Rigidity and sphere theorems for submanifolds. II. (English) Zbl 1005.53028

Shiohama, Katsuhiro; Xu, Hong Wei

Kyushu J. Math. 54, No. 1, 103-109 (2000).

The authors correct and develop results of an article in the same journal 48, No. 2, 291-306 (1994; Zbl 0826.53045). 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 \(D_{n}\) depending only on \(n\) such that \[ I(M, \varphi) \geq D_{n} \sum^{n - 1}_{i = 1} \beta_i,\quad n\leq 3,\quad I(M, \varphi) \geq D_{n} \left(\beta_1+3\sum^{n - 2}_{i = 2}\beta_i+\beta_{n-1}\right),\quad n\geq 4, \] where \(I(M, \varphi) = \int_M(S - n H^2)^{n/2}\), \(\;S\) is the squared norm of the second fundamental form, \(H\) is the mean curvature, and \(\beta_i\) is the \(i\)-th Betti number of \(M\). In particular, \(M\) is homeomorphic to \(S^n\) if \(I_0(M)<2D_n\). Moreover, some rigidity results in the above cited paper are improved. For instance, if an isometric immersion \(\varphi : M \to F^{n+p}(c)\) admits a parallel mean curvature normal field, and if there are numbers \(A_n\), \(B_{m}\) both depending only on \(n\), such that \[ I(M, \varphi) < A_n + B_{n} \sum^{n - 1}_{i = 1} \beta_i, \] then \(S=nH^2\) is constant, \(\varphi\) is totally umbilical and \(M\) is isometric to the standard \(n\)-sphere of constant sectional curvature \(H^2+c\).
The topological finiteness theorem is obtained by relaxing the assumption in the main theorem above: For a given integer \(n\geq 2\) and a constant \(A>0\) the class \(N(n,A):=\{M:\;I_0(M)<A\}\) of all compact Riemannian \(n\)-manifolds has at most finitely many homeomorphism types.

Reviewer: Vladimir Yu.Rovenskij (Haifa)

MSC:

53C20	Global Riemannian geometry, including pinching
53C24	Rigidity results

Keywords:

scalar curvature; space form; mean curvature; isometric immersion; Betti number; sphere theorem

Citations:

Zbl 0826.53045