Complete submanifolds in spheres (Q2752895)

The paper consists of 3 sections, two of them (Sec. 1 and 3) looking like surveys of earlier results concerning, respectively, submanifolds of unit spheres with parallel mean curvature vector and submanifolds of Euclidean spaces with constant scalar curvature. Section 2 contains proofs of some results concerning submanifolds of spheres with constant scalar curvature. For example, Thm. 2.3 says that if \(M\) (\(\dim M =n\)) is a complete hypersurface of the unit sphere \(S^{n+1}\), its scalar curvature is a constant equal to \(n(n-1)r\) and if \(M\) has two distinct principal curvatures, then \(r > 1 - 2/n\); moreover if \(r\neq (n-2)/(n-1)\) and the squared norm \(S\) of the second fundamental form of \(M\) is large enough NEWLINE\[NEWLINES\geq \frac{(n-1)(n(r-1)+2)}{(n-2)} + \frac{n-2} {n(r-1)+2},NEWLINE\]NEWLINE then \(M\) isometric to the Riemannian product of a circle and a sphere of dimension \(n-1\). Other results of Section 2 are similar to this one.NEWLINENEWLINEFor the entire collection see [Zbl 0960.00049].