Homogeneous geodesics in a three-dimensional Lie group

Rosa Anna Marinosci

Commentationes Mathematicae Universitatis Carolinae (2002)

  • Volume: 43, Issue: 2, page 261-270
  • ISSN: 0010-2628

Abstract

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O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.eȯne 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 ≥ 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.

How to cite

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Marinosci, Rosa Anna. "Homogeneous geodesics in a three-dimensional Lie group." Commentationes Mathematicae Universitatis Carolinae 43.2 (2002): 261-270. <http://eudml.org/doc/248985>.

@article{Marinosci2002,
abstract = {O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.eȯne 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 $\dim M\ge 3$. 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.},
author = {Marinosci, Rosa Anna},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {Riemannian manifold; homogeneous space; geodesics as orbits; homogeneous Riemannian manifold; 3-dimensional Lie group; invariant metric; homogeneous geodesic},
language = {eng},
number = {2},
pages = {261-270},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Homogeneous geodesics in a three-dimensional Lie group},
url = {http://eudml.org/doc/248985},
volume = {43},
year = {2002},
}

TY - JOUR
AU - Marinosci, Rosa Anna
TI - Homogeneous geodesics in a three-dimensional Lie group
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2002
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 43
IS - 2
SP - 261
EP - 270
AB - O. Kowalski and J. Szenthe [KS] proved that every homogeneous Riemannian manifold admits at least one homogeneous geodesic, i.eȯne 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 $\dim M\ge 3$. 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.
LA - eng
KW - Riemannian manifold; homogeneous space; geodesics as orbits; homogeneous Riemannian manifold; 3-dimensional Lie group; invariant metric; homogeneous geodesic
UR - http://eudml.org/doc/248985
ER -

References

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  2. Kajzer V.V., Conjugate points of left-invariant metrics on Lie groups, J. Soviet Math. 34 (1990), 32-44. (1990) Zbl0722.53049MR1106314
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  7. Kowalski O., Nikčević S., Vlášek Z., Homogeneous geodesics in homogeneous Riemannian manifolds. Examples, Reihe Mathematik, TU Berlin, No. 665/2000 (9 pages). MR1801906
  8. Kowalski O., Prüfer F., Vanhecke L., D'Atri spaces, in Topics in Geometry, Birkhäuser, Boston, 1996, pp.241-284. Zbl0862.53039MR1390318
  9. Kowalski O., Szenthe J., On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, Geom. Dedicata 81 (2000), 209-214. (2000) Zbl0980.53061MR1772203
  10. Kowalski O., Vanhecke L., Riemannian manifolds with homogeneous geodesics, Boll. Un. Mat. Ital. 5 (1991), 189-246. (1991) Zbl0731.53046MR1110676
  11. Milnor J., Curvatures of left-invariant metrics on Lie groups, Adv. Math. 21 (1976), 293-329. (1976) Zbl0341.53030MR0425012
  12. Tricerri F., Vanhecke L., Homogeneous structures on Riemannian manifolds, London Math. Soc. Lecture Note Series 83, Cambridge Univ. Press, Cambridge, 1983. Zbl0641.53047MR0712664

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