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 $M$ 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 $\gamma$.