Geometriae Dedicata

, Volume 113, Issue 1, pp 243–254

The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group

Article

Abstract

Let Mn be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on Mn of length at most 3 diameter(Mn). Moreover, this geodesic net is either a closed geodesic, consists of two geodesic loops emanating from the same point, or consists of three geodesic segments between the same endpoints. Geodesic nets can be viewed as the critical points of the length functional on the space of graphs immersed into a Riemannian manifold. One can also consider other natural functionals on the same space, in particular, the maximal length of an edge. We prove that either there exists a closed geodesic of length ≤ 2 diameter(Mn), or there exists a critical point of this functional on the space of immersed θ-graphs such that the value of the functional does not exceed the diameter of Mn. If n=2, then this critical θ-graph is not only immersed but embedded.

Keywords

closed geodesics geodesic nets Riemannian manifold surfaces 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Croke, C.B. 1988Area and the length of the shortest closed geodesicJ. Differential Geom.27121Google Scholar
  2. Hass, J., Morgan, F. 1996Geodesic nets on the 2-sphereProc. Amer. Math. Soc.12438433850CrossRefGoogle Scholar
  3. Maeda, M. 1994The length of a closed geodesic on a compact surfaceKyushu J. Math.48918Google Scholar
  4. Nabutovsky, A., Rotman, R. 2002The length of the shortest closed geodesic ion a 2-dimensional sphereInternat. Math. Res. Notices2312111222CrossRefGoogle Scholar
  5. Nabutovsky, A., Rotman, R. 2004Volume, diameter and the minimal mass of a stationary 1-cycleGeom. Funct. Anal.14748790CrossRefGoogle Scholar
  6. Rotman, R. The length of a shortest closed geodesic on a 2-dimensional sphere and coverings by metric balls, Geom. Dedicata (to appear)Google Scholar
  7. Rotman, R. The length of a shortest closed geodesic and the area of a 2-dimensional sphere, to appear in Proc. Amer. Math. SocGoogle Scholar
  8. Sabourau, S. 2004Filling radius and short closed geodesic on the 2-sphereBull. Soc. Math. France132105136Google Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of MathematicsThe Pennsylvania State UniversityUniversity ParkU.S.A.

Personalised recommendations