2]Sasaki, Shigeo and Goto, Morikuni, “Some theorems on holononiy groups of Riemannian manifolds”, Trans. Amer. Math. Soc. 80 (1955), 148–158.Google Scholar
[3]Walker., A.G., “On Ruse's spaces of recurrent curvature”, Proc. London Math. Soc. (2) 52 (1951), 36–64.Google Scholar
[4]Wong, Yung-Chow, “Recurrent tensors on a linearly connected differentiable manifold”, Trans. Amer. Math. Soc. 99 (1961), 325–341.CrossRefGoogle Scholar
[5]Wong, Yung-Chow, “Existence of linear connections with respect to which given tensor fields are parallel or recurrent”, Nagoya Math. J. 24 (1964), 67–108.Google Scholar