Locally Symmetric Affine Hypersurfaces

Affine complete locally convex hypersurfaces

An open problem in affine geometry is whether an affine complete locally uniformly convex hypersurface in Euclidean (n + 1)-space is Euclidean complete for n ≥ 2. In this paper we give the affirmative answer. As an application, it follows that an affine complete, affine maximal surface in R3 must be an elliptic paraboloid.

Hypersurfaces with Locally Symmetric Conormal Connection

This paper studies hypersurfaces admitting a locally symmetric connection which is induced by the Gauß conormal map in affine geometry. It is known that the rank of the shape operator is of importance to this topic. In dimension two we give new results for an arbitrary shape operator. In the case of a nondegenerate shape operator hypersurfaces with locally symmetric conormal connection can be t...

On hypersurfaces in a locally affine Riemannian Banach manifold II

In our previous work (2002), we proved that an essential second-order hypersurface in an infinite-dimensional locally affine Riemannian Banach manifold is a Riemannian manifold of constant nonzero curvature. In this note, we prove the converse; in other words, we prove that a hypersurface of constant nonzero Riemannian curvature in a locally affine (flat) semiRiemannian Banach space is an essen...

On Hypersurfaces in a Locally Affine Riemannian Banach Manifold

Injections into Affine Hypersurfaces )

Let X be a smooth aane variety of dimension n > 2. Assume that the group H 1 (X; Z) is a torsion group and that (X) = 1. Let Y be a projectively smooth aane hypersurface Y C n+1 of degree d > 1, which is smooth at innnity. Then there is no injective polynomial mapping f : X ! Y: This contradicts a result of Peretz 5].