Affine Focal Points for Locally Strictly Convex Surfaces in 4-Space

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• 作者：Juan J. Nuño-Ballesteros ; Marcelo J. Saia ; Luis F. Sánchez
• 关键词：Affine metric ; affine normal plane ; affine semiumbilical ; affine focal point
• 刊名：Results in Mathematics
• 出版年：2017
• 出版时间：February 2017
• 年：2017
• 卷：71
• 期：1-2
• 页码：357-376
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1420-9012
• 卷排序：71

文摘

We consider locally strictly convex surfaces M in affine 4-space. By using the metric of the transversal vector field on M we introduce a new affine normal plane and the familly of affine distance functions on M. We show that the singularities of the family of affine distance functions appear at points on the affine normal plane and the affine focal points correspond to degenerate singularities of this family. Moreover we show that if M is immersed in a locally strictly convex hypersurface, then the affine normal plane contains the affine normal vector to the hypersurface and conclude that any surface immersed in a locally strictly convex hypersphere is affine semiumbilical.