Projections of transnormal manifolds

Abstract

Transnormal manifolds are generalizations of convex hypersurfaces of constant width (see [7]). For such hypersurfaces it is known that every orthogonal projection onto a hyperplane has an outline which is of constant width, too. Orthogonal projections of transnormal manifolds have been studied by F.J. Craveiro de Carvalho [1] in the case when the projection is an immersion onto a convex hypersurface of constant width in a suitable affine subspace of the ambient Euclidean space.

Here it will be shown that transnormality is not preserved under orthogonal projection onto hyperplanes without assuming that the manifold has the topological type of a sphere. This implies another general version of the transnormal graph theorem (see [2]). Furthermore, in the case of closed transnormal curves in 3-space also non-orthogonal parallel projection onto planes cannot preserve transnormality as is shown in the last section.