  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.
