Complete orientable Riemannian manifolds with nonnegative curvature (Q2767420)

The aim of this paper is to discuss some properties of open Riemannian manifolds with non-negative curvature, the main results are the following:NEWLINENEWLINENEWLINE1. Suppose that \(M\) is a complete non-compact oriented Riemannian manifold of dimension \(n\), and denote by \(S\) its soul. If \(0<\text{codim} S= 2k+1<n\), then there are infinitely many closed geodesics on \(M\).NEWLINENEWLINENEWLINE2. Suppose \(M\) is given as above, and each geodesic normal to the soul \(S\) is a ray, then \(M=N\times R\), where \(N\) is a Riemannian manifold of dimension \(n-1\).NEWLINENEWLINENEWLINE3. Suppose that \(M\) is a complete non-compact oriented Riemannian manifold of dimension \(n\), and \(\text{codim} S=2\). If each geodesic normal to the soul \(S\) is a ray, then \(M=S\times R^2\).NEWLINENEWLINENEWLINEThe argumentation for these results is direct.