Homotopically invisible singular curves

Homotopically invisible singular curves Given a smooth manifold M and a totally nonholonomic distribution $$\Delta \subset TM $$ Δ ⊂ T M of rank $$d\ge 3$$ d ≥ 3 , we study the effect of singular curves on the topology of the space of horizontal paths joining two points on M. Singular curves are critical points of the endpoint map $$F\,{:}\,\gamma \mapsto \gamma (1)$$ F : γ ↦ γ ( 1 ) defined on the space $$\Omega $$ Ω of horizontal paths starting at a fixed point x. We consider a sub-Riemannian energy $$J\,{:}\,\Omega (y)\rightarrow \mathbb R$$ J : Ω ( y ) → R , where $$\Omega (y)=F^{-1}(y)$$ Ω ( y ) = F - 1 ( y ) is the space of horizontal paths connecting x with y, and study those singular paths that do not influence the homotopy type of the Lebesgue sets $$\{\gamma \in \Omega (y)\,|\,J(\gamma )\le E\}$$ { γ ∈ Ω ( y ) | J ( γ ) ≤ E } . We call them homotopically invisible. It turns out that for $$d\ge 3$$ d ≥ 3 generic sub-Riemannian structures in the sense of Chitour et al. (J Differ Geom 73(1):45–73, 2006) have only homotopically invisible singular curves. Our results can be seen as a first step for developing the calculus of variations on the singular space of horizontal curves (in this direction we prove a sub-Riemannian minimax principle and discuss some applications). http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Calculus of Variations and Partial Differential Equations Springer Journals

Loading next page...
 
/lp/springer_journal/homotopically-invisible-singular-curves-gMGrzNu9nn
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Mathematics; Analysis; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Theoretical, Mathematical and Computational Physics
ISSN
0944-2669
eISSN
1432-0835
D.O.I.
10.1007/s00526-017-1203-z
Publisher site
See Article on Publisher Site

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

Monthly Plan

  • Read unlimited articles
  • Personalized recommendations
  • No expiration
  • Print 20 pages per month
  • 20% off on PDF purchases
  • Organize your research
  • Get updates on your journals and topic searches

$49/month

Start Free Trial

14-day Free Trial

Best Deal — 39% off

Annual Plan

  • All the features of the Professional Plan, but for 39% off!
  • Billed annually
  • No expiration
  • For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588

$360/year

billed annually
Start Free Trial

14-day Free Trial