The length of a curve in a space of curvature

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 . Suppose its radius is less than the injectivity radius at the center of *M* and if . Denote by a circle of radius in the plane of constant curvature *K* and by the curvature of . Then any curve in *M* with curvature is not longer than a circular arc of curvature in whose ends are opposite points of . Any curve in *M* with total curvature not exceeding some ( if ) is not longer than the longest curve in with the same total curvature whose tangent vector rotates in a permanent direction.

**[1]**A. D. Aleksandrov and V. V. Strel′cov,*The isoperimetric problem and extimates of the length of a curve on a surface*, Trudy Mat. Inst. Steklov**76**(1965), 67–80 (Russian). MR**0202097****[2]**B. V. Dekster,*Estimates of the length of a curve*, J. Differential Geometry**12**(1977), no. 1, 101–117. MR**0470906****[3]**B. V. Dekster,*Upper estimates of the length of a curve in a Riemannian manifold with boundary*, J. Differential Geom.**14**(1979), no. 2, 149–166. MR**587544****[4]**Detlef Gromoll and Wolfgang Meyer,*On complete open manifolds of positive curvature*, Ann. of Math. (2)**90**(1969), 75–90. MR**0247590**, https://doi.org/10.2307/1970682**[5]**Ju. G. Rešetnjak,*Bound for the length of a rectifiable curve in 𝑛-dimensional space.*, Sibirsk. Mat. Ž.**2**(1961), 261–265 (Russian). MR**0125939**

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DOI:
https://doi.org/10.1090/S0002-9939-1980-0565353-X

Article copyright:
© Copyright 1980
American Mathematical Society