Geodesics in the Heisenberg Group

Piotr Hajłasz; Scott Zimmerman

Analysis and Geometry in Metric Spaces (2015)

  • Volume: 3, Issue: 1, page 325-337, electronic only
  • ISSN: 2299-3274

Abstract

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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.

How to cite

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Piotr Hajłasz, and Scott Zimmerman. "Geodesics in the Heisenberg Group." Analysis and Geometry in Metric Spaces 3.1 (2015): 325-337, electronic only. <http://eudml.org/doc/275961>.

@article{PiotrHajłasz2015,
abstract = {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.},
author = {Piotr Hajłasz, Scott Zimmerman},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {Heisenberg group; geodesics; Fourier series; isoperimetric inequality},
language = {eng},
number = {1},
pages = {325-337, electronic only},
title = {Geodesics in the Heisenberg Group},
url = {http://eudml.org/doc/275961},
volume = {3},
year = {2015},
}

TY - JOUR
AU - Piotr Hajłasz
AU - Scott Zimmerman
TI - Geodesics in the Heisenberg Group
JO - Analysis and Geometry in Metric Spaces
PY - 2015
VL - 3
IS - 1
SP - 325
EP - 337, electronic only
AB - 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.
LA - eng
KW - Heisenberg group; geodesics; Fourier series; isoperimetric inequality
UR - http://eudml.org/doc/275961
ER -

References

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  2. [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.  Zbl0862.53031
  3. [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. [4] D. Burago, Y. Burago, S. Ivanov, A course in metric geometry. Graduate Studies inMathematics, 33. AmericanMathematical Society, Providence, RI, 2001.  Zbl0981.51016
  5. [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.  Zbl1138.53003
  6. [6] H. Dym, H. P. McKean, Fourier series and integrals. Probability and Mathematical Statistics, No. 14. Academic Press, New York-London, 1972.  Zbl0242.42001
  7. [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. [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.  Zbl1048.46033
  9. [9] A. Hurwitz, Sur quelques applications géométriques des séries de Fourier, Ann. Ecole Norm. Sup. 19 (1902) 357–408.  Zbl33.0599.02
  10. [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. [11] R. Montgomery, A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs, 91. American Mathematical Society, Providence, RI, 2002.  Zbl1044.53022
  12. [12] R. Monti, Distances, boundaries and surface measures in Carnot-Carathéodory spaces, PhD thesis 2001. Available at http: //www.math.unipd.it/~monti/PAPERS/TesiFinale.pdf  Zbl1032.49045
  13. [13] R. Monti, Some properties of Carnot-Carathéodory balls in the Heisenberg group, Rend. MatA˙ cc. Lincei 11 (2000) 155–167.  Zbl1197.53064
  14. [14] I. J. Schoenberg, An isoperimetric inequality for closed curves convex in even-dimensional Euclidean spaces. Acta Math. 91 (1954), 143–164.  Zbl0056.15705

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