Harmonic maps into sub-Riemannian Lie groups

Erlend Grong; Irina Markina; Erlend Grong; Irina Markina

doi:10.3934/cam.2023025

Communications in Analysis and Mechanics

2023, Volume 15, Issue 3: 515-532. doi: 10.3934/cam.2023025

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Harmonic maps into sub-Riemannian Lie groups

Erlend Grong ^,,
Irina Markina

Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway

Received: 20 April 2023 Revised: 07 August 2023 Accepted: 16 August 2023 Published: 23 August 2023
58E20, 53C17

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.
- Sub-Riemannian manifolds,
- horizontal maps,
- harmonic maps,
- Darboux derivative
Citation: Erlend Grong, Irina Markina. Harmonic maps into sub-Riemannian Lie groups[J]. Communications in Analysis and Mechanics, 2023, 15(3): 515-532. doi: 10.3934/cam.2023025

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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.

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