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Hypersurfaces of constant curvature in space forms. (English) Zbl 0787.53046
Let $$M^ n$$ be a compact embedded hypersurface in the space form $$N^{m+1}(c)$$ ($$c=0$$ or $$c=-1$$) with boundary contained in a totally geodesic hyperplane, having one symmetric function $$S_{r+1}$$ of principal curvatures a positive constant. In this interesting paper the author obtains height estimates and derives some applications of them. As an example we note Ph. Hartman’s theorem [Trans. Am. Math. Soc. 245, 363-374 (1978; Zbl 0412.53027)]: Let $$M$$ be a complete embedded hypersurface of $$R^{m+1}$$ with $$S_{r+1}$$ a positive constant. If $$M$$ has nonnegative sectional curvature then $$M$$ is isometric to $$S^ p \times R^ \ell$$, $$S^ p$$ a round sphere. The proofs depend on obtaining some elliptic equations on $$M$$, by calculating the first variation of symmetric functions of principal curvatures.

##### MSC:
 53C40 Global submanifolds