Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



The Fary-Milnor theorem in Hadamard manifolds

Authors: Stephanie B. Alexander and Richard L. Bishop
Journal: Proc. Amer. Math. Soc. 126 (1998), 3427-3436
MSC (1991): Primary 57M25; Secondary 53C20
MathSciNet review: 1459103
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Fary-Milnor theorem is generalized: Let $\gamma $ be a simple closed curve in a complete simply connected Riemannian 3-manifold of nonpositive sectional curvature. If $\gamma $ has total curvature less than or equal to $4\pi $, then $\gamma $ is the boundary of an embedded disk. The example of a trefoil knot which moves back and forth abritrarily close to a geodesic segment shows that the bound $4\pi $ is sharp in any such space. The original theorem was for closed curves in Euclidean 3-space and the proof by integral geometry did not apply to spaces of variable curvature. Now, instead, a combinatorial proof has been devised.

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

  • [A1] A. D. Alexandrov, A theorem on triangles in a metric space and some of its applications, (This is translated into German and combined with more material in [A2]), Trudy Mat. Inst. Steklov 38 (1951), 5-23 (Russian). MR 14:198a
  • [A2] A. D. Alexandrov, Über eine Verallgemeinerung der Riemannschen Geometrie, Schr. Forschungsinst. Math. 1 (1957), 33-84. MR 19:304h
  • [AR] A. D. Alexandrov, Yu. G. Reshetnyak, General Theory of Irregular Curves, Kluwer Academic Publishers, Dordrecht, Boston, London, 1989. MR 92h:53003
  • [Ba] W. Ballmann, Lectures on Spaces of Nonpositive Curvature, DMV Seminar Band 25, Birkhäuser, Basel, 1995. MR 97a:53053
  • [BN] V. N. Berestovskii and I. G. Nikolaev, Multidimensional generalized Riemannian spaces, Geometry IV. Non-regular Riemannian Geometry. Encyclopaedia of Mathematical Sciences, Springer-Verlag, Berlin Heidelberg, 1993, pp. 165-244. CMP 94:08
  • [BHs] F. Brickell and C.C. Hsiung, The absolute total curvature of closed curves in Riemannian manifolds, J. Diff. Geom. 9 (1974), 177-193. MR 49:3795
  • [BHa] M. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, book to appear.
  • [F] I. Fàry, Sur la courbure totale d'une courbe gauche faisant un næud, Bulletin de la Soc. Math. de France 77 (1949), 128-138. MR 11:393h
  • [M] J. W. Milnor, On the Total Curvature of Knots, Ann.of Math. 52 (2) (1950), 248-257. MR 12:273c
  • [P] E. Pannwitz, Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten, Math. Annalen. 108 (1933), 629-672.
  • [S] C. Schmitz, The theorem of Fary and Milnor for Hadamard manifolds, Geom. Dedicata, to appear.
  • [Sz] J. Szenthe, On the total curvature of closed curves in Riemannian manifolds, Publ. Math. Debrechen 15 (1968), 99-105. MR 39:902
  • [T] Y. Tsukamoto, On the total absolute curvature of closed curves in manifolds of negative curvature, Math. Ann. 210 (1974), 313-319. MR 51:1670

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57M25, 53C20

Retrieve articles in all journals with MSC (1991): 57M25, 53C20

Additional Information

Stephanie B. Alexander
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Richard L. Bishop
Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Keywords: Knots, total curvature, CAT(0) spaces, Hadamard manifolds
Received by editor(s): October 2, 1996
Received by editor(s) in revised form: March 28, 1997
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society