### 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 χ, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ^{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.

Original language | English (US) |
---|---|

Pages (from-to) | 419-431 |

Number of pages | 13 |

Journal | Journal of Fixed Point Theory and Applications |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Jul 23 2010 |

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### All Science Journal Classification (ASJC) codes

- Modeling and Simulation
- Geometry and Topology
- Applied Mathematics

### Cite this

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*Journal of Fixed Point Theory and Applications*, vol. 7, no. 2, pp. 419-431. https://doi.org/10.1007/s11784-010-0015-y

**Existence and nonexistence of skew branes.** / Tabachnikov, Serge.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence and nonexistence of skew branes

AU - Tabachnikov, Serge

PY - 2010/7/23

Y1 - 2010/7/23

N2 - 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 χ, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ2/4. 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.

AB - 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 χ, then it is not a skew brane; generically, the number of oppositely oriented pairs of parallel tangent spaces is not less than χ2/4. 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.

UR - http://www.scopus.com/inward/record.url?scp=77957300805&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77957300805&partnerID=8YFLogxK

U2 - 10.1007/s11784-010-0015-y

DO - 10.1007/s11784-010-0015-y

M3 - Article

AN - SCOPUS:77957300805

VL - 7

SP - 419

EP - 431

JO - Journal of Fixed Point Theory and Applications

JF - Journal of Fixed Point Theory and Applications

SN - 1661-7738

IS - 2

ER -