No abstract

We study the geometry of the surfaces embedded in ~4 through their generic contacts with hyperplanes. The inflection points on them are shown to be the umbilic points of their families of height functions. As a consequence we prove that any genetic convexly embedded 2-sphere in ~4 has inflection points.

We introduce the notion of the lightcone Gauss-Kronecker curvature for a spacelike submanifold of codimention two in Minkowski space which is a generalization of the ordinary notion of Gauss curvature of hypersurfaces in Euclidean space. In the local sense, this curvature describes the contact of such submanifolds with lightlike hyperplanes. We study geometric properties of such curvatures and show a Gauss-Bonnet type theorem. As examples we have hypersurfaces in hyperbolic space, spacelike hypersurfaces in the lightcone and spacelike hypersurfaces in de Sitter space.

Some geometrical properties associated to the contact of submanifolds with hyperhorospheres in hyperbolic n-space are studied as an application of the theory of Legendrian singularities.

Abstract. We consider asymptotic line fields on generic surfaces in 4-space and show that they are globally defined on locally convex surfaces, and their singularities are the inflection points of the surface. As a consequence of the generalized Poincaré-Hopf formula, we obtain some relations between the number of inflection points in a generic surface and its Euler number. In particular, it follows that any 2-sphere, generically embedded as a locally convex surface in 4-space, has at least 4 inflection points.