Invariant harmonic unit vector fields on Lie groups

J. C. González-Dávila; L. Vanhecke

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 377-403
  • ISSN: 0392-4041

Abstract

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We provide a new characterization of invariant harmonic unit vector fields on Lie groups endowed with a left-invariant metric. We use it to derive existence results and to construct new examples on Lie groups equipped with a bi-invariant metric, on three-dimensional Lie groups, on generalized Heisenberg groups, on Damek-Ricci spaces and on particular semi-direct products. In several cases a complete list of such vector fields is given. Furthermore, for a lot of the examples we determine associated harmonic maps from the considered group into its unit tangent bundle equipped with the associated Sasaki metric.

How to cite

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González-Dávila, J. C., and Vanhecke, L.. "Invariant harmonic unit vector fields on Lie groups." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 377-403. <http://eudml.org/doc/194577>.

@article{González2002,
abstract = {We provide a new characterization of invariant harmonic unit vector fields on Lie groups endowed with a left-invariant metric. We use it to derive existence results and to construct new examples on Lie groups equipped with a bi-invariant metric, on three-dimensional Lie groups, on generalized Heisenberg groups, on Damek-Ricci spaces and on particular semi-direct products. In several cases a complete list of such vector fields is given. Furthermore, for a lot of the examples we determine associated harmonic maps from the considered group into its unit tangent bundle equipped with the associated Sasaki metric.},
author = {González-Dávila, J. C., Vanhecke, L.},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {unimodular; non-unimodular Lie groups; Sasaki metric},
language = {eng},
month = {6},
number = {2},
pages = {377-403},
publisher = {Unione Matematica Italiana},
title = {Invariant harmonic unit vector fields on Lie groups},
url = {http://eudml.org/doc/194577},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - González-Dávila, J. C.
AU - Vanhecke, L.
TI - Invariant harmonic unit vector fields on Lie groups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 377
EP - 403
AB - We provide a new characterization of invariant harmonic unit vector fields on Lie groups endowed with a left-invariant metric. We use it to derive existence results and to construct new examples on Lie groups equipped with a bi-invariant metric, on three-dimensional Lie groups, on generalized Heisenberg groups, on Damek-Ricci spaces and on particular semi-direct products. In several cases a complete list of such vector fields is given. Furthermore, for a lot of the examples we determine associated harmonic maps from the considered group into its unit tangent bundle equipped with the associated Sasaki metric.
LA - eng
KW - unimodular; non-unimodular Lie groups; Sasaki metric
UR - http://eudml.org/doc/194577
ER -

References

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  12. O'NEILL, B., Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983. Zbl0531.53051MR719023
  13. TSUKADA, K.- VANHECKE, L., Invariant minimal unit vector fields on Lie groups, Period. Math. Hungar., to appear. Zbl0973.53045MR1805310
  14. TSUKADA, K.- VANHECKE, L., Minimality and harmonicity for Hopf vector fields, Illinois J. Math., to appear. Zbl0997.53040MR1878613
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