圈与Clifford代数
Circles and Clifford Algebras
摘要：
考虑一个从实向量空间原点的邻域到欧几里得空间原点的邻域的光滑映射。假设这个映射把所有通过原点的线的芽都取为欧几里得圆、线或点的芽。我们证明，在一些简单的额外的假设下，这个地图需要所有的线通过原点到相同的圆作为一个霍普夫地图来自一个Clifford代数的表示。我们还描述了我们的结果和Hurwitz-Radon定理的平方和之间的联系。
英文摘要：
Consider a smooth map from a neighborhood of the origin in a real vector space to a neighborhood of the origin in a Euclidean space. Suppose that this map takes all germs of lines passing through the origin to germs of Euclidean circles, or lines, or a point. We prove that under some simple additional assumptions this map takes all lines passing though the origin to the same circles as a Hopf map coming from a representation of a Clifford algebra does. We also describe a connection between our result and the Hurwitz--Radon theorem about sums of squares.
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