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Proceedings of the American Mathematical Society

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

 

 

Local splitting theorems for Riemannian manifolds


Authors: M. Cai, G. J. Galloway and Z. Liu
Journal: Proc. Amer. Math. Soc. 120 (1994), 1231-1239
MSC: Primary 53C20
MathSciNet review: 1186984
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Abstract: In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold $ M$ has nonnegative Ricci curvature outside a compact set $ B$ and contains a line $ \gamma $ which does not intersect $ B$, then the line splits in a maximal neighborhood that is contained in $ \overline {M\backslash B} $. We use this result to give a simplified proof that $ M$ has a bounded number of ends. We also prove that if $ M$ has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood $ U$ of a geodesic $ \gamma $ which is a line in $ U$, then $ U$ splits along $ \gamma $.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1186984-2
Keywords: Ricci curvature, line, asymptotic ray
Article copyright: © Copyright 1994 American Mathematical Society