Existence and non-existence of skew branes
نویسنده
چکیده
A skew brane is a codimension 2 submanifold in affine space such that the tangent spaces at any two distinct points are not parallel. We show that if an oriented closed manifold has a non-zero Euler characteristic χ then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ/4. We give a version of this result for immersed surfaces in dimension 4. We construct examples of skew spheres of arbitrary odd dimensions, generalizing the construction of skew loops in 3-dimensional space due to Ghomi and Solomon. We conclude with two conjectures that are theorems in 1-dimensional case. MSC: 53C40, 57R40, 57R42
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