On differentiable manifolds with degenerate metrics
Published online by Cambridge University Press: 24 October 2008
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
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.
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