Line integration of Ricci curvature and conjugate points in Lorentzian and Riemannian manifolds |
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Authors: | Carmen Chicone Paul Ehrlich |
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Institution: | (1) Department of Mathematics, University of Missouri-Columbia, 65211 Columbia, MO, USA;(2) School of Mathematics, Institute for Advanced Study, 08540 Princeton, NJ, USA |
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Abstract: | Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.Partially supported by NSF grant MCS77-18723(02). |
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