A Bonnet–Myers type theorem for quaternionic contact structures

  • Davide BarilariEmail author
  • Stefan Ivanov


We prove a Bonnet–Myers type theorem for quaternionic contact manifolds of dimension bigger than 7. If the manifold is complete with respect to the natural sub-Riemannian distance and satisfies a natural Ricci-type bound expressed in terms of derivatives up to the third order of the fundamental tensors, then the manifold is compact and we give a sharp bound on its sub-Riemannian diameter.

Mathematics Subject Classification

53C17 53D25 



The first author has been supported by the ANR project SRGI “Sub-Riemannian Geometry and Interactions”, Contract Number ANR-15-CE40-0018. The second author has been partially supported by Contract DH/12/3/12.12.2017 and Contract 195/2016 with the Sofia University “St.Kl.Ohridski”.


  1. 1.
    Agrachev, A., Barilari, D., Rizzi, L.: Sub-Riemannian curvature in contact geometry. J. Geom. Anal. 27(1), 366–408 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Agrachev, A., Barilari, D., Rizzi, L.: Curvature: a variational approach. Mem. Am. Math. Soc. 256(1225), v+142 (2018)MathSciNetGoogle Scholar
  3. 3.
    Agrachev, A., Zelenko, I.: Geometry of Jacobi curves. I. J. Dyn. Control Syst. 8(1), 93–140 (2002)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Agrachev, A., Zelenko, I.: Geometry of Jacobi curves. II. J. Dyn. Control Syst. 8(2), 167–215 (2002)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Agrachev, A.A.: Some open problems. In: Stefani, G., Boscain, U., Gauthier, J.P., Sarychev, A., Sigalotti, M. (eds.) Geometric Control Theory and Sub-Riemannian Geometry. Springer INdAM Series, vol. 5. Springer, Cham (2014)Google Scholar
  6. 6.
    Agrachev, A.A., Barilari, D., Boscain, U.: A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge University Press, Cambridge (2019)zbMATHGoogle Scholar
  7. 7.
    Agrachev, A.A., Lee, P.W.Y.: Generalized Ricci curvature bounds for three dimensional contact subriemannian manifolds. Math. Ann. 360(1–2), 209–253 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Agrachev, A.A., Sachkov, Y.L.: Control theory from the geometric viewpoint, In: Volume 87 of Control Theory and Optimization, II of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin (2004)Google Scholar
  9. 9.
    Agrachev, A.A., Zelenko, I.: Geometry of Jacobi curves. I. J. Dyn. Control Syst. 8(1), 93–140 (2002)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Barilari, D., Rizzi, L.: Comparison theorems for conjugate points in sub-Riemannian geometry. ESAIM Control Optim. Calc. Var. 22(2), 439–472 (2016)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Barilari, D., Rizzi, L.: On Jacobi fields and a canonical connection in sub-Riemannian geometry. Arch. Math. (Brno) 53(2), 77–92 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Baudoin, F., Garofalo, N.: Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries. J. Eur. Math. Soc. 19, 151–219 (2017).
  13. 13.
    Baudoin, F., Kim, B., Wang, J.: Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves. Commun. Anal. Geom. (2014) (to appear) Google Scholar
  14. 14.
    Baudoin, F., Wang, J.: Curvature dimension inequalities and subelliptic heat kernel gradient bounds on contact manifolds. Potential Anal. 40(2), 163–193 (2014)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Baudoin, F., Wang, J.: The subelliptic heat kernels of the quaternionic Hopf fibration. Potential Anal. 41(3), 959–982 (2014)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Bellaïche, A.: The tangent space in sub-Riemannian geometry. In: Sub-Riemannian Geometry, Volume 144 of Progress in Mathematics, pp. 1–78. Birkhäuser, Basel (1996)Google Scholar
  17. 17.
    Biquard, O.: Métriques d’Einstein asymptotiquement symétriques. Astérisque (265):vi+109 (2000)Google Scholar
  18. 18.
    Capria, M.M., Salamon, S.M.: Yang–Mills fields on quaternionic spaces. Nonlinearity 1(4), 517–530 (1988)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Davidov, J., Ivanov, S., Minchev, I.: The twistor space of a quaternionic contact manifold. Q. J. Math. (Oxford) 63(4), 873–890 (2012)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Duchemin, D.: Quaternionic contact structures in dimension 7. Ann. Inst. Fourier (Grenoble) 56(4), 851–885 (2006)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Folland, G.: Subelliptic estimates and function spaces on nilpotent lie groups. Ark. Math. 13(2), 161–207 (1975)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Folland, G.B., Stein, E.M.: Estimates for the \({\bar{\partial }}_{b}\) complex and analysis on the Heisenberg group. Commun. Pure Appl. Math. 27, 429–522 (1974)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Garofalo, N., Vassilev, D.: Symmetry properties of positive entire solutions of Yamabe type equations on groups of Heisenberg type. Duke Math. J. 106(3), 411–448 (2001)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Hladky, R.K.: The topology of quaternionic contact manifolds. Ann. Glob. Anal. Geom. 47(1), 99–115 (2015)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hughen, W.K.: The sub-Riemannian geometry of three-manifolds. ProQuest LLC, Ann Arbor, MI, Thesis (Ph.D.). Duke University (1995)Google Scholar
  26. 26.
    Iliev, B.Z.: Handbook of Normal Frames and Coordinates. Progress in Mathematical Physics, vol. 42. Birkhäuser Verlag, Basel (2006)CrossRefGoogle Scholar
  27. 27.
    Ivanov, S., Minchev, I., Vassilev, D.: The qc Yamabe problem on 3-Sasakian manifolds and the quaternionic Heisenberg group. arXiv:1504.03142
  28. 28.
    Ivanov, S., Minchev, I., Vassilev, D.: Quaternionic contact Einstein manifolds. to appear in Math. Res. Lett. (2015). arXiv:1306.0474
  29. 29.
    Ivanov, S., Minchev, I., Vassilev, D.: Extremals for the Sobolev inequality on the seven-dimensional quaternionic Heisenberg group and the quaternionic contact Yamabe problem. J. Eur. Math. Soc. (JEMS) 12(4), 1041–1067 (2010)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Ivanov, S., Minchev, I., Vassilev, D.: The optimal constant in the \(l^2\) Folland–Stein inequality on the quaternionic Heisenberg group. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 11(3), 635–662 (2012)MathSciNetzbMATHGoogle Scholar
  31. 31.
    Ivanov, S., Minchev, I., Vassilev, D.: Quaternionic contract Einstein structures and the quaternionic contact Yamabe problem. Mem. Am. Math. Soc. 231(1086), vi+82 (2014)zbMATHGoogle Scholar
  32. 32.
    Ivanov, S., Petkov, A., Vassilev, D.: The sharp lower bound of the first eigenvalue of the sub-Laplacian on a quaternionic contact manifold. J. Geom. Anal. 24(2), 756–778 (2014)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Ivanov, S., Vassilev, D.: Conformal quaternionic contact curvature and the local sphere theorem. J. Math. Pures Appl. (9) 93(3), 277–307 (2010)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Ivanov, S.P., Vassilev, D.N.: Extremals for the Sobolev Inequality and the Quaternionic Contact Yamabe Problem. World Scientific Publishing Co. Pte. Ltd., Hackensack (2011)CrossRefGoogle Scholar
  35. 35.
    Jean, F.: Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning. Springer Briefs in Mathematics. Springer, Cham (2014)Google Scholar
  36. 36.
    Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, Volume of Mathematical Surveys and Monographs. American Mathematical Society, Providence (2002)zbMATHGoogle Scholar
  37. 37.
    Myers, S.B.: Riemannian manifolds with positive mean curvature. Duke Math. J. 8, 401–404 (1941)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Ohta, S.-I.: On the measure contraction property of metric measure spaces. Comment. Math. Helv. 82(4), 805–828 (2007)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Rifford, L.: Sub-Riemannian Geometry and Optimal Transport. Springer Briefs in Mathematics. Springer, Cham (2014)CrossRefGoogle Scholar
  40. 40.
    Rizzi, L., Silveira, P.: Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds. J. Inst. Math. Jussieu (2017).
  41. 41.
    Rumin, M.: Formes différentielles sur les variétés de contact. J. Differ. Geom. 39(2), 281–330 (1994)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Tanno, S.: Killing vectors on contact Riemannian manifolds and fiberings related to the Hopf fibrations. Tôhoku Math. J. 2(23), 313–333 (1971)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Wang, W.: The Yamabe problem on quaternionic contact manifolds. Ann. Mat. Pura Appl. (4), 186(2), 359–380, (2007)Google Scholar
  44. 44.
    Zelenko, I., Li, C.: Differential geometry of curves in Lagrange Grassmannians with given Young diagram. Differ. Geom. Appl. 27(6), 723–742 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive Gauche, UMR CNRS 7586Université Paris-DiderotParis Cedex 13France
  2. 2.Faculty of Mathematics and InformaticsUniversity of SofiaSofiaBulgaria
  3. 3.Institute of Mathematics and InformaticsBulgarian Academy of SciencesSofiaBulgaria

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