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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.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
##### Citations:
Zbl 0185.25103; Zbl 0216.44001
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