A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres
A characterization of quadric constant Gauss-Kronecker curvature hypersurfaces of spheres
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DOI：
10.4310/ajm.2015.v19.n2.a3
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发表时间：
2015
影响因子：
0.6
通讯作者：
O. Perdomo;G. Wei
中科院分区：
文献类型：
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作者：
O. Perdomo;G. Wei
设M是一个具有常Gauss-Kronecker曲率G的完备可定向超曲面。对任意v ∈ R，定义如下两个真实的函数lv，fv：M → R，其中lv（x）= λ x，v ∈ R，fv（x）= λ v（x），v ∈ R，其中v：M → S是M的高斯映射.本文证明了：若n = 3，对于某个非零向量v ∈ R和某个真实的实数λ，lv = λfv，则M要么是全脐的（一个欧氏球面），要么是欧氏球面的一个乘积.我们还将用一个例子来说明在我们刚才提到的结果中需要完备性条件。我们还证明了，如果n = 4，lv = λfv，对某个非零向量v ∈ R和某个真实的数λ，且（λ − 1）+（G − 1）= 0，则M要么是全脐的（一个欧氏球面），要么是欧氏球面的一个卡累利加积.此外，我们将给出一个例子，S中具有常数Gauss-Kronecker曲率的完备超曲面满足条件lv = λfv，对于某个非零v，它既不是全脐超曲面，也不是欧氏球面的Carnival积。
Let M ⊂ S be a complete orientable hypersurface with constant Gauss-Kronecker curvature G. For any v ∈ R, 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 a Gauss map of M . In this paper, we show that if n = 3, lv = λfv for some nonzero vector v ∈ R 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 and some real number λ and (λ − 1) + (G − 1) = 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 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.