Centro-affine hypersurface immersions with parallel cubic form
Centro-affine hypersurface immersions with parallel cubic form
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DOI：
10.1007/s13366-014-0216-4
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发表时间：
2012-08
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通讯作者：
R. Hildebrand
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作者：
R. Hildebrand
我们考虑非简并中心仿射超曲面浸没，其立方形式与仿射度量的 Levi-Civita 连接平行。中心在原点且具有平行立方形式的真仿射超球面的象射族与科赫圆锥域之间存在双射对应关系，科赫圆锥域是实半简单约当代数中由可逆元素组成的最大连通集。域内函数的每一级曲面都是一个仿射完备、欧几里得完备真仿射超球面，具有平行立方形式，中心位于原点。另一方面，每个具有平行立方形式且中心位于原点的真仿射超球面都可以表示为这样的水平面。基于半简单实乔丹代数的分类，我们提供了具有平行立方形式的真仿射超球面的完整分类。具有平行立方形式的中心仿射超曲面浸没与更广泛的实单位乔丹代数相关。每一个这样的浸入都可以扩展到仿射完全浸入，其圆锥壳是实单位乔丹代数中可逆元素集合中单位元素的连通分量。我们的方法还可以用于研究具有平行立方形式的其他类别的超曲面。
We consider non-degenerate centro-affine hypersurface immersions inwhose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, and Köchers conic-domains, which are the maximal connected sets consisting of invertible elements in a real semi-simple Jordan algebra. Every level surface of thefunction in an-domain is an affine complete, Euclidean complete proper affine hypersphere with parallel cubic form and with center in the origin. On the other hand, every proper affine hypersphere with parallel cubic form and with center in the origin can be represented as such a level surface. We provide a complete classification of proper affine hyperspheres with parallel cubic form based on the classification of semi-simple real Jordan algebras. Centro-affine hypersurface immersions with parallel cubic form are related to the wider class of real unital Jordan algebras. Every such immersion can be extended to an affine complete one, whose conic hull is the connected component of the unit element in the set of invertible elements in a real unital Jordan algebra. Our approach can be used to study also other classes of hypersurfaces with parallel cubic form.