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Local rigidity theorems for minimal hypersurfaces. (English) Zbl 0174.24901

One purpose of the paper is to determine all minimal hypersurfaces \(M\) of the unit sphere \(S^{n+1}\) satisfying certain additional conditions. Let \(\kappa=\frac 1{n(n-1)} \sum g^{ij}R_{ij}\) be the scalar curvature of \(M\). Let \(M_{k, n-k} = S^k(\sqrt{k/n})\times S^{n-k}(\sqrt{n-k/n})\), where \(S^p(r)= \{x\in\mathbb R^{p+1}\mid \sum x_i^2 =r^2\}\). The author proves
(I) If \(\kappa=(n-2)/(n-1)\), then \(M\) is locally \(M_{k, n-k}\).
(II) If the Ricci tensor of \(M\) is parallel, then \(M\) is locally \(M_{k, n-k}\).
(I) is proved independently by S. S. Chern, M. P. do Carmo and S. Kobayashi [Functional analysis and related fields, Conf. Chicago 1968, 59–75 (1970; Zbl 0216.44001)] and by K. Nomizu and B. Smyth [J. Differ. Geom. 3, 367–377 (1969; Zbl 0196.25103)]. In proving (I) and (II), the author gives more general proposition:
(III) If the second fundamental form is parallel, then \(M\) is locally \(M_{k, n-k}\). Another purpose is to generalize (III) to the case where the ambient manifold is a real space form and determine all hypersurfaces with parallel Ricci tensor and constant mean curvature.
Let \(D^p(r) = \{x\in\mathbb R^{p+1}\mid \sum x_i^2 - x_{p+1}^2 = -r^2, x_{p+1} >0\}\) and \(F^0 = \{x\in D^{n+1}(R)\mid x_{n+1} = x_n + R\}\).
(IV) Let \(M\) be a hypersurface of \(\tilde M\) with parallel second fundamental form. Then
(1) if \(\tilde M =S^{n+1}(R)\), then \(M\) is locally \(S^k(r) \times S^{n-k}(sqrt{R^2 - r^2)\),
(2) if \(\tilde M = \mathbb R^{n+1}\), then \(M\) is locally \(S^k(r)\times \mathbb R^{n-k}\) and
(3) if \(\tilde M = D^{n+1}(R)\), then \(M\) is locally \(S^k(r) \times D^{n-k}(\sqrt{R^2 + r^2)\) or \(F^n\).
Making use of (IV) the author proves finally the following result:
(V) Let \(M\) be a hypersurface of the real space form with parallel Ricci tensor and with constant mean curvature. Then the conclusion is the same as (IV).

MSC:

53C24 Rigidity results
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