Geodesics in the Heisenberg Group

Piotr Hajłasz 1  and Scott Zimmerman 1
  • 1 Department of Mathematics, University of Pittsburgh, 301 Thackeray Hall, Pittsburgh, PA 15260, USA


We provide a new and elementary proof for the structure of geodesics in the Heisenberg group Hn. The proof is based on a new isoperimetric inequality for closed curves in R2n.We also prove that the Carnot- Carathéodory metric is real analytic away from the center of the group.

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  • [1] L. Ambrosio, S. Rigot, Optimal mass transportation in the Heisenberg group, J. Funct. Anal. 208 (2004), 261–301.

  • [2] A. Bellaïche, The tangent space in sub-Riemannian geometry, in: A. Bellaïche, J.J. Risler (Eds.), Sub-Riemannian geometry, Progress in Mathematics, Vol. 144, Birkhäuser, Basel, 1996, pp. 1–78.

  • [3] V. N. Berestovskii, Geodesics of nonholonomic left-invariant intrinsic metrics on the Heisenberg group and isoperimetric curves on the Minkowski plane. Siberian Math. J. 35 (1994), 1–8.

  • [4] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry. Graduate Studies inMathematics, 33. AmericanMathematical Society, Providence, RI, 2001.

  • [5] L. Capogna, S. D. Pauls, D. Danielli, J. T. Tyson, An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem, Progress in Mathematics, Vol. 259. Birkhäuser Basel. 2007.

  • [6] H. Dym, H. P. McKean, Fourier series and integrals. Probability and Mathematical Statistics, No. 14. Academic Press, New York-London, 1972.

  • [7] B. Gaveau, Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents. Acta Math. 139 (1977), 95–153.

  • [8] P. Hajłasz, Sobolev spaces on metric-measure spaces, in Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002), 173–218, Contemp. Math., 338, Amer. Math. Soc., Providence, RI, 2003.

  • [9] A. Hurwitz, Sur quelques applications géométriques des séries de Fourier, Ann. Ecole Norm. Sup. 19 (1902) 357–408.

  • [10] S. G. Krantz, H. R. Parks, A primer of real analytic functions. Second edition. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Boston, Inc., Boston, MA, 2002.

  • [11] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs, 91. American Mathematical Society, Providence, RI, 2002.

  • [12] R. Monti, Distances, boundaries and surface measures in Carnot-Carathéodory spaces, PhD thesis 2001. Available at http: //

  • [13] R. Monti, Some properties of Carnot-Carathéodory balls in the Heisenberg group, Rend. MatA˙ cc. Lincei 11 (2000) 155–167.

  • [14] I. J. Schoenberg, An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces. Acta Math. 91 (1954), 143–164.


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