| Title:
|
Harmonic and Minimal Unit Vector Fields on the Symmetric Spaces $G_2$ and $G_2/SO(4)$ (English) |
| Author:
|
Verhóczki, László |
| Language:
|
English |
| Journal:
|
Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica |
| ISSN:
|
0231-9721 |
| Volume:
|
51 |
| Issue:
|
1 |
| Year:
|
2012 |
| Pages:
|
101-109 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The exceptional compact symmetric spaces $G_2$ and $G_2/SO(4)$ admit cohomogeneity one isometric actions with two totally geodesic singular orbits. These singular orbits are not reflective submanifolds of the ambient spaces. We prove that the radial unit vector fields associated to these isometric actions are harmonic and minimal. (English) |
| Keyword:
|
harmonic unit vector field |
| Keyword:
|
minimal unit vector field |
| Keyword:
|
Lie group |
| Keyword:
|
Riemannian symmetric space |
| Keyword:
|
isometric action |
| MSC:
|
53C35 |
| MSC:
|
53C40 |
| MSC:
|
53C42 |
| MSC:
|
53C43 |
| MSC:
|
57S15 |
| idZBL:
|
Zbl 06204924 |
| idMR:
|
MR3060012 |
| . |
| Date available:
|
2012-06-25T08:26:38Z |
| Last updated:
|
2014-03-12 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/142877 |
| . |
| Reference:
|
[1] Berndt, J., Vanhecke, L., Verhóczki, L.: Harmonic and minimal unit vector fields on Riemannian symmetric spaces. Illinois J. Math. 47 (2003), 1273–1286. Zbl 1045.53036, MR 2037003 |
| Reference:
|
[2] Boeckx, E., Vanhecke, L.: Harmonic and minimal radial vector fields. Acta Math. Hungar. 90 (2001), 317–331. Zbl 1012.53040, MR 1910716, 10.1023/A:1010687231629 |
| Reference:
|
[3] Boeckx, E., Vanhecke, L.: Isoparametric functions and harmonic and minimal unit vector fields. In: Fernández, M., Wolf, J. A. (eds.) Global differential geometry: The mathematical legacy of Alfred Gray Contemp. Math., 288, Amer. Math. Soc., Providence, RI, 2001, 20–31. Zbl 1004.53046, MR 1870997 |
| Reference:
|
[4] Gil-Medrano, O., Llinares-Fuster, E.: Minimal unit vector fields. Tôhoku Math. J. 54 (2002), 71–84. Zbl 1006.53053, MR 1878928, 10.2748/tmj/1113247180 |
| Reference:
|
[5] Helgason, S.: Differential geometry, Lie groups, and symmetric spaces. Academic Press, New York, 1978. Zbl 0451.53038, MR 0514561 |
| Reference:
|
[6] Kollross, A.: A classification of hyperpolar and cohomogeneity one actions. Trans. Amer. Math. Soc. 354 (2002), 571–612. Zbl 1042.53034, MR 1862559, 10.1090/S0002-9947-01-02803-3 |
| Reference:
|
[7] Leung, D. S. P.: On the classification of reflective submanifolds of Riemannian symmetric spaces. Indiana Univ. Math. J. 24 (1974), 327–339. Zbl 0296.53039, MR 0367873, 10.1512/iumj.1975.24.24029 |
| Reference:
|
[8] Postnikov, M.: Lectures in Geometry. Semester V. Lie groups and Lie algebras. Mir Publishers, Moscow, 1986. MR 0905471 |
| Reference:
|
[9] Verhóczki, L.: The exceptional compact symmetric spaces $G_2$ and $G_2/SO(4)$ as tubes. Monatsh. Math. 141 (2004), 323–335. Zbl 1058.53041, MR 2053657, 10.1007/s00605-002-0036-8 |
| Reference:
|
[10] Wiegmink, G.: Total bending of vector fields on Riemannian manifolds. Math. Ann. 303 (1995), 325–344. Zbl 0834.53034, MR 1348803, 10.1007/BF01460993 |
| . |
