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Existence and non-existence of skew branes

S. Tabachnikov·Yu. Tyurina
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Abstract

Following recent work by Ghomi, Solomon and Tabachnikov, we study geometry and topology of skew branes. A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any pair of distinct points are not parallel. We prove that if an oriented closed manifold has a non-zero Euler characteristic $ξ$ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than $(ξ^2)/4$. We also construct examples of skew odd-dimensional spheres and skew two-dimensional tori.

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S. Tabachnikov, Yu. Tyurina. 2005-04-23. Existence and non-existence of skew branes. https://arxiv.org/abs/math/0504484

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