Mathematische Zeitschrift

, Volume 287, Issue 1–2, pp 1–38 | Cite as

Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group

  • Zoltán M. Balogh
  • Jeremy T. TysonEmail author
  • Eugenio Vecchi


We use a Riemannnian approximation scheme to define a notion of intrinsic Gaussian curvature for a Euclidean \(C^{2}\)-smooth surface in the Heisenberg group \(\mathbb {H}\) away from characteristic points, and a notion of intrinsic signed geodesic curvature for Euclidean \(C^{2}\)-smooth curves on surfaces. These results are then used to prove a Heisenberg version of the Gauss–Bonnet theorem. An application to Steiner’s formula for the Carnot–Carathéodory distance in \(\mathbb {H}\) is provided.


Heisenberg group Sub-Riemannian geometry Riemannian approximation Gauss–Bonnet theorem Steiner formula 

Mathematics Subject Classification

Primary 53C17 Secondary 53A35 52A39 



Research for this paper was conducted during visits of the second and third authors to the University of Bern in 2015 and 2016. The hospitality of the Institute of Mathematics of the University of Bern is gratefully acknowledged. The authors would also like to thank Luca Capogna for many valuable conversations on these topics and for helpful remarks concerning the proof of Theorem 1.1. The authors would also like to thank the referee for a careful reading of the paper and for the numerous useful comments, ideas and suggestions which have improved the paper.


  1. 1.
    Agrachev, A., Boscain, U., Sigalotti, M.: A Gauss–Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discrete Contin. Dyn. Syst. 20(4), 801–822 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agrachev, A.A.: A Gauss–Bonnet formula for contact sub-Riemannian manifolds. Dokl. Akad. Nauk. 381(5), 583–585 (2001)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Arcozzi, N., Ferrari, F.: Metric normal and distance function in the Heisenberg group. Math. Z. 256(3), 661–684 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Balogh, Z.M.: Size of characteristic sets and functions with prescribed gradient. J. Reine Angew. Math. 564, 63–83 (2003)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Balogh, Z.M., Ferrari, F., Franchi, B., Vecchi, E., Wildrick, K.: Steiner’s formula in the Heisenberg group. Nonlinear Anal. 126, 201–217 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bao, D., Chern, S .S.: A note on the Gauss–Bonnet theorem for Finsler spaces. Ann. Math. 143(2), 233–252 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Capogna, L., Citti, G., Manfredini, M.: Uniform estimates of fundamental solution and regularity of vanishing viscosity solutions of mean curvature equations in \(H^n\). In: Subelliptic PDE’s and Applications to Geometry and Finance. Lecture Notes Seminar Interdisciplinare Matematica (S.I.M.), vol. 6, pp. 107–117. Potenza (2007)Google Scholar
  8. 8.
    Capogna, L., Citti, G., Manfredini, M.: Regularity of mean curvature flow of graphs on Lie groups free up to step 2. Nonlinear Anal. 126, 437–450 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Capogna, L., Danielli, D., Pauls, S.D., Tyson, J.T.: An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem. Progress in Mathematics, 259th edn. Birkhäuser Verlag, Basel (2007)zbMATHGoogle Scholar
  10. 10.
    Capogna, L., Pauls, S.D., Tyson, J.T.: Convexity and horizontal second fundamental forms for hypersurfaces in Carnot groups. Trans. Am. Math. Soc. 362(8), 4045–4062 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cheng, J.-H., Hwang, J.-F.: Properly embedded and immersed minimal surfaces in the Heisenberg group. Bull. Aust. Math. Soc. 70(3), 507–520 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cheng, J.-H., Hwang, J.-F., Malchiodi, A., Yang, P.: Minimal surfaces in pseudohermitian geometry. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5). 4(1), 129–177 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Cheng, J.-H., Hwang, J.-F., Malchiodi, A., Yang, P.: A Codazzi-like equation and the singular set for \(C^1\) smooth surfaces in the Heisenberg group. J. Reine Angew. Math. 671, 131–198 (2012)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Chiu, H., Lai, S.: The fundamental theorems for curves and surfaces in 3d Heisenberg group.
  15. 15.
    Chiu, H.-L., Lai, S.-H.: The fundamental theorem for hypersurfaces in Heisenberg groups. Calc. Var. Partial Differ. Equ. 54(1), 1091–1118 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chousionis, V., Magnani, V., Tyson, J.T.: On the classification of uniform measures in the Heisenberg group. (In preparation)Google Scholar
  17. 17.
    Citti, G., Manfredini, M.: Uniform estimates of the fundamental solution for a family of hypoelliptic operators. Potential Anal. 25(2), 147–164 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Danielli, D., Garofalo, N., Nhieu, D.-M.: Notions of convexity in Carnot groups. Commun. Anal. Geom. 11(2), 263–341 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Danielli, D., Garofalo, N., Nhieu, D.M.: Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv. Math. 215(1), 292–378 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Danielli, D., Garofalo, N., Nhieu, D.-M.: A partial solution of the isoperimetric problem for the Heisenberg group. Forum Math. 20(1), 99–143 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Danielli, D., Garofalo, N., Nhieu, D.M.: Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group. Proc. Am. Math. Soc. 140(3), 811–821 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Diniz, M.M., Veloso, J.M.M.: Gauss–Bonnet theorem in sub-Riemannian Heisenberg space. J. Dyn. Control Syst. 22(4), 807–820 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    do Carmo, M.P.; Differential Geometry of Curves and Surfaces. Prentice-Hall, Inc., Englewood Cliffs, NJ. Translated from the Portuguese (1976)Google Scholar
  24. 24.
    do Carmo, M.P.: Riemannian Geometry. Mathematics: Theory and Applications. Birkhäuser Boston, Inc., Boston, MA. Translated from the second Portuguese edition by Francis Flaherty (1992)Google Scholar
  25. 25.
    Hladky, R.K., Pauls, S.D.: Constant mean curvature surfaces in sub-Riemannian geometry. J. Differ. Geom. 79(1), 111–139 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lee, J.M.: Riemannian Manifolds. Graduate Texts in Mathematics, 176th edn. Springer, NewYork (1997)CrossRefGoogle Scholar
  27. 27.
    Ni, Y.: Sub-Riemannian constant mean curvature surfaces in the Heisenberg group as limits. Ann. Mat. Pura Appl. 183(4), 555–570 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Pauls, S.D.: Minimal surfaces in the Heisenberg group. Geom. Dedic. 104, 201–231 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Ritoré, M., Rosales, C.: Rotationally invariant hypersurfaces with constant mean curvature in the Heisenberg group \(\mathbb{H}^n\). J. Geom. Anal. 16(4), 703–720 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Zoltán M. Balogh
    • 1
  • Jeremy T. Tyson
    • 2
    Email author
  • Eugenio Vecchi
    • 3
  1. 1.Mathematisches InstitutUniversität BernBernSwitzerland
  2. 2.Department of MathematicsUniversity of IllinoisUrbanaUSA
  3. 3.Dipartimento di MatematicaUniversità di BolognaBolognaItaly

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