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Research Article | Open Access

Harmonic maps into sub-Riemannian Lie groups

Erlend Grong( ), Irina Markina
Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway
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Abstract

We define harmonic maps between sub-Riemannian manifolds by generalizing known definitions for Riemannian manifolds. We establish conditions for when a horizontal map into a Lie group with a left-invariant metric structure is a harmonic map. We show that sub-Riemannian harmonic maps can be abnormal or normal, just as sub-Riemannian geodesics. We illustrate our study by presenting the equations for harmonic maps into the Heisenberg group.

Keywords

Sub-Riemannian manifoldshorizontal mapsharmonic mapsDarboux derivative
CLC number: 58E20, 53C17

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Communications in Analysis and Mechanics
Volume 15 Issue 3,
September 2023
Pages 515-532
DOI: 10.3934/cam.2023025

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Cite this article:
Grong E, Markina I. Harmonic maps into sub-Riemannian Lie groups. Communications in Analysis and Mechanics, 2023, 15(3): 515-532. https://doi.org/10.3934/cam.2023025

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Received: 20 April 2023
Revised: 07 August 2023
Accepted: 16 August 2023
Published: 15 September 2023
©2023 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)