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
Abstract: We are given a immersion , and is a point where is one-dimensional. We have shown that there is a tensor intrinsic to 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 , on have the property that intrinsic covariant derivative extends smoothly to all of . Second, given a curve in containing determines which parallel vector fields along the curve extend smoothly through . As an application we locally characterize product and flat metric singularities.
- 1. John K. Beem and Paul E. Ehrlich, Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Math., vol. 67, Marcel Dekker, Inc., New York, 1981. MR 619853
-  M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Springer-Verlag, New York-Heidelberg, 1973. Graduate Texts in Mathematics, Vol. 14. MR 0341518
-  M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173
-  S. Kobayasi and K. Nomizu, Foundations of differential geometry, Interscience, New York, 1969.
-  Marek Kossowski, Fold singularities in pseudo-Riemannian geodesic tubes, Proc. Amer. Math. Soc. 95 (1985), no. 3, 463–469. MR 806088, https://doi.org/10.1090/S0002-9939-1985-0806088-X
-  -, First order PDE with singular solutions, Indiana Univ. Math. J. 35 (1986).
-  Barrett O’Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1983. With applications to relativity. MR 719023
- J. Beem, and P. Ehrlich, Global Lorentzian geometry, Dekker, New York, 1981. MR 619853 (82i:53051)
- M. Golubitsky and V. Guillemin, Stable mappings and their singularities, Graduate Texts in Math., Vol. 14, Springer-Verlag, 1973. MR 0341518 (49:6269)
- 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)
- S. Kobayasi and K. Nomizu, Foundations of differential geometry, Interscience, New York, 1969.
- M. Kossowski, Fold singularities in pseudo Riemannian geodesic tubes, Proc. Amer. Math. Soc. 95 (1985), 463-469. MR 806088 (87f:58023)
- -, First order PDE with singular solutions, Indiana Univ. Math. J. 35 (1986).
- B. O'Neill, Semi Riemannian geometry, Academic Press, 1983. MR 719023 (85f:53002)