The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group
 Alexander Nabutovsky,
 Regina Rotman
 … show all 2 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
Let M ^{ n } be a closed Riemannian manifold with a nontrivial second homology group. In this paper we prove that there exists a geodesic net on M ^{ n } of length at most 3 diameter(M ^{ n }). 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(M^{ n }), or there exists a critical point of this functional on the space of immersed θgraphs such that the value of the functional does not exceed the diameter of M ^{ n }. If n=2, then this critical θgraph is not only immersed but embedded.
 Croke, C.B. (1988) Area and the length of the shortest closed geodesic. J. Differential Geom. 27: pp. 121
 Hass, J., Morgan, F. (1996) Geodesic nets on the 2sphere. Proc. Amer. Math. Soc. 124: pp. 38433850 CrossRef
 Maeda, M. (1994) The length of a closed geodesic on a compact surface. Kyushu J. Math. 48: pp. 918
 Nabutovsky, A., Rotman, R. (2002) The length of the shortest closed geodesic ion a 2dimensional sphere. Internat. Math. Res. Notices 23: pp. 12111222 CrossRef
 Nabutovsky, A., Rotman, R. (2004) Volume, diameter and the minimal mass of a stationary 1cycle. Geom. Funct. Anal. 14: pp. 748790 CrossRef
 Rotman, R. The length of a shortest closed geodesic on a 2dimensional sphere and coverings by metric balls, Geom. Dedicata (to appear)
 Rotman, R. The length of a shortest closed geodesic and the area of a 2dimensional sphere, to appear in Proc. Amer. Math. Soc
 Sabourau, S. (2004) Filling radius and short closed geodesic on the 2sphere. Bull. Soc. Math. France 132: pp. 105136
 Title
 The Minimal Length of a Closed Geodesic Net on a Riemannian Manifold with a Nontrivial Second Homology Group
 Journal

Geometriae Dedicata
Volume 113, Issue 1 , pp 243254
 Cover Date
 20050601
 DOI
 10.1007/s1071100551771
 Print ISSN
 00465755
 Online ISSN
 15729168
 Publisher
 Kluwer Academic Publishers
 Additional Links
 Topics
 Keywords

 closed geodesics
 geodesic nets
 Riemannian manifold
 surfaces
 Industry Sectors
 Authors

 Alexander Nabutovsky ^{(1)}
 Regina Rotman ^{(1)}
 Author Affiliations

 1. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, U.S.A.