Geodesic connectedness of semi-Riemannian manifolds

Details Geodesic connectedness of semi-Riemannian manifolds Geodesic connectedness of semi-Riemannian manifolds

2000-05-04

arxiv.org/abs/math/0005039v2

Mathematics

Differential Geometry

14 pages, Latex; revised version with minor changes; a shortened version of 12 pages will appear at Nonlinear Anal. (proceedin

Scientific paper

The question whether a Riemannian manifold is geodesically connected can be studied from geometrical as well as variational methods, and accurate results can be obtained by using the associated distance and related properties of the positive-definiteness. It is natural to state this problem in manifolds with (possibly non-smooth) boundary, and some conditions on this boundary has been studied. For Lorentzian manifolds, the results cannot be so general, and very different techniques has been introduced which are satisfactory for some classes of Lorentzian manifolds: spaceforms, disprisoning and pseudoconvex manifolds, stationary, globally hyperbolic or multiwarped spacetimes... Some of them are appliable to semi-Riemannian manifolds with higher index or even to manifolds with just an affine connection. Our purpose is to review these semi-Riemannian techniques, discussing the results and possible extensions.

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