Existence and nonexistence of skew branes

Research output: Contribution to journalArticle

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 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.

Original languageEnglish (US)
Pages (from-to)419-431
Number of pages13
JournalJournal of Fixed Point Theory and Applications
Volume7
Issue number2
DOIs
StatePublished - Jul 23 2010

Fingerprint

Branes
Skew
Nonexistence
Tangent Space
Affine Space
Euler Characteristic
Submanifolds
Codimension
Odd
Distinct
Closed
Arbitrary
Theorem

All Science Journal Classification (ASJC) codes

  • Modeling and Simulation
  • Geometry and Topology
  • Applied Mathematics

Cite this

@article{f669c752a5c3496494e059dac78cda96,
title = "Existence and nonexistence of skew branes",
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 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.",
author = "Serge Tabachnikov",
year = "2010",
month = "7",
day = "23",
doi = "10.1007/s11784-010-0015-y",
language = "English (US)",
volume = "7",
pages = "419--431",
journal = "Journal of Fixed Point Theory and Applications",
issn = "1661-7738",
publisher = "Springer Science + Business Media",
number = "2",

}

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

In: Journal of Fixed Point Theory and Applications, Vol. 7, No. 2, 23.07.2010, p. 419-431.

Research output: Contribution to journalArticle

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 -