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Some results of Chern-do Carmo-Kobayashi type and the length of second fundamental form. (English) Zbl 0219.53047
Let \(M\) be an \(n\)-dimensional manifold immersed in an \((n+p)\)-dimensional Riemannian manifold of constant sectional curvature. \(M\) is called pseudo-umbilical if the second fundamental form \(\text{II}_H\) of \(M\) in the direction of the mean curvature normal and the first fundamental form \(\text{I}\) of \(M\) are proportional. Following the lines of a paper of S.-S. Chern, M. P. do Carmo and S. Kobayashi [Functional analysis and related fields, Conf. Chicago 1968, 59–75 (1970; Zbl 0216.44001)] the author derives an integral formula for pseudo-umbilical submanifolds of \(N\) and proves the following result:
The Veronese surface in \(S^x\), the standard embeddings \(M_{m,n-m}\) of \(S^m((m/n)^{1/2} \times S^{n-m}(((n - m)/n)^{1/2})\) into \(S^{n+1}(1)\) and the “small” \(n\)-spheres in \(S^{n+1}\) are the only compact, not totally geodesic, pseudo-umbilical submanifolds of dimension \(n\) in \(S^{n+p}\) for which the integrand of the integral formula vanishes identically. Under the assumption that \(\lambda\) defined by \(\text{II}_H =\lambda \text{I}\) be constant on \(M\) the author gets a corresponding local rigidity theorem. S. S. Chernet al. (loc. cit.) proved analogous results in the case \(\lambda=0\). S. Braidi and C.-C. Hsiung independently derived a slightly more general integral formula [Math. Z. 115, 235–251 (1970; Zbl 0185.25103)] and obtained related results.

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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