Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-27T08:32:23.206Z Has data issue: false hasContentIssue false
  • English
  • Français

Closed geodesics in Riemannian manifolds with convex boundary*

Published online by Cambridge University Press:  14 November 2011

Anna Maria Candela  and
Addolorata Salvatore
Anna Maria Candela
Affiliation:
Dipartimento di Matematica, via E. Orabona 4, 70125 Bari, Italy
Addolorata Salvatore
Affiliation:
Dipartimento di Matematica, via E. Orabona 4, 70125 Bari, Italy
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we look for closed geodesies on a noncomplete Riemannian manifold ℳ. We prove that if ℳ has convex boundary, then there exists at least one closed nonconstant geodesic on it.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 124 , Issue 6 , 1994 , pp. 1247 - 1258
Copyright
Copyright © Royal Society of Edinburgh 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Benci, V.. Normal modes of a Lagrangian system constrained in a potential well. Ann. Inst. H. Poincare Anal. Non Lineaire 1 (1984), 379–400.CrossRefGoogle Scholar
2Benci, V., Fortunato, D. and Giannoni, F.. On the existence of multiple geodesies in static space-times. Ann. Inst. H. Poincare Anal. Non Lineaire 8 (1991), 79–102.Google Scholar
3Benci, V., Fortunato, D. and Giannoni, F.. On the existence of periodic trajectories in static Lorentz manifolds with singular boundary. “Nonlinear Analysis”, a tribute in honour of G. Prodi, Quaderni della Scuola Norm. Sup. Pisa, eds Ambrosetti, A. and Marino, A., 109–133 (Pisa: Scuola Normale Superiore di Pisa, 1991).Google Scholar
4Benci, V., Fortunato, D. and Giannoni, F.. On the existence of geodesies in static Lorentz manifolds with singular boundary. Ann. Scuola Norm. Sup. Pisa, Ser. IV 19 (1992), 255–289.Google Scholar
5Benci, V., Fortunato, D. and Giannoni, F.. A remark on closed geodesies on Riemannian manifolds (Preprint).Google Scholar
6Benci, V. and Giannoni, F.. On the existence of closed geodesies on noncompact Riemannian manifolds. Duke Math. J. 68 (1992), 195–215.CrossRefGoogle Scholar
7Fadell, E. and Husseini, S.. Category of loop spaces of open subsets in Euclidean Space. Nonlinear Anal. 17(1991), 1153–1161.CrossRefGoogle Scholar
8Gordon, W. B.. The existence of geodesies joining two given points. J. Differential Geom. 9 (1974), 443–450.CrossRefGoogle Scholar
9Gromoll, D. and Meyer, W.. Periodic geodesies on compact Riemannian manifolds. J. Differential Geom. 3(1969), 493–510.CrossRefGoogle Scholar
10Klingenberg, W.. Lectures on Closed Geodesies (Berlin: Springer, 1978).CrossRefGoogle Scholar
11Masiello, A.. Metodi variazionali in geometria Lorentziana (Tesi di Dottorato, Pisa, 1991).Google Scholar
12Nash, J.. The embedding problem for Riemannian manifold. Ann. Math. 63 (1956), 20–63.CrossRefGoogle Scholar
13O'Neill, B.. Semi-Riemannian Geometry with Applications to Relativity (New York: Academic Press, 1983).Google Scholar
14Salvatore, A.. On the existence of infinitely many periodic solutions on non-complete Riemannian manifolds (To appear on J. Diff. Eq.).Google Scholar
15Schwartz, J. T.. Nonlinear Functional Analysis, Notes on Mathematics and Applications (New York; Gordon and Breach, 1969).Google Scholar
16Struwe, M.. Variational Methods. Applications to Nonlinear Partial Differential Equations and Hamiltonian Systems (Berlin: Springer, 1990).Google Scholar
17Thorbergsson, G.. Closed geodesies on non-compact Riemannian manifolds. Math. Z. 159 (1978), 249–258.CrossRefGoogle Scholar