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Proceedings of the American Mathematical Society

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Pseudo-Riemannian metric singularities and the extendability of parallel transport

Author: Marek Kossowski
Journal: Proc. Amer. Math. Soc. 99 (1987), 147-154
MSC: Primary 53C50; Secondary 53C40, 58A12
MathSciNet review: 866445
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Abstract: We are given a $ C\infty $ immersion $ i:N \to (M,\langle \;\rangle )$, and $ p \in N$ is a point where $ N_p^ \bot \cap {T_p}N$ is one-dimensional. We have shown that there is a tensor $ {\text{I}}{{\text{I}}_p}:{T_p}N \times {T_p}N \times \operatorname{Rad}_p \to {\mathbf{R}}$ intrinsic to $ (N,{i^*}\langle \;\rangle )$ which determines an extrinsic feature of the immersion. The purpose of this paper is to show that II controls the following two intrinsic properties. First, II determines which pairs of vector fields $ X$, $ Y$ on $ N$ have the property that intrinsic covariant derivative $ {\nabla _x}Y$ extends smoothly to all of $ N$. Second, given a curve in $ N$ containing $ p,{\text{I}}{{\text{I}}_p}$ determines which parallel vector fields along the curve extend smoothly through $ p$. As an application we locally characterize product and flat metric singularities.

References [Enhancements On Off] (What's this?)

  • 1. J. Beem, and P. Ehrlich, Global Lorentzian geometry, Dekker, New York, 1981. MR 619853 (82i:53051)
  • [1] M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Math., Vol. 14, Springer-Verlag, 1973. MR 0341518 (49:6269)
  • [2] M. W. Hirsh, C. C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin and New York, 1977. MR 0501173 (58:18595)
  • [3] S. Kobayasi and K. Nomizu, Foundations of differential geometry, Interscience, New York, 1969.
  • [4] M. Kossowski, Fold singularities in pseudo Riemannian geodesic tubes, Proc. Amer. Math. Soc. 95 (1985), 463-469. MR 806088 (87f:58023)
  • [5] -, First order PDE with singular solutions, Indiana Univ. Math. J. 35 (1986).
  • [6] B. O'Neill, Semi Riemannian geometry, Academic Press, 1983. MR 719023 (85f:53002)

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Article copyright: © Copyright 1987 American Mathematical Society

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