Global behavior of curves in a space of positive curvature
Author:
B. V. Dekster
Journal:
Proc. Amer. Math. Soc. 85 (1982), 419-426
MSC:
Primary 53C40; Secondary 53C20
DOI:
https://doi.org/10.1090/S0002-9939-1982-0656116-7
MathSciNet review:
656116
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is well known that any geodesic in a complete noncompact space of positive curvature goes to infinity. In this paper, we prove that this is true for more general curves and estimate how fast they go to infinity in terms of their curvature and curvature of the space.
- [1] Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, https://doi.org/10.2307/1970819
- [2] B. V. Dekster, Estimates of the length of a curve, J. Differential Geometry 12 (1977), no. 1, 101–117. MR 470906
- [3] B. V. Dekster, Upper estimates of the length of a curve in a Riemannian manifold with boundary, J. Differential Geometry 14 (1979), no. 2, 149–166. MR 587544
- [4] B. V. Dekster and I. Kupka, Asymptotics of curvature in a space of positive curvature, J. Differential Geometry 15 (1980), no. 4, 553–568 (1981). MR 628344
- [5] Detlef Gromoll and Wolfgang Meyer, On complete open manifolds of positive curvature, Ann. of Math. (2) 90 (1969), 75–90. MR 247590, https://doi.org/10.2307/1970682
- [6] 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
-dimensional space, Sibirsk. Mat. Ž. 2 (1961), 261-265. (Russian) Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C40, 53C20
Retrieve articles in all journals with MSC: 53C40, 53C20
Additional Information
DOI:
https://doi.org/10.1090/S0002-9939-1982-0656116-7
Article copyright:
© Copyright 1982
American Mathematical Society
