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

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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,*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)**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
53C21,
53C40

Retrieve articles in all journals with MSC: 53C21, 53C40

Additional Information

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

Article copyright:
© Copyright 1980
American Mathematical Society