Geodesic Connectedness of Semi-riemannian Manifolds

The problem of geodesic connectedness in semi-Riemannian manifolds (i.e. the question whether each two points of the manifold can be joined by a geodesic) has been widely studied from very different viewpoints. Our purpose is to review these semi-Riemannian techniques, and possible extensions. In the Riemannian case, it is natural to state this problem on (incomplete) manifolds with (possibly non-smooth) boundary, and we will discuss different conditions on this boundary in the remainder of this Section. In this case, geometrical as well as variational methods are appliable, and accurate results can be obtained by using the associated distance and related properties of positive-definiteness. For Lorentzian manifolds, the results cannot be so general, and very different techniques have been introduced which are satisfactory for some particular classes of Lorentzian manifolds. We start by considering several geometrical notions appliable to affine manifolds and, thus, to all semi-Riemannian manifolds, Section 2. Recall that variational methods are not appliable to affine manifolds, at least in a standard way: geodesics are the critical points of the action functional, but a metric tensor must be provided for the definition of this functional. In Section 3 some general facts on geodesic connectedness of Lorentzian manifolds are pointed out, and classical results about connectedness of spaceforms, which rely in the properties of actions of isommetry groups, are summarized. In Section 4 we discuss variational methods applied to Lorentzian manifolds, which have been shown to be useful, mainly, to study stationary and splitting manifolds, with or without boundary. Finally, in Section 5 recent results, based on topological arguments and appliable to multiwarped spacetimes, are explained.