A Characterization of Quadric Constant Scalar Curvature Hypersurfaces of Spheres

Perdómo, Oscar; Wei, Guoxin

doi:10.1007/s12220-013-9399-4

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“…Also in [1] an example of a complete non-isoparametric hypersurface that satisfies f v = λl v was presented. After this result, it looked like to fulfill only one of the conditions was easy but only the isoparametric hypersurfaces of order 1 and 2 were the only ones that satisfy an algebraic equation for the principal curvatures and the additional condition f v = λl v for some nonzero vector v. In [10], the authors were able to show that, as expected, the conditions constant scalar curvature plus f v = λl v for some nonzero vector on a complete oriented hypersurface of S 4 imply that the hypersurface must be isoparametric of order 1 or 2. The authors proved that the completeness condition was needed this time by showing a non isoparametric hypersurface in S 4 with constant scalar curvature that satisfies the condition f v = λl v for some nonzero vector v. Recall that these non complete examples do not exist if we replace the condition, constant scalar curvature by the condition constant mean curvature.…”

Section: Introductionmentioning

confidence: 99%

A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres

Perdómo

Wei

2015

Asian Journal of Mathematics

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Let M ⊂ S n+1 be a complete orientable hypersurface with constant Gauss-Kronecker curvature G. For any v ∈ R n+2 , let us define the following two real functions lv, fv : M → R on M by lv(x) = x, v and fv(x) = ν(x), v with ν : M → S n+1 a Gauss map of M. In this paper, we show that if n = 3, lv = λfv for some nonzero vector v ∈ R 5 and some real number λ, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. We will also show with an example that the completeness condition is needed in the result we just mentioned. We also show that if n = 4, lv = λfv for some nonzero vector v ∈ R 6 and some real number λ and (λ 2 − 1) 2 + (G − 1) 2 = 0, then M is either totally umbilical (a Euclidean sphere) or M is a cartesian product of Euclidean spheres. Moreover, we will give an example of a complete hypersurface in S 5 with constant Gauss-Kronecker curvature that satisfies the condition lv = λfv for some non zero v, which is neither a totally umbilical hypersurface nor a cartesian product of Euclidean spheres.

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Section: Introductionmentioning

confidence: 99%