The exponential map of a \(C^{1,1}\)-metric. (English) Zbl 1291.53016
Summary: Given a pseudo-Riemannian metric of regularity \(C^{1,1}\) on a smooth manifold, we prove that the corresponding exponential map is a bi-Lipschitz homeomorphism locally around any point. We also establish the existence of totally normal neighborhoods in an appropriate sense. The proofs are based on regularization, combined with methods from comparison geometry.
MSC:
| 53B20 | Local Riemannian geometry |
| 53B21 | Methods of local Riemannian geometry |
| 53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |
References:
| [1] | Cheeger, J.; Gromov, M.; Taylor, M., Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., 17, 15-53 (1982) · Zbl 0493.53035 |
| [2] | Chen, B.-L.; LeFloch, P., Injectivity radius of Lorentzian manifolds, Commun. Math. Phys., 278, 679-713 (2008) · Zbl 1156.53040 |
| [3] | Chruściel, P. T., Elements of causality theory |
| [4] | Dieudonne, J., Treatise on analysis, Bull. Am. Math. Soc., 3, 1, Part 1 (1980) |
| [5] | do Carmo, M. P., Riemannian Geometry (1992), Birkhäuser: Birkhäuser Boston, MA · Zbl 0752.53001 |
| [6] | Grosser, M.; Kunzinger, M.; Oberguggenberger, M.; Steinbauer, R., Geometric Theory of Generalized Functions, Math. Appl., vol. 537 (2001), Kluwer · Zbl 0998.46015 |
| [7] | Harris, S. G., A triangle comparison theorem for Lorentz manifolds, Indiana Univ. Math. J., 31, 289-308 (1982) · Zbl 0496.53042 |
| [8] | Jost, J., Riemannian Geometry and Geometric Analysis, Universitext (2011), Springer · Zbl 1227.53001 |
| [9] | Kulkarni, R. S., The values of sectional curvature in indefinite metrics, Comment. Math. Helv., 54, 1, 173-176 (1979) · Zbl 0393.53022 |
| [10] | O’Neill, B., Semi-Riemannian Geometry. With Applications to Relativity, Pure Appl. Math., vol. 103 (1983), Academic Press: Academic Press New York · Zbl 0531.53051 |
| [11] | Whitehead, J. H.C., Convex regions in the geometry of paths, Q. J. Math., Oxf. Ser., 3, 33-42 (1932) · Zbl 0004.13102 |
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