| Title:
|
Homogeneous geodesics in a three-dimensional Lie group (English) |
| Author:
|
Marinosci, Rosa Anna |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
43 |
| Issue:
|
2 |
| Year:
|
2002 |
| Pages:
|
261-270 |
| . |
| Category:
|
math |
| . |
| Summary:
|
O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.e\. one geodesic which is an orbit of a one-parameter group of isometries. In [KNV] the related two problems were studied and a negative answer was given to both ones: (1) Let $M=K/H$ be a homogeneous Riemannian manifold where $K$ is the largest connected group of isometries and $\dim M\geq 3$. Does $M$ always admit more than one homogeneous geodesic? (2) Suppose that $M=K/H$ admits $m = \dim M$ linearly independent homogeneous geodesics through the origin $o$. Does it admit $m$ mutually orthogonal homogeneous geodesics? In this paper the author continues this study in a three-dimensional connected Lie group $G$ equipped with a left invariant Riemannian metric and investigates the set of all homogeneous geodesics. (English) |
| Keyword:
|
Riemannian manifold |
| Keyword:
|
homogeneous space |
| Keyword:
|
geodesics as orbits |
| MSC:
|
53C20 |
| MSC:
|
53C22 |
| MSC:
|
53C30 |
| idZBL:
|
Zbl 1090.53038 |
| idMR:
|
MR1922126 |
| . |
| Date available:
|
2009-01-08T19:21:44Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/119318 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
|
[Kw] Kowalski O.: Generalized symmetric spaces.Lecture Notes in Math. 805, Springer-Verlag, Berlin, Heidelberg, New York, 1980. Zbl 0543.53041, MR 0579184 |
| Reference:
|
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| Reference:
|
[KNV] Kowalski O., Nikčević S., Vlášek Z.: Homogeneous geodesics in homogeneous Riemannian manifolds. Examples.Reihe Mathematik, TU Berlin, No. 665/2000 (9 pages). MR 1801906 |
| Reference:
|
[KPV] Kowalski O., Prüfer F., Vanhecke L.: D'Atri spaces.in Topics in Geometry, Birkhäuser, Boston, 1996, pp.241-284. Zbl 0862.53039, MR 1390318 |
| Reference:
|
[KS] Kowalski O., Szenthe J.: On the existence of homogeneous geodesics in homogeneous Riemannian manifolds.Geom. Dedicata 81 (2000), 209-214. Zbl 0980.53061, MR 1772203 |
| Reference:
|
[KV] Kowalski O., Vanhecke L.: Riemannian manifolds with homogeneous geodesics.Boll. Un. Mat. Ital. 5 (1991), 189-246. Zbl 0731.53046, MR 1110676 |
| Reference:
|
[M] Milnor J.: Curvatures of left-invariant metrics on Lie groups.Adv. Math. 21 (1976), 293-329. Zbl 0341.53030, MR 0425012 |
| Reference:
|
[TV] Tricerri F., Vanhecke L.: Homogeneous structures on Riemannian manifolds.London Math. Soc. Lecture Note Series 83, Cambridge Univ. Press, Cambridge, 1983. Zbl 0641.53047, MR 0712664 |
| . |
