Flexibility for tangent and transverse immersions in Engel manifolds Articles uri icon

authors

  • DEL PINO, ALVARO
  • PRESAS MATA, FRANCISCO

publication date

  • January 2019

start page

  • 215

end page

  • 238

issue

  • 1

volume

  • 32

International Standard Serial Number (ISSN)

  • 1139-1138

Electronic International Standard Serial Number (EISSN)

  • 1988-2807

abstract

  • We study the space of immersions of S1 that are tangent to an Engel structure D. We show that the full h-principle holds as soon as one excludes the closed orbits of W, the characteristic foliation of D. This is sharp: we elaborate on work of Bryant and Hsu to show that curves tangent to W sometimes form additional isolated components that cannot be detected at a formal level. We then show that this is an exceptional phenomenon: if D is C-generic, curves tangent to W are not isolated anymore. These results, in conjunction with an argument due to M. Gromov, prove that a full h-principle holds for immersions transverse to the Engel structure.

keywords

  • engel structure; h-principle; horizontal curve; existence