Existence and nonexistence of skew branes |
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Authors: | Serge Tabachnikov |
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Institution: | (1) School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA;(2) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA |
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Abstract: | 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 nonzero Euler characteristic c{\chi}, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less
than c2/4{\chi^2{/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 (2002). We conclude
with two conjectures that are theorems in 1-dimensional case. |
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Keywords: | |
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