Homogeneous geodesics in homogeneous Riemannian manifolds
by
Stana Nikcevic
University of Belgrade, Yugoslavia

Oral Communication

We study the geodesics which are the orbits of one-parameter groups of isometries (called homogeneous geodesics) on homogeneous Riemannian spaces. There exist special homogeneous Riemannian manifolds on which every geodesic is homogeneous with respect to the largest connected group of isometries. In these spaces, so-called g.o. spaces, we analyze the algebraic structure of certain minimal sets of vectors of the corresponding Lie algebra which generate all geodesics through a fixed point. These sets are called geodesic graphs. (See [1]).

A closely related problem is the following: How many homogeneous geodesics (if any) can be found in a general homogeneous Riemannian manifold? V. V. Kajzer proved that there is at least one homogeneous geodesic on a Lie group with a left-invariant metric. Then O. Kowalski and J. Szenthe [3] generalized this result to the general homogeneous Riemannian manifolds as follows:

Theorem A: Let (M, <, >)=G/H be a homogeneous Riemannian manifold (i.e. G acts transitivelly and effectively as a group of isometries). Then G/H admits at least one homogeneous geodesic through the origin o in M.

Theorem B: If, in addition, the group G is semi-simple, then G/H admits m mutually orthogonal homogeneous geodesics through the origin o, where m=dimM.

Now, some natural problems arise.

Problem 1: Let (M, <, >)=G/H be a homogeneous Riemannian manifold where G denotes the largest connected group of isometries and dimM >= 3. Does M always admit more than one homogeneous geodesic, up to a reparametrization?

Problem 2: Suppose that (M, <, >)=G/H admits m=dimM linearly independent homogeneous geodesics through the origin o. Does it admit m mutually orthogonal homogeneous geodesics?

We show in [2] that the answers to both problems are negative. From our examplis we shall also see that a solvable Lie group with a left invariant metric can admit m mutually orthogonal homogeneous geodesics, whereas there is a homogeneous space (G/H, <, >), where G is neither solvable nor semi-simple, which admits m linearly independent homogeneous geodesics through a point but not m orthogonal ones.

References

[1] O. Kowalski and S. Z. Nikcevi\' c, On geodesic graphs of Riemannian g.o. spaces, Arch. Math. 73 (1999), 223-234.

[2] O. Kowalski, S. Z. Nikcevi\' c and Z. Vl 'asek, Homogenous geodesics in homogeneous Riemannian manifolds - examples, preprint Reihe Mathematik, TU Berlin 665/2000.

[3] O. Kowalski and J. Szenthe, On the existence of homogeneous geodesics in homogeneous Riemannian manifolds, to appear in Geom. Dedicata.

Mathematical Institute, SANU,

Knez Mihailova 35, p.p. 367,

11 000 Beograd,

Yugoslavia

e-mail: stanan@mi.sanu.ac.yu

Date received: May 26, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cadq-58.