Rigidity and sphere theorems for submanifolds. (English) Zbl 0826.53045

Shiohama, Katsuhiro; Xu, Hongwei

Kyushu J. Math. 48, No. 2, 291-306 (1994).

The main result of the paper is the following: Let \(F^{n+p} (c)\) be an \((n + p)\)-dimensional space form of non-negative curvature \(c\). If \(\varphi : M \to F^{n + p} (c)\) is an isometric immersion of a compact connected Riemannian \(n\)-dimensional manifold, then there is a number \(D_{n,p}\) depending on \(n\) and \(p\) such that \[ \int (S - n H^2)^{n/2} \geq D_{n,p} \left( \sum^{n - 1}_{i = 1} \beta_i\right), \] where \(S\) is the squared norm of the second fundamental form, \(H\) is the mean curvature and \(\beta_i\) is the \(i\)-th Betti number of \(M\).
Moreover, some rigidity results are proved. For instance, if an isometric immersion \(\varphi : M \to \mathbb{R}^{n +p} (c)\) has parallel mean curvature vector field, and there are numbers \(A_n\), \(B_{n,p}\) such that if \[ \int (S - nH^2)^{n/2} < A_n + B_{n,p} \left (\sum^{n - 1}_{i = 1} \beta_i \right), \] then \(\varphi\) is totally umbilic and \(M\) is isometric to a standard \(n\)-sphere.

Reviewer: B.Opozda (Kraków)

MSC:

53C40	Global submanifolds
53C20	Global Riemannian geometry, including pinching
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

scalar curvature; space form; mean curvature; isometric immersion; parallel mean curvature vector