A note on the first Betti number of submanifolds with nonnegative Ricci curvature in codimension two (Q1191490)

Let \(f: M^ n\to \mathbb{R}^{n+2}\), \(n\geq 3\), be an isometric immersion of a complete Riemannian manifold \(M^ n\) with nonnegative Ricci curvature. The author proves the following results: 1. If \(M^ n\) is compact, then either \(M^ n\) has nonnegative sectional curvature or the fundamental group \(\pi_ 1(M^ n)\) is finite. 2. If \(M^ n\) is non-compact with sectional curvature bounded above, then the first Betti number \(b_ 1(M^ n)\leq n-3\).