Skip to main content
SpringerLink
Log in

On the subRiemannian cut locus in a model of free two-step Carnot group

  • Published:
Calculus of Variations and Partial Differential Equations Aims and scope Submit manuscript

Abstract

We characterize the subRiemannian cut locus of the origin in the free Carnot group of step two with three generators, giving a new, independent proof of a result by Myasnichenko (J Dyn Control Syst 8(4):573-597, 2002). We also calculate explicitly the cut time of any extremal path and the distance from the origin of all points of the cut locus. Furthermore, by using the Hamiltonian approach, we show that the cut time of strictly normal extremal paths is a smooth explicit function of the initial velocity covector. Finally, using our previous results, we show that at any cut point the distance has a corner-like singularity.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Notes

  1. After the submission of this paper, a precise conjecture for the cut locus on the free group in higher dimension has been formulated by Rizzi and Serres [27].

  2. That is, the property \(F(r\alpha ,r\beta ,r\zeta ,\varphi )=\delta _r F(\alpha ,\beta ,\zeta , \varphi )\) for all \(r>0\), \(\alpha ,\beta ,\zeta \) admissible and \(\varphi >0\).

References

  1. Agrachev, A., Barilari, D., Boscain, U.: On the Hausdorff volume in sub-Riemannian geometry. Calc. Var. Partial Differ. Equ. 43(3–4), 355–388 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  2. Agrachev, A., Barilari, D., Boscain, U.: Notes on Riemannian and sub-Riemanniann geometry (2016). (preprint)

  3. Agrachev, A., Bonnard, B., Chyba, M., Kupka, I.: Sub-Riemannian sphere in Martinet flat case. ESAIM Control Optim. Calc. Var. 2, 377–448 (1997). (electronic)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arcozzi, N., Ferrari, F.: Metric normal and distance function in the Heisenberg group. Math. Z. 256(3), 661–684 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Agrachev, A.A., Gentile, A., Lerario, A.: Geodesics and horizontal-path spaces in Carnot groups. Geom. Topol. 19(3), 1569–1630 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  6. Agrachev, A., Lee, P.: Optimal transportation under nonholonomic constraints. Trans. Am. Math. Soc. 361(11), 6019–6047 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Autenried, C., Godoy Molina, M.: The sub-Riemannian cut locus of \(H\)-type groups. Math. Nachr. 289(1), 4–12 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  8. Ambrosio, L., Rigot, S.: Optimal mass transportation in the Heisenberg group. J. Funct. Anal. 208(2), 261–301 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ardentov, A.A., Sachkov, Y.L.: Extremal trajectories in the nilpotent sub-Riemannian problem on the Engel group. Mat. Sb. 202(11), 31–54 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ardentov, A.A., Sachkov, Y.L.: Cut time in sub-Riemannian problem on Engel group. ESAIM Control Optim. Calc. Var. 21(4), 958–988 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Barilari, D., Boscain, U., Gauthier, J.P.: On 2-step, corank 2, nilpotent sub-Riemannian metrics. SIAM J. Control Optim. 50(1), 559–582 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  12. Barilari, D., Boscain, U., Neel, R.W.: Small-time heat kernel asymptotics at the sub-Riemannian cut locus. J. Differ. Geom. 92(3), 373–416 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  13. Barilari, D., Boscain, U., Neel, R.W.: Heat kernel asymptotics on sub-Riemannian manifolds with symmetries and applications to the bi-Heisenberg group. arXiv:1606.01159 (2016)

  14. Cordero-Erausquin, D., McCann, R.J., Schmuckenschläger, M.: A Riemannian interpolation inequality à la Borell Brascamp and Lieb. Invent. Math. 146(2), 219–257 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cannarsa, P., Rifford, L.: Semiconcavity results for optimal control problems admitting no singular minimizing controls. Ann. l’Institut Henri Poincare (C) Nonlinear Anal. 25(4), 773–802 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Le Donne, E., Montgomery, R., Ottazzi, A., Pansu, P., Vittone, D.: Sard property for the endpoint map on some carnot groups. Ann. l’Institut Henri Poincare (C) Nonlinear Anal. 33(6), 1639–1666 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Figalli, A., Rifford, L.: Mass transportation on sub-Riemannian manifolds. Geom. Funct. Anal. 20(1), 124–159 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. Donne, E.L., Golo, S.N.: Regularity properties of spheres in homogeneous groups. Trans. Amer. Math. Soc. arXiv:1509.03881 (2015, to appear)

  19. Liu, W, Sussman, H.J.: Shortest paths for sub-Riemannian metrics on rank-two distributions. Mem. Am. Math. Soc. 118(564), 1–104 (1995)

  20. Martini, A., Müller, D.: \(L^p\) spectral multipliers on the free group \(N_{3,2}\). Studia Math. 217(1), 41–55 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  21. Montanari, A., Morbidelli, D.: On the lack of semiconcavity of the sub-Riemannian distance in a class of Carnot groups. J. Math. Anal. Appl. 444(2), 1652–1674 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Monti, R: Some properties of Carnot-Carathéodory balls in the Heisenberg group. Atti Accad. Naz. Lincei Cl. Sci. Fisc. Mat. Nat. Rend. Lincei (9) Mat. Appl 11 (2000), no. 3, 155–167 (2001)

  23. Monroy-Pérez, F., Anzaldo-Meneses, A.: The step-2 nilpotent \((n, n(n+1)/2)\) sub-Riemannian geometry. J. Dyn. Control Syst. 12(2), 185–216 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  24. Myasnichenko, O.: Nilpotent \((3,6)\) sub-Riemannian problem. J. Dyn. Control Syst. 8(4), 573–597 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Podobryaev, A.V., Sachkov, Y.L.: Cut locus of a left invariant Riemannian metric on SO\(_{3}\) in the axisymmetric case. J. Geom. Phys. 110, 436–453 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  26. Rifford, L.: Sub-Riemannian Geometry and Optimal Transport. Springer, BCAM—Basque Center for Applied Mathematics, Cham, Bilbao (2014). English

    Book  MATH  Google Scholar 

  27. Rizzi, L., Serres, U.: On the cut locus of free, step two Carnot groups. Proc. Amer. Math. Soc. arXiv:1610.01596 (2016, to appear)

  28. Rifford, L., Trélat, E.: On the stabilization problem for nonholonomic distributions. J. Eur. Math. Soc. (JEMS) 11(2), 223–255 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  29. Sachkov, Y.L.: Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calc. Var. 17(2), 293–321 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We thank Luca Rizzi and Yuri Sachkov who kindly gave us several useful bibliographic information on the state of the art of the subject. We thank the anonymous referee for raising some useful questions, which led us to an improvement of the first draft of the paper. The authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM)

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università di Bologna, Bologna, Italy

    Annamaria Montanari & Daniele Morbidelli

Authors
  1. Annamaria Montanari
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniele Morbidelli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniele Morbidelli.

Additional information

Communicated by L.Ambrosio.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Montanari, A., Morbidelli, D. On the subRiemannian cut locus in a model of free two-step Carnot group. Calc. Var. 56, 36 (2017). https://doi.org/10.1007/s00526-017-1149-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00526-017-1149-1

Mathematics Subject Classification

Search

Navigation