Carnot-Carathéodory and Korányi-Geodesics in the Heisenberg Group

Authors

  • Josh Ascher University of Pittsburgh
  • Armin Schikorra University of Pittsburgh

DOI:

https://doi.org/10.33043/3DEM6HFD3V

Abstract

We discuss the Heisenberg group H1, the three-dimensional space R^3 equipped with one of two equivalent metrics, the Korányi- and Carnot-Carathéodory metric. We show that the notion of length of curves for both metrics coincide, and that shortest curves, so-called geodesics, exist.

Author Biographies

Josh Ascher, University of Pittsburgh

Josh Ascher is an undergraduate student studying math and computer science at the University of Pittsburgh. He is in his senior year and plans to attend graduate school where he will conduct research in theoretical computer science. Josh hopes to one day become a professor so that he can continue pursuing research, as well as teaching the next generation of researchers.

Armin Schikorra, University of Pittsburgh

Armin Schikorra received his Ph.D. from RWTH Aachen University in 2010 and is currently associate professor at the University of Pittsburgh. He works in partial differential equations which are often motivated from the geometric calculus of variations, one example being harmonic maps between manifolds and various generalizations thereof. In particular he is interested in regularity theory of such local or nonlocal equations.

References

L. Capogna, D. Danielli, S. D. Pauls, and J. T. Tyson. An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem, volume 259 of Progress in Mathematics. Birkh¨auser Verlag, Basel, 2007.

M. Gromov. Carnot-Carathéodory spaces seen from within. In Sub-Riemannian geometry, volume 144 of Progr. Math., pages 79–323. Birkh¨auser, Basel, 1996.

P. Hajłlasz and S. Zimmerman. Geodesics in the Heisenberg group. Anal. Geom. Metr. Spaces, 3(1):325–337, 2015.

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

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Published

2026-03-03

How to Cite

Ascher, J., & Schikorra, A. (2026). Carnot-Carathéodory and Korányi-Geodesics in the Heisenberg Group. Mathematics Exchange, 16(1), 85–103. https://doi.org/10.33043/3DEM6HFD3V

Issue

Vol. 16 No. 1 (2022): Mathematics Exchange, Fall 2022

Section

Articles

License

Copyright (c) 2026 Josh Ascher, Armin Schikorra

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This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.