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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Ball covering on manifolds with nonnegative Ricci curvature near infinity


Author: Zhong-dong Liu
Journal: Proc. Amer. Math. Soc. 115 (1992), 211-219
MSC: Primary 53C20; Secondary 53C21
MathSciNet review: 1068127
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Abstract: Let $ M$ be a complete open Riemannian manifold with nonnegative Ricci curvature outside a compact set $ B$. We show that the following ball covering property (see [LT]) is true provided that the sectional curvature has a lower bound:

For a fixed $ {p_0} \in M$, there exist $ N > 0$ and $ {r_0} > 0$ such that for $ r \geq {r_0}$, there exist $ {p_1},{\kern 1pt} \ldots ,{p_k} \in \partial {B_{p0}}(2r),k \leq N$ , with

$\displaystyle \bigcup\limits_{j = 1}^k {{B_{{p_j}}}(r) \supset \partial {B_{{p_0}}}(2r).} $

Furthermore $ N$ and $ {r_0}$ depend only on the dimension, the lower bound on the sectional curvature, and the radius of the ball at $ {p_0}$ that contains $ B$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1992-1068127-7
PII: S 0002-9939(1992)1068127-7
Keywords: Ricci curvature, end, volume comparison, geodesic ball
Article copyright: © Copyright 1992 American Mathematical Society