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Proceedings of the American Mathematical Society

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



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
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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Article copyright: © Copyright 1980 American Mathematical Society

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