Affine hypersurfaces with parallel cubic form
Affine hypersurfaces with parallel cubic form
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DOI：
10.1017/s0027763000005006
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发表时间：
1994-09
影响因子：
0.8
通讯作者：
F. Dillen;L. Vrancken;Şahnur Yaprak
中科院分区：
文献类型：
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作者：
F. Dillen;L. Vrancken;Şahnur Yaprak
众所周知，非简并仿射超曲面M上存在一个正则横向向量场。这个向量场称为仿射法线。与该仿射法线相关的第二种基本形式称为仿射度量。如果 M 是局部强凸的，则该仿射度量是黎曼度量。而且，利用仿射法线和高斯公式，可以在 M 上引入仿射连接 ∇，称为诱导仿射连接。因此，M 上通常有两种不同的连接：一种是诱导连接 ∇，另一种是仿射度量 h 的 Levi Civita 连接。差分张量 K 的定义为 K(X, Y) = KXY — ∇XY — XY。立方形式 C 由 C = ∇h 定义，并与差张量相关。
As is well known, there exists a canonical transversal vector field on a non-degenerate affine hypersurface M. This vector field is called the affine normal. The second fundamental form associated to this affine normal is called the affine metric. If M is locally strongly convex, then this affine metric is a Riemannian metric. And also, using the affine normal and the Gauss formula one can introduce an affine connection ∇ on M which is called the induced affine connection. Thus there are in general two different connections on M: one is the induced connection ∇ and the other is the Levi Civita connection of the affine metric h. The difference tensor K is defined by K(X, Y) = KXY — ∇XY — XY. The cubic form C is defined by C = ∇h and is related to the difference tensor by .