Zur Differentialgeometrie der \(n\)-dimensionalen Kugel- und Linienmannigfaltigkeiten im \((n+1)\)-dimensionalen Euklidischen Raum. (On the differential geometry of \(n\)-dimensional sphere- and line-manifolds in \((n+1)\)-dimensional Euclidean space). (German) Zbl 0716.53019

In an earlier article [Cas. Pestovani Mat. 114, 45-52 (1989; Zbl 0668.53003)], the author studied n-dimensional manifolds of spheres in the euclidean space \(E_{n+1}\). The adjoint line manifold is defined for such a manifold. The properties of special manifold of spheres are found. In the article under consideration certain sections of planes with the manifold of spheres are considered. These sections form the directional characteristics \(^{on}v,...,^{on-p}v\) (n\(\geq 3\), \(0\leq p\leq n-3)\) of the manifold of spheres. A special parametric representation connected with the directional characteristics is defined. The fundamental tensors are calculated and their properties are found. A special manifold of spheres with the same manifold of centers and fixed spheres as the considered manifold of spheres is studied. The properties of this pair are found. To this manifold the adjoint line manifolds are studied, too. Especially the properties of the focal manifolds of these manifolds are found.
Reviewer: J.Vala


53A25 Differential line geometry


Zbl 0668.53003
Full Text: DOI