Local splitting theorems for Riemannian manifolds

Abstract

In this paper we establish two local versions of the Cheeger-Gromoll Splitting Theorem. We show that if a complete Riemannian manifold M M has nonnegative Ricci curvature outside a compact set B B and contains a line γ \gamma which does not intersect B B , then the line splits in a maximal neighborhood that is contained in M ∖ B ¯ \overline {M\backslash B} . We use this result to give a simplified proof that M M has a bounded number of ends. We also prove that if M M has sectional curvature which is nonnegative (and bounded from above) in a tubular neighborhood U U of a geodesic γ \gamma which is a line in U U , then U U splits along γ \gamma .