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

DOI:
https://doi.org/10.1090/S0002-9939-98-04423-2

MathSciNet review:
1459103

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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.

- 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 [A. D. Alexandrov,*Über eine Verallgemeinerung der Riemannschen Geometrie*, Schr. Forschungsinst. Math.**1**(1957), 33-84]), Trudy Mat. Inst. Steklov**38**(1951), 5-23 (Russian). - A. D. Alexandrov,
*Über eine Verallgemeinerung der Riemannschen Geometrie*, Schr. Forschungsinst. Math.**1**(1957), 33-84. - A. D. Alexandrov and Yu. G. Reshetnyak,
*General theory of irregular curves*, Mathematics and its Applications (Soviet Series), vol. 29, Kluwer Academic Publishers Group, Dordrecht, 1989. Translated from the Russian by L. Ya. Yuzina. MR**1117220** - Werner Ballmann,
*Lectures on spaces of nonpositive curvature*, DMV Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995. With an appendix by Misha Brin. MR**1377265** - 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. - F. Brickell and C. C. Hsiung,
*The total absolute curvature of closed curves in Riemannian manifolds*, J. Differential Geometry**9**(1974), 177–193. MR**339032** - M. Bridson and A. Haefliger,
*Metric Spaces of Non-positive Curvature*, book to appear. - I. Fàry,
*Sur la courbure totale d’une courbe gauche faisant un næud*, Bulletin de la Soc. Math. de France**77**(1949), 128-138. - J. W. Milnor,
*On the Total Curvature of Knots*, Ann.of Math.**52**(2) (1950), 248-257. - E. Pannwitz,
*Eine elementargeometrische Eigenschaft von Verschlingungen und Knoten*, Math. Annalen.**108**(1933), 629-672. - C. Schmitz,
*The theorem of Fary and Milnor for Hadamard manifolds*, Geom. Dedicata, to appear. - J. Szenthe,
*On the total curvature of closed curves in Riemannian manifolds*, Publ. Math. Debrecen**15**(1968), 99–105. MR**239545** - Yôtarô Tsukamoto,
*On the total absolute curvature of closed curves in manifolds of negative curvature*, Math. Ann.**210**(1974), 313–319. MR**365418**, DOI https://doi.org/10.1007/BF01434285

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Additional Information

**Stephanie B. Alexander**

Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Email:
sba@math.uiuc.edu

**Richard L. Bishop**

Affiliation:
Department of Mathematics, University of Illinois, Urbana, Illinois 61801

Email:
bishop@math.uiuc.edu

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