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.
In questo lavoro viene presentata una nuova caratterizzazione dei campi vettoriali armonici unitari sui gruppi di Lie dotati di metrica invariante a sinistra. Ciò permette di dedurre risultati di esistenza e nuovi esempi di tali campi, in particolare sui gruppi di Lie con metrica bi-invariante, sui gruppi di Lie di dimensione 3, sui gruppi di Heisenberg generalizzati, sugli spazi di Damek-Ricci e su particolari prodotti semi-diretti. In diversi casi si ottiene l'elenco completo di tutti i campi di questo tipo; in molti esempi vengono determinate le applicazioni armoniche associate, il cui dominio è il gruppo considerato e il codominio è il relativo fibrato tangente unitario, con metrica di Sasaki.
@article{BUMI_2002_8_5B_2_377_0,
author = {J. C. Gonz\'alez-D\'avila and L. Vanhecke},
title = {Invariant harmonic unit vector fields on Lie groups},
journal = {Bollettino dell'Unione Matematica Italiana},
volume = {5-A},
year = {2002},
pages = {377-403},
zbl = {1097.53033},
mrnumber = {1911197},
language = {en},
url = {http://dml.mathdoc.fr/item/BUMI_2002_8_5B_2_377_0}
}
González-Dávila, J. C.; Vanhecke, L. Invariant harmonic unit vector fields on Lie groups. Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002) pp. 377-403. http://gdmltest.u-ga.fr/item/BUMI_2002_8_5B_2_377_0/
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