1. Introduction
Homogeneous spaces are a natural generalization of symmetric spaces and they keep many of their nice properties. One of them is the existence of a transitive group of transformations, which are sometimes called symmetries. The importance of geodesic curves is well known in mathematics and also in physics and homogeneous geodesics are, moreover, orbits of these symmetries. In physics, they are related with relative equilibria. In Riemannian geometry, homogeneous geodesics were studied by many authors and many results were obtained, see the recent survey paper [
1] by the author. In recent years, homogeneous geodesics attained interest in Finsler geometry. In the present paper, we shall focus on the existence of homogeneous geodesics in homogeneous Finsler manifolds and on an interesting phenomenon related with nonreversibility of general Finsler metrics and consequent nonreversibility of homogeneous geodesics.
The existence of at least one homogeneous geodesic in arbitrary homogeneous Riemannian manifold was proved by O. Kowalski and J. Szenthe in [
2]. In the papers [
3,
4], it was proved that this result is optimal, namely, examples of homogeneous Riemannian metrics on solvable Lie groups were constructed which admit just one homogeneous geodesic through any point. Generalization of this existence result to pseudo-Riemannian geometry was proved by the author using a different approach in the broader context of homogeneous affine manifolds in [
5]. This affine approach was used by the author also in [
6] to prove that an even-dimensional Lorentzian manifold admits a light-like homogeneous geodesic.
Generalization of this existence result to Finsler geometry was proved in the series of papers [
7] by Z. Yan and S. Deng for Randers metrics, [
8] by the author for odd-dimensional Finsler metrics, [
9] by the author for Berwald or reversible Finsler metrics, [
10] by Z. Yan and L. Huang in general. In this last paper, an original approach by O. Kowalski and J. Szenthe is modified and a purely Finslerian construction is used. However, due to the nonreversibility of general Finsler metrics, it was conjectured by the author in [
11] that the result and its proofs in the nonreversible situation are not optimal. In comparison with Riemannian geometry, the situation is rather delicate. In the context of Finsler geometry, the trajectory of the unique homogeneous geodesic in a Riemannian manifold should be regarded as two geodesics—they have the same trajectory, their initial vectors are
X and
and they have opposite parametrizations. For a general homogeneous Finsler manifold, the initial vectors of the two homogeneous geodesics may be non-opposite. In the paper [
11], examples of invariant Randers metrics which admit just two homogeneous geodesics were constructed. The initial vectors of these geodesics are
and
, for certain vectors
.
In the present paper, the mentioned proofs are revised and refined. The complete and selfcontained proof of the existence of two homogeneous geodesics through an arbitrary point in arbitrary homogeneous Finsler manifold is given. Some constructions from [
2,
10,
12] are used.
2. Basic Settings
A Minkowski norm on the vector space is a nonnegative function which is smooth on , positively homogeneous ( for any ) and whose Hessian is positively definite on . Variables are the components of a vector with respect to a basis B of and putting to a subscript refers to the partial derivative. The pair is called a Minkowski space. The tensor whose components are is the fundamental tensor. We recall the well known formulas
A Finsler metric on a differentiable manifold M is a function F on which is differentiable on and such that its restriction to any tangent space is a Minkowski norm. The pair is called a Finsler manifold. On a Finsler manifold, functions depend differentiably on and on .
Let M be a Finsler manifold . If some connected Lie group G acts transitively on M by isometries, then M is called a homogeneous manifold. We remark that a homogeneous manifold may admit more presentations as a homogeneous space in the form , corresponding to various transitive isometry groups.
Homogeneous manifold M can be identified with the homogeneous space . Here H is the isotropy group of the origin . A homogeneous Finsler space is a reductive homogeneous space in the following sense: Denote by and the Lie algebras of the groups G and H, respectively, and consider the representation of H on . There exists a reductive decomposition where is a vector subspace with the property . For a fixed reductive decomposition it is natural to identify with the tangent space via the projection . Using this identification, from the Minkovski norm and its fundamental tensor on , we obtain the -invariant Minkowski norm and the -invariant fundamental tensor on .
We further recall the slit tangent bundle
, which is defined as
. Using the restriction of the projection
to
, we construct the pullback vector bundle
over
. The Chern connection is the unique linear connection on
which is torsion free and almost
g-compatible. See some monograph, for example [
13] by D. Bao, S.-S. Chern and Z. Shen or [
14] by S. Deng for details. Using the Chern connection, the derivative along a curve
can be defined. A regular differentiable curve
with tangent vector field
T is a
geodesic if it holds
. In particular, for a geodesic of constant speed it holds
.
A geodesic
through the point
p is homogeneous if it is an orbit of a one-parameter group of isometries. Explicitly, if there exists a nonzero vector
such that
for all
. Such a vector
X is called a geodesic vector. Geodesic vectors are characterized by the geodesic lemma, proved in Riemannian geometry by O. Kowalski and L. Vanhecke in [
15] and generalized to Finsler geometry by D. Latifi in [
16].
Lemma 1 ([
16])
. Let be a homogeneous Finsler space with a reductive decomposition . A nonzero vector is geodesic if and only if it holdswhere the subscript indicates the projection of a vector from to .