On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature

Details On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature On Loops Representing Elements of the Fundamental Group of a Complete Manifold with Nonnegative Ricci Curvature

1999-04-19

arxiv.org/abs/math/9904096v1

Indiana Journal of Mathematics, 50 (2001) no. 4, 1867-1883

Mathematics

Differential Geometry

19 pages

Scientific paper

This paper concerns complete noncompact manifolds with nonnegative Ricci curvature. Roughly, we say that M has the loops to infinity property if given any noncontractible closed curve, C, and given any compact set, K, there exists a closed curve contained in M\K which is homotopic to C. The main theorems in this paper are the following. Theorem I: If M has positive Ricci curvature then it has the loops to infinity property. Theorem II: If M has nonnegative Ricci curvature then it either has the loops to infinity property or it is isometric to a flat normal bundle over a compact totally geodesic submanifold and its double cover is split. Theorem III: Let M be a complete riemannian manifold with the loops to infinity property along some ray starting at a point, p. Let D containing p be a precompact region with smooth boundary and S be any connected component of the boundary containing a point, q, on the ray. Then the map from the fundamental group of S based at q to the fundamental group of Cl(D) based at p induced by the inclusion map is onto.

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