Ruled hypersurfaces of Euclidean space
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.
scientific article; zbMATH DE number 4040723