Affine hypersurfaces with parallel cubic forms
Affine hypersurfaces with parallel cubic forms
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DOI：
10.2748/tmj/1178227697
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发表时间：
1990-03
影响因子：
0.5
通讯作者：
N. Bokan;K. Nomizu;U. Simon
中科院分区：
文献类型：
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作者：
N. Bokan;K. Nomizu;U. Simon
在本文中，我们首先研究张量 VC 和 FC 的对称性，其中 V 是诱导仿射连接，C 是 R + 中非简并仿射超曲面 M 的立方形式。特别是，我们研究平行立方形式的超曲面，即 PC=0。在 « = 2 的情况下，已知此条件可表征 Cayley 曲面（Nomizu 和 Pinkall [3]）。我们获得了一类更一般的仿射曲面和超曲面。另一方面，对于仿射超曲面 M", «>3，条件 VR = Q（即平行曲率张量场）意味着 M 是不正确的仿射超球面或二次曲面（Verheyen 和 Verstraelen [7]）。我们将通过证明仿射超曲面的条件 PΛ = 0 意味着 FR = Q 来提供此结果的推广。回想一下，对于黎曼流形，条件 VR = Q （事实上，对于某个整数 k，VR = Q）意味着 FR = Q，但这样的结果一般不适用于仿射连接。我们的研究显示了这些结果在三次形式和曲率张量场的协变微分上的共同背景。
In this note, we first investigate the symmetry properties of the tensors VC and FC, where V is the induced affine connection and C is the cubic form of a nondegenerate affine hypersurface M in R + . In particular, we study hypersurfaces with parallel cubic form, i.e. PC=0. In the case « = 2, this condition is known to characterize a Cayley surface (Nomizu and Pinkall [3]). We obtain a certain class of more general affine surfaces and hypersurfaces. On the other hand, for an affine hypersurface M", «>3, condition VR = Q (i.e. parallel curvature tensor field) implies that M is an improper affine hypersphere or a quadric (Verheyen and Verstraelen [7]). We shall provide a generalization of this result by proving that the condition PΛ = 0 implies FR = Q for an affine hypersurface. Recall that, for a Riemannian manifold, the condition VR = Q (in fact, VR = Q for some integer k) implies FR = Q but that such a result does not hold for an affine connection in general. Our study shows the common background for these results on the covariant differentials of the cubic form and those of the curvature tensor field.