Geodesic connectedness of affine manifolds
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Geodesic connectedness of affine manifolds Ivan P. Costa e Silva1 · José L. Flores2 Received: 21 May 2020 / Accepted: 31 July 2020 © Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract We discuss new sufficient conditions under which an affine manifold (M, ∇) is geodesically connected. These conditions are shown to be essentially weaker than those discussed in groundbreaking work by Beem and Parker and in recent work by Alexander and Karr, with the added advantage that they yield an elementary proof of the main result. Keywords Affine manifolds · Semi-Riemannian manifolds · Geodesic connectivity Mathematics Subject Classification Primary 53C22
1 Introduction The geodesic connectedness of a (connected) Riemannian manifold (M, g) is of course directly linked with geodesic completeness via the Hopf–Rinow theorem, in which the existence of minimal geodesics connecting any two points of M is established. Even in the absence of geodesic completeness, the geodesic connectedness of Riemannian manifolds is fairly well understood [2, 19] By contrast, it has long been known that for indefinite semiRiemannian manifolds, where no analogue of the Hopf–Rinow theorem exists, the situation is much subtler. A famous example by Bates [3] has shown that even complete and compact affine manifolds may fail to be geodesically connected. Even if one only considers the more restricted (but important) class of Lorentzian manifolds, it is well known that de Sitter and antide Sitter spaces are examples of geodesically complete, maximally symmetric Lorentz
* Ivan P. Costa e Silva [email protected] José L. Flores [email protected] 1
Department of Mathematics, Universidade Federal de Santa Catarina, Florianópolis, SC, Brazil
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Departamento de Álgebra, Geometría y Topología, Facultad de Ciencias, Universidad de Málaga, Campus Universitario de Teatinos, 29071 Málaga, Spain
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I. P. Costa e Silva, J. L. Flores
manifolds which are not geodesically connected. The underlying manifolds in these examples, however, are not compact. Yet, compact Lorentz tori can still be found which also fail to be geodesically connected. These examples, in turn, are not geodesically complete.1 Indeed, to the best of our knowledge it remains an open problem to ascertain whether a connected Lorentzian manifold which is both compact and geodesically complete is geodesically connected. On the flip side, powerful variational tools have been developed starting with the seminal work of Benci and Fortunato [8, 9] to tackle the problem of geodesic connectedness in important special cases of Lorentzian manifolds. These special cases not only include stationary (i.e., endowed with a complete timelike Killing vector field), but also time-dependent orthogonal splitting Lorentzian manifolds. But an important limitation of these methods ultimately arises from the fact that the natural energy functional on the space of curves does not, in general, satisfy any natural compactness
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