Harmonic maps into sub-Riemannian Lie groups
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...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
AIMS Press
2023-08-01
|
| Series: | Communications in Analysis and Mechanics |
| Subjects: | |
| Online Access: | https://www.aimspress.com/article/doi/10.3934/cam.2023025?viewType=HTML |
| _version_ | 1797361232071622656 |
|---|---|
| author | Erlend Grong Irina Markina |
| author_facet | Erlend Grong Irina Markina |
| author_sort | Erlend Grong |
| collection | DOAJ |
| description | 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. |
| first_indexed | 2024-03-08T15:50:57Z |
| format | Article |
| id | doaj.art-9e6277b5ebe34635b85dcbbd20743f84 |
| institution | Directory Open Access Journal |
| issn | 2836-3310 |
| language | English |
| last_indexed | 2024-03-08T15:50:57Z |
| publishDate | 2023-08-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Communications in Analysis and Mechanics |
| spelling | doaj.art-9e6277b5ebe34635b85dcbbd20743f842024-01-09T05:55:05ZengAIMS PressCommunications in Analysis and Mechanics2836-33102023-08-0115351553210.3934/cam.2023025Harmonic maps into sub-Riemannian Lie groupsErlend Grong 0Irina Markina1Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, NorwayDepartment of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, NorwayWe 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.https://www.aimspress.com/article/doi/10.3934/cam.2023025?viewType=HTMLsub-riemannian manifoldshorizontal mapsharmonic mapsdarboux derivative |
| spellingShingle | Erlend Grong Irina Markina Harmonic maps into sub-Riemannian Lie groups Communications in Analysis and Mechanics sub-riemannian manifolds horizontal maps harmonic maps darboux derivative |
| title | Harmonic maps into sub-Riemannian Lie groups |
| title_full | Harmonic maps into sub-Riemannian Lie groups |
| title_fullStr | Harmonic maps into sub-Riemannian Lie groups |
| title_full_unstemmed | Harmonic maps into sub-Riemannian Lie groups |
| title_short | Harmonic maps into sub-Riemannian Lie groups |
| title_sort | harmonic maps into sub riemannian lie groups |
| topic | sub-riemannian manifolds horizontal maps harmonic maps darboux derivative |
| url | https://www.aimspress.com/article/doi/10.3934/cam.2023025?viewType=HTML |
| work_keys_str_mv | AT erlendgrong harmonicmapsintosubriemannianliegroups AT irinamarkina harmonicmapsintosubriemannianliegroups |