Ruled hypersurfaces of Euclidean space
type: Item
instance of: scholarly article
title: Ruled hypersurfaces of Euclidean space
zbMATH Open document ID: 0638.53008
DOI: 10.1216/RMJ-1987-17-4-697
author: Kinetsu Abe
author: David E. Blair
published in: Rocky Mountain Journal of Mathematics
publication date: 1987-01-01T00:00:00Z
review text: Let M be a hypersurface immersed in Euclidean space \(E^{n+1}\) (n\(\geq 2)\) and satisfying the condition: through each point \(x\in M\), there exists an entire (n-1)-plane contained in M. Then it may appear that M is foliated by these (n-1)-planes. In the general case, however, if M is not foliated by the given (n-1)-planes, M contains points where these planes intersect. The authors' main result is to show that in a neighbourhood of such a point M has the product structure of an open set in \(E^{n-2}\) and a piece of a ruled surface in \(E^ 3\). If these intersections are dense, M is the product of \(E^{n-2}\) and a doubly ruled surface. Finally the example of a special hypersurface M in \(E^ 4\) shows that M may be foliated by (n-1)-planes but not have a product structure with \(E^{n-2}\) as a factor.
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53A25
zbMATH DE Number: 4040723
zbMATH Keywords: ruled hypersurface
zbMATH Keywords: relative null space
zbMATH Keywords: product structure
zbMATH Keywords: doubly ruled surface
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4040723