Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the number of geodesic segments
connecting two points
on manifolds of non-positive curvature

Author: Paul Horja
Journal: Trans. Amer. Math. Soc. 349 (1997), 5021-5030
MSC (1991): Primary 53C22; Secondary 53C70
MathSciNet review: 1401773
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove that on a complete Riemannian manifold $M$ of dimension $n$ with sectional curvature $K_M < 0$, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least $2n+1$ geodesic segments. A simpler version of the argument shows that if one of the points is fixed and $K_M \leq 0$ then the two points are connected by at least $n+1$ geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.

References [Enhancements On Off] (What's this?)

  • [1] A.D. Alexandrov, V.N. Berestovskii & I.G. Nikolaev, Generalized Riemannian spaces, Russian Math. Surveys 41 (1986), no. 3, 1-54. MR 88e:53103
  • [2] W. Ballmann, M. Gromov & V.Schroeder, Manifolds of nonpositive curvature, Progress in Math., Vol. 61, Birkhäuser, Boston, 1985. MR 87h:53050
  • [3] R. Benedetti & C. Petronio, Lectures on hyperbolic geometry, Universitext, Springer, Berlin, 1992. MR 94e:57015
  • [4] R.L. Bishop & B. O'Neill, Manifolds of negative curvature, Transactions of the AMS 145 (1969) 1-49. MR 40:4891
  • [5] E. Ghys & P. de la Harpe (Eds.), Sur les groupes hyperboliques d'après Mikhael Gromov, Progress in Math., Vol. 83, Birkhäuser, Boston, 1990. MR 92f:53050
  • [6] H. Karcher, Schnittort und konvexe Mengen in vollständigen Riemannschen Mannigfaltigkeiten, Math. Ann. 177 (1968) 105-121. MR 37:2131
  • [7] R. Schoen & S.T. Yau, Compact group actions and the topology of manifolds with non-positive curvature, Topology 18 (1979) 361-380. MR 81a:53044; MR 83k:53061

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 53C22, 53C70

Retrieve articles in all journals with MSC (1991): 53C22, 53C70

Additional Information

Paul Horja
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320

Received by editor(s): October 27, 1995
Received by editor(s) in revised form: June 6, 1996
Article copyright: © Copyright 1997 American Mathematical Society

American Mathematical Society