Vanishing of homology groups, Ricci estimate for submanifolds and applications
Let \(x: M^n\to Q_c^{n+m}\) be an isometric immersion of a Riemannian manifold \(M^n\) (superscripts denote dimensions) into a complete simply-connected space \(Q_c^{n+m}\) of constant sectional curvature \(c\). The authors obtain, among other results, an estimate for the Ricci curvature of \(M^n\) and, when \(M^n\) is compact, a criterion for the vanishing of its homology groups. Here is an interesting corollary of the Ricci curvature estimate. Let \(M^n\), \(n\geq 3\), be a compact, minimal submanifold of the unit sphere \(S^{n+m}\). Assume that the norm squared \(|A|^2\) of the second fundamental form \(A\) satisfies \(|A|^2\leq n\) and that the fundamental group of \(M^n\) is infinite. Then \(M^n\) is a Clifford torus \(S^1\times S^{n-1}\) in a totally geodesic submanifold \(S^{n+1}\subset S^{n+m}\).
scientific article; zbMATH DE number 1742079