Homogeneous Geodesics Research Articles

ON THE EXISTENCE AND EXAMPLES OF HOMOGENEOUS GEODESICS IN GENERALIZED m-KROPINA SPACE

In this paper, we find a necessary and sufficient condition for a non-zero vector to be a geodesic vector in homogeneous generalized m-Kropina space. Further, we prove the existence of at least one homogeneous geodesic. However, it is conjectured that the outcomes and proofs in the case of Finsler geometry are not ideal, since general Finsler metrics are non-reversible. In Finsler geometry, the trajectory of unique homogeneous geodesic should be regarded as two geodesics with initial vectors X and -X. Hence, we construct an (n + 1)-dimensional and a 4-dimensional space to find homogeneous geodesics explicitly.

Homogeneous Geodesics of $4$-dimensional Solvable Lie Groups

We study homogeneous geodesics in $4$-dimensional solvable Lie groups $\mathrm{Sol}_0^4$, $\mathrm{Sol}_1^4$, $\mathrm{Sol}_{m,n}$ and $\mathrm{Nil}_4$.

Geodesics and magnetic curves in the 4-dim almost Kähler model space F4

Abstract We study geodesics and magnetic trajectories in the model space F 4 {{\rm{F}}}^{4} . The space F 4 {{\rm{F}}}^{4} is isometric to the 4-dim simply connected Riemannian 3-symmetric space due to Kowalski. We describe the solvable Lie group model of F 4 {{\rm{F}}}^{4} and investigate its curvature properties. We introduce the symplectic pair of two Kähler forms on F 4 {{\rm{F}}}^{4} . Those symplectic forms induce invariant Kähler structure and invariant strictly almost Kähler structure on F 4 {{\rm{F}}}^{4} . We explore some typical submanifolds of F 4 {{\rm{F}}}^{4} . Next, we explore the general properties of magnetic trajectories in an almost Kähler 4-manifold and characterize Kähler magnetic curves with respect to the symplectic pair of Kähler forms. Finally, we study homogeneous geodesics and homogeneous magnetic curves in F 4 {{\rm{F}}}^{4} .

The minimal number of homogeneous geodesics depending on the signature of the Killing form

The minimal number of homogeneous geodesics depending on the signature of the Killing form

Homogeneous geodesics in sub-Riemannian geometry

We study homogeneous geodesics of sub-Riemannian manifolds, i.e., normal geodesics that are orbits of one-parametric subgroups of isometries. We obtain a criterion for a geodesic to be homogeneous in terms of its initial momentum. We prove that any weakly commutative sub-Riemannian homogeneous space is geodesic orbit, that means all geodesics are homogeneous. We discuss some examples of geodesic orbit sub-Riemannian manifolds. In particular, we show that geodesic orbit Carnot groups are only groups of step 1 and 2. Finally, we get a broad condition for existence of at least one homogeneous geodesic.

Two-step homogeneous geodesics in pseudo-Riemannian manifolds

Given a homogeneous pseudo-Riemannian space $$(G/H,\langle \ , \ \rangle),$$ a geodesic $$\gamma :I\rightarrow G/H$$ is said to be two-step homogeneous if it admits a parametrization $$t=\phi (s)$$ (s affine parameter) and vectors X, Y in the Lie algebra $${\mathfrak{g}}$$ , such that $$\gamma (t)=\exp (tX)\exp (tY)\cdot o$$ , for all $$t\in \phi (I)$$ . As such, two-step homogeneous geodesics are a natural generalization of homogeneous geodesics (i.e., geodesics which are orbits of a one-parameter group of isometries). We obtain characterizations of two-step homogeneous geodesics, both for reductive homogeneous spaces and in the general case, and undertake the study of two-step g.o. spaces, that is, homogeneous pseudo-Riemannian manifolds all of whose geodesics are two-step homogeneous. We also completely determine the left-invariant metrics $$\langle \ ,\ \rangle$$ on the unimodular Lie group $$SL(2,{{\mathbb{R}}})$$ such that $$\big (SL(2,{{\mathbb{R}}}),\langle \ ,\ \rangle \big )$$ is a two-step g.o. space.

Homogeneous geodesics and natural reductivity of homogeneous Gödel-type spacetimes

We prove that all geodesics of homogeneous Gödel-type metrics are homogeneous. This result makes natural to ask whether these spaces are naturally reductive, and a positive answer is provided for all of them through the study of their homogeneous Lorentzian structures.

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous $(\alpha,\beta)$-spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any $(\alpha,\beta)$-homogeneous space. Also, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.

The Existence of Two Homogeneous Geodesics in Finsler Geometry

The existence of a homogeneous geodesic in homogeneous Finsler manifolds was positively answered in previous papers. However, the result is not optimal. In the present paper, this result is refined and the existence of at least two homogeneous geodesics in any homogeneous Finsler manifold is proved. In a previous paper, examples of Randers metrics which admit just two homogeneous geodesics were constructed, which shows that the present result is the best possible.

Homogeneous Randers spaces admitting just two homogeneous geodesics

Homogeneous Randers spaces admitting just two homogeneous geodesics

Riemannian generalized C-spaces with homogeneous geodesics

We investigate homogeneous geodesics in a class of homogeneous spaces G/K' called generalized C-spaces. We give necessary conditions so that a G-invariant metric on G/K' is a g.o. metric.

On homogeneous geodesics and weakly symmetric spaces

In this paper, we establish a sufficient condition for a geodesic in a Riemannian manifold to be homogeneous, i.e. an orbit of an $1$-parameter isometry group. As an application of this result, we provide a new proof of the fact that every weakly symmetric space is geodesic orbit manifold, i.e. all its geodesics are homogeneous. We also study general properties of homogeneous geodesics, in particular, the structure of the closure of a given homogeneous geodesic. We present several examples where this closure is a torus of dimension $\geq 2$ which is (respectively, is not) totally geodesic in the ambient manifold. Finally, we discuss homogeneous geodesics in Lie groups supplied with left-invariant Riemannian metrics.

On Invariant Matsumoto Metrics

In this paper, we consider invariant Matsumoto metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces. We first study the intersection between automorphism and isometry groups of these spaces. Then, we study geodesic vectors and investigate the set of all homogeneous geodesics on homogeneous spaces and nilpotent Lie groups of dimension five.

Variational aspects of homogeneous geodesics on generalized flag manifolds and applications

We study conjugate points along homogeneous geodesics in generalized flag manifolds. This is done by analyzing the second variation of the energy of such geodesics. We also give an example of how the homogeneous Ricci flow can evolve in such way to produce conjugate points in the complex projective space \({\mathbb {C}}P^{2n+1} = \text {Sp}(n+1)/(\text {U}(1)\times \text {Sp}(n))\).

Riemannian M-spaces with homogeneous geodesics

We investigate homogeneous geodesics in a class of homogeneous spaces called $M$-spaces, which are defined as follows. Let $G/K$ be a generalized flag manifold with $K=C(S)=S\times K_1$, where $S$ is a torus in a compact simple Lie group $G$ and $K_1$ is the semisimple part of $K$. Then the {\it associated $M$-space} is the homogeneous space $G/K_1$. These spaces were introduced and studied by H.C. Wang in 1954. We prove that for various classes of $M$-spaces the only g.o. metric is the standard metric. For other classes of $M$-spaces we give either necessary, or necessary and sufficient conditions, so that a $G$-invariant metric on $G/K_1$ is a g.o. metric. The analysis is based on properties of the isotropy representation $\mathfrak{m}=\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the flag manifold $G/K$ (as Ad$(K)$-modules) and corresponding decomposition $\mathfrak{n}=\mathfrak{s}\oplus\mathfrak{m}_1\oplus \cdots\oplus \mathfrak{m}_s$ of the tangent space of the $M$-space $G/K_1$ (as Ad$(K_1)$-modules).

The affine approach to homogeneous geodesics in homogeneous Finsler spaces

In the recent paper [Yan, Z.: Existence of homogeneous geodesics on homogeneous Finsler spaces of odd dimension, Monatsh. Math. 182,1, 165–171 (2017)], it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. However, the proof contains a serious gap. The situation is a bit delicate, because the statement is correct. In the present paper, the incorrect part in this proof is indicated. Further, it is shown that homogeneous geodesics in homogeneous Finsler spaces can be studied by another method developed in earlier works by the author for homogeneous affine manifolds. This method is adapted for Finsler geometry and the statement is proved correctly.

Homogeneous Geodesics in Generalized Wallach Spaces

We classify generalized Wallach spaces which are g.o. spaces. We also investigate homogeneous geodesics in generalized Wallach spaces for any given invariant Riemannian metric and we give some examples.

Riemannian g.o. metrics in certain M-spaces

Let G/K be a generalized flag manifold with K=C(S)=S×K1, where S is a torus in a compact simple Lie group G and K1 is the semisimple part of K. Then the associated M-space is the homogeneous space G/K1. These spaces were introduced and studied by H.C. Wang in 1954. We mainly investigate homogeneous geodesics in M-spaces which correspond to generalized flag manifolds with two isotropy summands and we also give some results for M-spaces corresponding to flag manifolds with at least three isotropy summands, thus extending some previous results. Finally, we prove that there are several M-spaces which admit nonnaturally reductive g.o. metrics.

Homogeneous manifolds whose geodesics are orbits. Recent results and some open problems

A homogeneous Riemannian manifold $(M=G/K, g)$ is called a space with homogeneous geodesics or a $G$-g.o. space if every geodesic $\gamma (t)$ of $M$ is an orbit of a one-parameter subgroup of $G$, that is $\gamma(t) = \exp(tX)\cdot o$, for some non zero vector $X$ in the Lie algebra of $G$. We give an exposition on the subject, by presenting techniques that have been used so far and a wide selection of previous and recent results. We also present some open problems.

Two-step Homogeneous Geodesics in Homogeneous Spaces

We study geodesics of the form $\gamma(t)=\pi(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $\pi:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $\gamma(t)=\pi(\exp (tX))$, $X\in \fr{g}$). We obtain sufficient conditions on a homogeneous space implying the existence of such geodesics for $X,Y\in \fr{m}=T_o(G/K)$. We use these conditions to obtain examples of Riemannian homogeneous spaces $G/K$ so that all geodesics of $G/K$ are of the above form. These include total spaces of homogeneous Riemannian submersions endowed with one parameter families of fiber bundle metrics, Lie groups endowed with special one parameter families of left-invariant metrics, generalised Wallach spaces, generalized flag manifolds, and $k$-symmetric spaces with $k$-even, equipped with certain one-parameter families of invariant metrics.