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Closed geodesies on compact Lorentzian manifolds of splitting type

Published online by Cambridge University Press:  14 November 2011

Flavia Antonacci  and
Rosell Sampalmieri
Flavia Antonacci
Affiliation:
Dipartimento di Matematica, Università degli Studi di Roma III, Largo S. Leonardo Murialdo 1, Italy e-mail: flavia@matrm3.mat.uniroma3.it
Rosell Sampalmieri
Affiliation:
Facoltà di Ingegneria, Dipartimento di Energetica, Università dell'Aquila, Monteluco di Roio, L'Aquila, Italy e-mail: sampalm@ing.univaq.it
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Abstract

In this paper, we consider the problem of the existence of a spacelike closed geodesic on compact Lorentzian manifolds. Tipler and Galloway proved that, under suitable topological properties of the manifold, there exists a closed timelike geodesic. In their proofs, they use the hypothesis that the time coordinate of one timelike geodesic has derivative always different from zero. This clearly fails for spacelike geodesies. Using variational methods and applying the relative category theory, we prove the existence of a closed spacelike geodesic on a compact manifold of splitting type. Observe that, thanks to the previous results, the existence of at least two geometrically distinct closed geodesies on follows.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 3 , 1998 , pp. 447 - 462
Copyright
Copyright © Royal Society of Edinburgh 1998

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References

1Benci, V. and Fortunate, D. On the existence of infinitely many geodesies on space—time manifolds. Adv. Math, (to appear).Google Scholar
2Benci, V. and Giannoni, F.. On the existence of closed geodesies on noncompact Riemannianmanifolds. Duke Math. J. 68 (1992), 195–215.CrossRefGoogle Scholar
3Candela, A. M., Giannoni, F. and Masiello, A.. Multiple critical points for indefinite functionals and application (Preprint).Google Scholar
4Fadell, E.. Lectures in cohomological index theories of G-spaces with applications to the critical point theory (Sem. Dip. Mat. Università della Calabria, Cosenza, 1985).Google Scholar
5Fadell, E. and Husseini, S.. Relative category, products and coproducts (Preprint, 1995).CrossRefGoogle Scholar
6Fournier, G. and Willem, M.. Relative category and the calculus of variations. In Variational Problems, eds.Beresticky, H., Coron, J. M. and Ekeland, I., 95–104 (Basel: Birkhäuser, 1990).Google Scholar
7Galloway, G. J.. Closed timelike geodesies. Trans. Amer. Math. Soc. 285 (1984), 379–88.CrossRefGoogle Scholar
8Galloway, G. J.. Compact Lorentzian manifolds without closed nonspacelike geodesies. Proc.Amer. Math. Soc. 98 (1986), 119–23.CrossRefGoogle Scholar
9Giannoni, F. and Masiello, A.. Geodesies on product Lorentzian manifolds. Ann. Inst. H. Poincaré, Anal. Non Linéaire 12 (1995), 27–60.CrossRefGoogle Scholar
10Hawking, S. W. and Ellis, G. F.. The Large Scale Structure of Space-Time (London: Cambridge University Press, 1973).CrossRefGoogle Scholar
11Klingenberg, W.. Lectures on Closed Geodesies, Lecture Notes in Mathematics 230 (Berlin: Springer, 1978).CrossRefGoogle Scholar
12Masiello, A.. On the existence of a closed geodesic in stationary Lorentz manifolds. J. Differential Equations 104 (1993), 48–59.CrossRefGoogle Scholar
13Masiello, A.. Variational Methods in Lorentzian Geometry, Pitman Research Notes in Mathematics 59 (Harlow: Longman, 1994).Google Scholar
14Nash, J.. The embedding problem for Riemannian manifolds. Ann. Math. 63 (1956), 20–63.CrossRefGoogle Scholar
15O'Neill, B.. Semi-Riemannian Geometry with Applications to Relativity (New York: Academic Press, 1983).Google Scholar
16Palais, R.. Morse theory on Hilbert manifolds. Topology 2 (1963), 299–340.CrossRefGoogle Scholar
17Piccione, P. and Sampalmieri, R.. Geodesical connectedness of compact manifolds. Dynam. Systems Appl. (to appear).Google Scholar
18Rabinowitz, P. H.. Min–Methods in Critical Point Theory with Applications to Differential Equations, CMBS Regional Conference Societies in Mathematics 65 (Providence, RI: American Mathematical Society, 1984).Google Scholar
19Szulkin, A.. A relative category and applications to critical point theory for strongly indefinitefunctionals. Nonlinear Anal. 15 (1990), 725–39.CrossRefGoogle Scholar
20Thorbergsson, G.. Closed geodesies on noncompact Riemannian manifolds. Math. Z. 159 (1978), 249–58.CrossRefGoogle Scholar
21Tipler, F. J.. Existence of closed timelike geodesies in Lorentz spaces. Proc. Amer. Math. Soc. 76 (1979), 145–7.CrossRefGoogle Scholar