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On differentiable manifolds with degenerate metrics

Published online by Cambridge University Press:  24 October 2008

Michael Crampin
Michael Crampin
Affiliation:
Department of Mathematics, King's College, London
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Extract

A metric on a differentiable manifold induces a bilinear form on the tangent space at each point of the manifold. The set of tangent vectors orthogonal, with respect to this bilinear form, to the whole tangent space is a vector subspace of the tangent space: the metric is called non-degenerate or degenerate at the point according as the subspace is or is not empty. This paper is concerned with the geometry of manifolds having everywhere degenerate metrics.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 2 , April 1968 , pp. 307 - 316
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

† See, for example, Kobayashi, and Nomizu, , Foundations of Differential Geometry, Chapter 1.Google Scholar

† Sternberg, , Lectures, in Differential GeometryGoogle Scholar, Chapter 7. Singer, and Sternberg, , The Infinite Groups of Lie and Cartan, J. Analyse Math. 15, 1965. In the case treated here the group is of infinite type.CrossRefGoogle Scholar

† Kobayashi and Nomizu, loc. cit., Chapter 2.

† There are in general higher order structure functions, the second of which corresponds roughly to the curvature. See Sternberg.

† Kobayashi and Nomizu, loc. cit., Chapter 2.

† This result is similar to the de Rham decomposition theorem in the case of a positive definite metric. The proof is adapted from the corresponding proof in Kobayashi and Nomizu, loc. cit., Chapter 4.