Sacksteder和Bourgain例子的等价性
Equivalence of Examples of Sacksteder and Bourgain
摘要：
在欧几里得空间$R^4$中寻找切向退化子流形（具有退化高斯映射的子流形）的例子，这些子流形是非圆柱形的且没有奇点，这是微分几何的一个重要问题。这种超曲面的第一个例子是由Sacksteder在1960年构造的。在1995年吴发表了一个例子的非圆柱切退化代数超曲面在$R^4$的高斯映射是秩2，这也是没有奇点。这个例子是由Bourgain构建的（但没有发布）。本文分析了Bourgain的例子，证明了Bourgain超曲面的奇点与Sacksteder超曲面的奇点一样，都位于空间$R^4$的无穷远超平面上，且这两个超曲面是局部等价的.
英文摘要：
Finding examples of tangentially degenerate submanifolds (submanifolds with degenerate Gauss mappings) in an Euclidean space $R^4$ that are noncylindrical and without singularities is an important problem of differential geometry. The first example of such a hypersurface was constructed by Sacksteder in 1960. In 1995 Wu published an example of a noncylindrical tangentially degenerate algebraic hypersurface in $R^4$ whose Gauss mapping is of rank 2 and which is also without singularities. This example was constructed (but not published) by Bourgain. In this paper, the authors analyze Bourgain's example, prove that, as was the case for the Sacksteder hypersurface, singular points of the Bourgain hypersurface are located in the hyperplane at infinity of the space $R^4$, and these two hypersurfaces are locally equivalent.
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