The differentiable sphere theorem for manifolds with positive Ricci curvature (Q2880662)

The authors prove the following theorems:NEWLINENEWLINETheorem 1.1. Let \(M^n\) be an \(n\)-dimensional, \(n\geq 3\), complete submanifold in an \(n+p\)-dimensional Riemannian manifold \(N^{n+p}\) with codimension \(p(\geq 0)\). If NEWLINE\[NEWLINE \sup_{M}\left[S-\frac{n^2H^2}{n-1}-\frac 53\left(\overline{Ric}_{\min}^{(k)}-\left(k-\frac 65\right)\overline{K}_{\max}\right)\right]<0, NEWLINE\]NEWLINE for some integer \(k\in[2,n+p-1]\), where \(H\) and \(S\) are the mean curvature and the squared length of the second fundamental form of \(M\) respectively, \(\overline{Ric}_{\min}^{(k)}\) is the minimum of the \(k\)-th Ricci curvature of \(N\), \(\overline{K}_{\max}\) the maximum of the sectional curvatures of \(N\), then \(M\) is differentiable to a spherical space form. In particular, if \(M\) is simply connected, then \(M\) is diffeomorphic to \(S^n\).NEWLINENEWLINETheorem 1.2. Let \(M^n\) be an \(n\)-dimensional, \(n\geq 3\), complete submanifold in an \(n+p\)-dimensional Riemannian manifold \(N^{n+p}\) with codimension \(p(\geq 0)\). If NEWLINE\[NEWLINE \sup_{M}\left[S-\frac{5\sqrt{2}}{3} \left(\overline{Ric}_{\min}^{(k)}-\left(k-\frac 65\right)\overline{K}_{\max}\right)\right]<0, NEWLINE\]NEWLINE for some integer \(k\in[2,n+p-1]\), then \(M\) is diffeomorphic to a spherical space form. In particular, if \(M\) is simply connected, then \(M\) is diffeomorphic to \(S^n\).NEWLINENEWLINETheorem 1.3. Let \(M^n\) be an \(n\)-dimensional, \(n\geq 3\), compact Riemannian manifold. If \(Ric_{\min}>(n-1)\tau_nK_{\max}\), where \(\displaystyle \tau_n=1-6/5(n-1)\), then \(M\) is diffeomorphic to a spherical space form. In particular, if \(M\) is simply connected, then \(M\) is diffeomorphic to \(S^n\).NEWLINENEWLINEThe authors propose two conjectures to sharpen the conditions of Theorem 1.3. A similar topological sphere theorem for manifolds of positive Ricci curvature was given by \textit{Z. M. Shen}, Indiana Univ. Math. J. 38, No. 1, 229--233 (1989; Zbl 0678.53032)].