Hypersurfaces in ${\mathbb{S}}^4$ that are of $L_k$-2-type
Bull. Korean Math. Soc. 2016 Vol. 53, No. 3, 885-902
Published online May 31, 2016
Pascual Lucas and H\'ector-Fabi\'an Ram\'\i rez-Ospina
Campus de Espinardo, Universidad Nacional de Colombia
Abstract : In this paper we begin the study of $L_k$-2-type hypersurfaces of a hypersphere $\S^{n+1}\subset\R^{n+2}$ for $k\geq 1$. Let $\psi:\M\rightarrow\S^{4}$ be an orientable $H_k$-hypersurface, which is not an open portion of a hypersphere. Then $\M$ is of $L_k$-2-type if and only if $\M$ is a Clifford tori $\S^1(r_1)\times\S^2(r_2)$, $r_1^2+r_2^2=1$, for appropriate radii, or a tube $T^r(V^2)$ of appropriate constant radius $r$ around the Veronese embedding of the real projective plane $\R P^2(\sqrt3)$.
Keywords : linearized operator $L_k$, $L_k$-finite-type hypersurface, higher order mean curvatures, Newton transformations
MSC numbers : 53C40, 53B25
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