On affine hypersurfaces with everywhere nondegenerate Second Quadratic Form

Abstract

Consider a closed connected hypersurface in ${\mathbb{R}}^n$ with constant signature (k,l) of the second quadratic form, and approaching a quadratic cone at infinity. This hypersurface divides ${\mathbb{R}}^n$ into two pieces. We prove that one of them contains a k-dimensional subspace, and another contains a l-dimensional subspace, thus proving an affine version of Arnold hypothesis. We construct an example of a surface of negative curvature in ${\mathbb{R}}^3$ with slightly different asymptotical behavior for which the previous claim is wrong.

18pp