Spherical rigidities of submanifolds in Euclidean spaces (Q1884807)

The \(n\)-dimensional complete immersed submanifolds \(M\) in an Euclidean space \(E^{n+p}\) are studied. The first main theorem asserts that such an \(M\) with everywhere nonzero mean curvature \(H\) is diffeomorphic to a sphere \(S^n(c)\) if one of the following conditions is satisfied: \[ S\leq {n^2 H^2\over n-1},\quad n^2 H^2\leq{(n- 1)R\over n-2}, \] where \(S\) and \(R\) denote the squared norm of the second fundamental form of \(M\) and the scalar curvature of \(M\), respectively. The second main theorem asserts that if for such an \(M\) with \(n>2\) and constant mean curvature \(H\) \(S\leq n^2 H^2/(n- 1)\) is satisfied, then \(M\) is isometric to the totally umbilical sphere \(S^n(c)\), the totally geodesic Euclidean space \(E^n\), or the generalized cylinder \(S^{n-1}(c)\times E^1\). In preliminaries the fundamental formulae for a connected submanifold \(M\) in \(E^{n+p}\) are given by means of an orthonormal frame adapted to \(M\) and dual coframe bundles, and exterior calculus. Also, the generalized maximum principle (by Omori and Yau) is formulated for further use. Then a codimension reduction theorem is proven as a preparation.