A differentiable sphere theorem inspired by rigidity of minimal submanifolds (Q418451)

Let \(F^{n+p}(c)\) be an \(n+p\) dimensional space form of constant sectional curvature \(c\geq0\). If \(c=0\), then \(F^{n+p}\) can be taken to be a Euclidean space; if \(c>0\), then \(F^{n+p}\) is a sphere of some radius. The authors present a vanishing theorem for the fundamental group of a submanifold of \(F^{n+p}\). Let \(H\) be the mean curvature of \(M\):NEWLINENEWLINETheorem. Suppose that \(M\) is an \(n\)-dimensional compact submanifold in an \(n+p\) dimensional space form \(F^{n+p}(c)\) with \(c\geq0\). If the Ricci curvature of \(M\) satisfies \(\mathrm{Ric}_M>{1\over2}(n-1)c+{1\over 8}n^2H^2\), then \(\pi_1(M)=0\).NEWLINENEWLINEThe authors also obtain a refined version of a rigidity theorem of \textit{N. Ejiri} [J. Math. Soc. Japan 31, 251--256 (1979; Zbl 0396.53026)].NEWLINENEWLINETheorem. Suppose that \(M\) is an \(n\)-dimensional compact submanifold in an \(n+p\) dimensional space form \(F^{n+p}(c)\) with \(c\geq0\) and \(n\geq3\). If the Ricci curvature of \(M\) satisfies \(\mathrm{Ric}_M>(n-2)c+{1\over8}n^2H^2\), then \(M\) is diffeomorphic to \(S^n\).NEWLINENEWLINE The authors show that their differentiable sphere theorem is sharp; they use a method of Ricci flow in the proof of these results.