Summary
In this paper we study the geodesic connectedness of some open connected subsets (regions) of a Lorentzian manifold, using a convexity property of the boundary of such regions. Necessary and sufficient conditions on the metric are given for the convexity of the boundary of such regions. Finally it is presented a result on the geodesic connectedeness of the whole manifold which relates the asymptotic behaviour of the coefficients of the metric to the convexity of the boundary of a family of regions which cover the manifold.
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J. K. Beem -R. E. Parker,Pseudoconvexity and Geodesic connectedness, Ann. Mat. Pura Appl. (IV),CLV (1989), pp. 137–142.
V. Benci -D. Fortunato,Existence of geodesic for the Lorentz metric of a stationary gravitational field, Ann. Inst. H. Poincaré, Analyse non Lineaire,7 (1990), pp. 27–35.
V.Benci - D.Fortunato,On the existence of infinitely many geodesics on space-time manifolds, Adv. Math., to appear.
V. Benci -D. Fortunato -F. Giannoni,On the existence of multiple geodesics in static space-times, Ann. Inst. H. Poincaré, Analyse non Lineaire,8 (1991), pp. 79–102.
V.Benci - D.Fortunato - F.Giannoni,Geodesic on static Lorentz manifolds with convex boundary, in:Proc. Variational methods in Hamiltonian systems and elliptic equations, (M. Girardi - M. Matzeu ed.), pp. 21–41, Pitman Research Notes in Mathematics,243 (1992).
V.Benci - D.Fortunato - F.Giannoni,On the existence of geodesics in static Lorentz manifolds with nonsmooth boundary, Ann. Sc. Norm. Sup. Pisa, in press.
V.Benci - D.Fortunato - A.Masiello,Geodesics in Lorentzian manifolds, preprint Dip. Mat. Univ. Bari, 5/92.
V.Benci - D.Fortunato - A.Masiello,On the geodesic connectedeness of Lorentzian manifolds, preprint Dip. Univ. Bari, 9/92.
R. Geroch,Domains of dependence, J. Math. Phys.,11 (1970), pp. 437–449.
F. Giannoni,Geodesic on non static Lorentz manifolds of Reissner-Nordström type, Math. Ann.,291 (1991), pp. 383–401.
F. Giannoni -A. Masiello,On the existence of geodesics on stationary Lorentz manifolds with convex boundary, J. Funct. Anal.,101 (1991), pp. 340–369.
F.Giannoni - A.Masiello,Geodesics on Lorentzian manifolds with quasi-convex boundary, preprint.
S. W.Hawking - G. F.Ellis,The Large Scale Structure of Space-Time, Cambridge Univ. Press (1973).
L. Landau -E. Lifchitz,Theorie des champs, Mir, Moscou (1970).
A.Masiello,Some results on the geodesic connectedeness of Lorentzian manifolds, in:Proc. Variational Methods and Nonlinear Analysis (A. Ambrosetti - K. C. Chang Eds.), to appear.
J. Nash,The embedding problem for Riemannian manifolds, Ann. Math.,63 (1956), pp. 20–63.
B. O'Neill,Semi-Riemannian Geometry with Applications to Relativity, Academic Press Inc., New York-London (1983).
R. S. Palais,Morse Theory on Hilbert manifolds, Topology,2 (1963), pp. 299–340.
R. Penrose,Techniques of differential topology in relativity, Conf. Board Math. Sci.,7 (S.I.A.M., Philadelphia (1972).
P. H.Rabinowitz,MinMax methods in critical point theory with applications to Differential Equations, CBMS Reg. Conf. Soc. in Math. n.65, A.M.S. (1984).
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Sponsored by M.U.R.S.T. (research funds 40%–60%).
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Masiello, A. Convex regions of Lorentzian manifolds. Annali di Matematica pura ed applicata 167, 299–322 (1994). https://doi.org/10.1007/BF01760337
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DOI: https://doi.org/10.1007/BF01760337