Geodesic
Invariant harmonic unit vector fields on Lie groups
Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 377-403.

Voir la notice de l'article provenant de la source Biblioteca Digitale Italiana di Matematica

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_a6,
     author = {Gonz\'alez-D\'avila, J. C. and Vanhecke, L.},
     title = {Invariant harmonic unit vector fields on {Lie} groups},
     journal = {Bollettino della Unione matematica italiana},
     pages = {377--403},
     publisher = {mathdoc},
     volume = {Ser. 8, 5B},
     number = {2},
     year = {2002},
     zbl = {1097.53033},
     mrnumber = {1198875},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a6/}
}
TY  - JOUR
AU  - González-Dávila, J. C.
AU  - Vanhecke, L.
TI  - Invariant harmonic unit vector fields on Lie groups
JO  - Bollettino della Unione matematica italiana
PY  - 2002
SP  - 377
EP  - 403
VL  - 5B
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a6/
LA  - en
ID  - BUMI_2002_8_5B_2_a6
ER  - 
%0 Journal Article
%A González-Dávila, J. C.
%A Vanhecke, L.
%T Invariant harmonic unit vector fields on Lie groups
%J Bollettino della Unione matematica italiana
%D 2002
%P 377-403
%V 5B
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a6/
%G en
%F BUMI_2002_8_5B_2_a6
González-Dávila, J. C.; Vanhecke, L. Invariant harmonic unit vector fields on Lie groups. Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 377-403. http://geodesic.mathdoc.fr/item/BUMI_2002_8_5B_2_a6/

[1] E. Abbena-S. Garbiero-L. Vanhecke, Einstein-like metrics on three-dimensional Riemannian homogeneous manifolds, Simon Stevin, 66 (1992), 173-182. | MR | Zbl

[2] J. Berndt-F. Tricerri-L. Vanhecke, Generalized Heisenberg groups and Damek-Ricci harmonic spaces, Lecture Notes in Math., 1598, Springer-Verlag, Berlin, Heidelberg, New York, 1995. | MR | Zbl

[3] E. Boeckx-L. Vanhecke, Harmonic and minimal radial vector fields, Acta Math. Hungar., to appear. | MR | Zbl

[4] E. Boeckx-L. Vanhecke, Harmonic and minimal vector fields on tangent and unit tangent bundles, Differential Geom. Appl., 13 (2000), 77-93. | MR | Zbl

[5] O. Gil-Medrano, Relationship between volume and energy of unit vector fields, Differential Geom. Appl., to appear. | MR | Zbl

[6] O. Gil-Medrano-E. Llinares-Fuster, Minimal unit vector fields, preprint, 1998. | MR | Zbl

[7] J. C. González-Dávila-L. Vanhecke, Examples of minimal unit vector fields, Global Anal. Geom., 18 (2000), 385-404. | MR | Zbl

[8] J. C. González-Dávila-L. Vanhecke, Minimal and harmonic characteristic vector fields on three-dimensional contact metric manifolds, J. Geom., to appear. | MR | Zbl

[9] Y. Haraguchi, Sur une généralisation des structures de contact, Thèse, Univ. du Haute Alsace, Mulhouse, 1981.

[10] A. Kaplan, Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc., 258 (1980), 147-153. | MR | Zbl

[11] J. Milnor, Curvature of left invariant metrics on Lie groups, Adv. in Math., 21 (1976), 293-329. | MR | Zbl

[12] B. O'Neill, Semi-Riemannian geometry with applications to relativity, Academic Press, New York, 1983. | MR | Zbl

[13] K. Tsukada-L. Vanhecke, Invariant minimal unit vector fields on Lie groups, Period. Math. Hungar., to appear. | MR | Zbl

[14] K. Tsukada-L. Vanhecke, Minimality and harmonicity for Hopf vector fields, Illinois J. Math., to appear. | fulltext mini-dml | MR | Zbl

[15] G. Wiegmink, Total bending of vector fields on Riemannian manifolds, Math. Ann., 303 (1995), 325-344. | MR | Zbl

[16] C. M. Wood, On the energy of a unit vector field, Geom. Dedicata, 64 (1997), 319-330. | MR | Zbl