Abstract
We show that the space of geodesics of a Hadamard manifold of dimension n is symplectomorphic to the cotangent bundle of the sphere of dimension n−1. This enables us to apply the techniques of symplectic geometry in cotangent bundles to the study of the extrinsic geometry of hypersurfaces and wave fronts in Hadamard manifolds.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alvarez, J. C.: Thése, Rutgers University, 1995.
Alvarez, J. C.: Contact topology, taut immersions and Hilbert's fourth problem, to appear in Symplectic and Contact Topology of Knots and Curves, Progress in Math. Sci., Amer. Math. Soc., Providence, 1995.
Bao, D., Chern, S.S.: On a notable connection in Finsler geometry, Houston J. Math. 19 (1993), 135-180.
Besse, A.L.: Manifolds all of Whose Geodesics are Closed, Springer-Verlag, New York, 1978.
Chekanov, Y.: Critical points of quasifunctions and generating families of legendrian manifolds, Funct. Anal. Appl. 30 (1996), 118-128.
Egloff, D.: Some new developments in Finsler geometry, Inaugural-Dissertation, Universität Freiburg (Swiss).
Ferrand, E.: On a Theorem of Chekanov, Banach Center Publications 39, 1997, pp. 39-48.
Foulon, P.: FrGéométrie des équations différentielles du second ordre, Ann. Inst. H. Poincaré, Physique Théorique 45 (1986), 1-28.
Gallot, S., H ulin, D., Lafontaine, J.: Riemannian Geometry, 2nd edition, Universitext, Springer-Verlag, New York, 1993.
Iglesias, P.: FrSur les géodésiques qui coupent un convexe en courbure négative ou nulle, Ann. Fac. Sci. Toulouse 6(1) (1992), 39-42.
Keilhauer, G.: A note on the space of geodesics, Rev. Union Math. Arg. 36 (1990), 164-173.
Maillot, H.: FrVariétés de géodésiques (cas non compact), Bull. Sci. Math. 118 (1994), 343-383.
Otal, J.P.: FrSur la géométrie symplectique de l'espace des géodésiques d'une variété àcourbure négative, Rev. Math. Iberoamericana 8 (1992), 441-456.
Reznikov, A.: Affine symplectic geometry I: Applications to geometric inequalities, Israel J. Math. 80 (1992), 207-224.
Rights and permissions
About this article
Cite this article
Ferrand, E. Sur la structure symplectique de la variété des géodésiques d'un espace de Hadamard. Geometriae Dedicata 68, 79–89 (1997). https://doi.org/10.1023/A:1004945807350
Issue Date:
DOI: https://doi.org/10.1023/A:1004945807350