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 γ which does not intersect B, then the line splits in a maximal neighborhood that is contained in M \ 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 γ which is a line in U, then U splits along γ.
Original language | English (US) |
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Pages (from-to) | 1231-1239 |
Number of pages | 9 |
Journal | Proceedings of the American Mathematical Society |
Volume | 120 |
Issue number | 4 |
DOIs | |
State | Published - Apr 1994 |
Keywords
- Asymptotic ray
- Line
- Ricci curvature
ASJC Scopus subject areas
- Mathematics(all)
- Applied Mathematics