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On affine hypersurfaces with everywhere nondegenerate Second Quadratic Form

A. Khovanskii·D. Novikov
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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.

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A. Khovanskii, D. Novikov. 2002-03-19. On affine hypersurfaces with everywhere nondegenerate Second Quadratic Form. https://arxiv.org/abs/math/0203202

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