On affine hypersurfaces with everywhere nondegenerate second quadratic form

An Arnold conjecture claims that a real projective hypersurface with second quadratic form of constant signature $(k,l)$ should separate two projective subspaces of dimension $k$ and $l$ correspondingly. We consider affine versions of the conjecture dealing with hypersurfaces approaching at infinity two shifted halves of a standard cone. We prove that if the halves intersect, then the hypersurface does separate two affine subspaces. In the case of non-intersecting half-cones we construct an example of a surface of negative curvature in $\mathbb R^3$ bounding a domain without a line inside.