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Proceedings of the American Mathematical Society

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The length of a curve in a space of curvature $ \leq K$


Author: B. V. Dekster
Journal: Proc. Amer. Math. Soc. 79 (1980), 271-278
MSC: Primary 53C21; Secondary 53C40
DOI: https://doi.org/10.1090/S0002-9939-1980-0565353-X
MathSciNet review: 565353
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Abstract: Let M be a compact ball in a Riemannian manifold with sectional curvatures $ \leqslant K$. Suppose its radius $ {R_0}$ is less than the injectivity radius at the center of M and $ {R_0} < \pi /2\sqrt K $ if $ K > 0$. Denote by $ {M_0}$ a circle of radius $ {R_0}$ in the plane of constant curvature K and by $ \kappa $ the curvature of $ \partial {M_0}$. Then any curve in M with curvature $ \leqslant \chi < \kappa $ is not longer than a circular arc of curvature $ \chi $ in $ {M_0}$ whose ends are opposite points of $ \partial {M_0}$. Any curve in M with total curvature not exceeding some $ \tau > 0$ ( $ \tau = \pi /2$ if $ K \leqslant {\kappa ^2}$) is not longer than the longest curve in $ {M_0}$ with the same total curvature whose tangent vector rotates in a permanent direction.


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  • [1] A. D. Aleksandrov and V. V. Strel'cov, Isoperimetric problem and estimates of the length of a curve on a surface, Proc. Steklov Inst. Math. 76 (1965), 81-99. MR 0202097 (34:1971)
  • [2] B. V. Dekster, Estimates of the length of a curve, J. Differential Geometry 12 (1977), 101-117. MR 0470906 (57:10650)
  • [3] -, Upper estimates of the length of a curve in a Riemannian manifold with boundary, J. Differential Geometry 14 (1979). MR 587544 (82b:53051)
  • [4] D. Gromoll and W. Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75-90. MR 0247590 (40:854)
  • [5] Ju. G. Rešetnyak, Bound for the length of a rectifiable curve in n-dimensional space, Sibirsk. Mat. Ž. 2 (2) (1961), 261-265. (Russian) MR 0125939 (23:A3236)

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DOI: https://doi.org/10.1090/S0002-9939-1980-0565353-X
Article copyright: © Copyright 1980 American Mathematical Society

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