Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection
Centroaffine surfaces with parallel or recurrent cubic form relative to the induced connection
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DOI：
10.1007/s13366-016-0305-7
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发表时间：
2017-06
期刊：
影响因子：
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通讯作者：
A. Fujioka;Kunio Hamamoto;Y. Nakai
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文献类型：
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作者：
A. Fujioka;Kunio Hamamoto;Y. Nakai
仿射空间中超曲面的立方形式是仿射微分几何中的一个重要不变量。众所周知，仿射空间中许多有趣的超曲面类别都具有立方形式的特征。例如，在等仿射微分几何中，二次曲面被描述为具有消失立方形式的非简并超曲面，这被称为 Pick-Berwald 定理。在本文中，我们将根据立方形式和诱导连接给出非简并中心仿射曲面的一些分类结果。首先，我们证明立方形式相对于诱导连接平行的非简并中心仿射曲面是一块以原点为中心的椭球体或双曲面。接下来，我们证明一个非简并中心仿射曲面，其立方形式的无痕部分相对于诱导连接是循环的，是某个中心仿射最小直纹曲面的一部分，使得 Pick 函数消失并且中心仿射度量的曲率等于 1。
The cubic form of a hypersurface in the affine space is an important invariant in affine differential geometry. It is known that many interesting classes of hypersurfaces in the affine space are characterized by the cubic form. For example, in equiaffine differential geometry, quadrics are characterized as nondegenerate hypersurfaces with vanishing cubic form, which is known as theorem of Pick–Berwald. In this paper, we shall give some classification results for nondegenerate centroaffine surfaces in terms of the cubic form and the induced connection. First, we prove that a nondegenerate centroaffine surface whose cubic form is parallel relative to the induced connection is a piece of an ellipsoid or a hyperboloid centered at the origin. Next, we prove that a nondegenerate centroaffine surface whose traceless part of the cubic form is recurrent relative to the induced connection is a piece of a certain centroaffine minimal ruled surface such that the Pick function vanishes and the curvature of the centroaffine metric equals to 1.