arXiv research

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…

Study on homogeneous geodesics in sub-Riemannian geometry.

Minimal number of geodesics in Finsler manifolds with indefinite Killing form is at least four.

The paper extends two-step homogeneous geodesics to homogeneous Finsler 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 $(α,β)$-spaces, specially, homogeneous Kropina spaces. We show that any homoge…

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 ho…

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous pseudo-Riemannian manifold admits a homogeneous geodesic through arbitrary point. …

Classifies totally geodesic submanifolds in specific geometric spaces.

In this paper, we study geodesics and geodesic vectors for homogeneous exponential Finsler space and homogeneous infinite series Finsler space. Further, we find necessary and sufficient condition for a non-zero vector in these homogeneous spaces to be a geodesic vector.

Homogeneous magnetic paths found in Heisenberg space.

We prove that the Penrose limit of a spacetime along a homogeneous geodesic is a homogeneous plane wave spacetime and that the Penrose limit of a reductive homogeneous spacetime along a homogeneous geodesic is a Cahen--Wallach space. We then consider several homogenous examples to show that these results are indeed sha…

Geodesic orbit metrics proven on specific homogeneous spaces.

In a recent paper, it was claimed that any homogeneous Finsler space of odd dimension admits a homogeneous geodesic through any point. For the proof, the algebraic method dealing with the reductive decomposition of the Lie algebra of the isometry group was used. However, the proof contains a serious gap. In the present…

Study geodesic complexity in homogeneous Riemannian manifolds.

Study finds homogeneous spaces with geodesic orbits but no integrable distributions.

Geodesic orbit and weakly symmetric properties in spray geometry.

In this paper, we study the set of homogeneous geodesics of a leftinvariant Finsler metric on Lie groups. We first give a simple criterion that characterizes geodesic vectors. As an application, we study some geometric properties of bi-invariant Finsler metrics on Lie groups. In particular a necessary and sufficient co…

We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2π$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Fin…

Study shows magnetic trajectories in Berger spheres are homogeneous.

We study geodesics of the form $γ(t)=π(\exp(tX)\exp(tY))$, $X,Y\in \fr{g}=\operatorname{Lie}(G)$, in homogeneous spaces $G/K$, where $π:G\rightarrow G/K$ is the natural projection. These curves naturally generalise homogeneous geodesics, that is orbits of one-parameter subgroups of $G$ (i.e. $γ(t)=π(\exp (tX))$, $X\in …

Classifies totally geodesic submanifolds in Hopf-Berger spheres.

Homogeneous magnetic trajectories in a special linear group proven.

We show that a simply connected Riemannian homogeneous space M which admits a totally geodesic hypersurface F is isometric to either (a) the Riemannian product of a space of constant curvature and a homogeneous space, or (b) the warped product of the Euclidean space and a homogeneous space, or (c) the twisted product o…

We classify totally geodesic and parallel hypersurfaces of four-dimensional non-reductive homogeneous pseudo-Riemannian manifolds.

The paper studies geodesic orbit properties in Finsler spaces.

Study geodesic orbit metrics on specific homogeneous spaces.

This paper provides a characterization of homogeneous curves on a geometric flag manifold which are geodesic with respect to any invariant metric. We call such curves homogeneous equigeodesics. We also characterize homogeneous equigeodesics whose associated Killing field is closed, hence, the corresponding geodesics is…

In this paper we consider some properties of the three-dimensional homogeneous SO(2)-isotropic Riemannian manifolds. In particular, we determine the geodesics, the totally geodesic surfaces, the totally umbilical surfaces and the geodesics of the rotational surfaces.

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^…

Found a 6D Lie group with a closed geodesic.

Study geodesic orbit metrics in quaternionic Stiefel manifolds.

Two metrics on a manifold are geodesically equivalent if sets of their unparameterized geodesics coincide. In this paper we show that if two left $G$-invariant metrics of arbitrary signature on homogenous space $G/H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connecti…

Totally geodesic submanifolds in hyperbolic space up to codimension two.

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.

Derives smooth homogeneous structures for low-rank tensors.

The paper classifies geodesic orbit spaces with simple isotropy groups.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

We prove that a normal homogeneous space with the property that every Jacobi field along a geodesic vanishing at two points is the restriction of a Killing field along that geodesic is a globally symmetric space.

Let $ξ=(G\times_{K} \mathcal{G} / \mathcal{K}, ρ_ξ, \emph{G} / \emph{K},\mathcal{G} / \mathcal{K})$ be the associated bundle and $τ_{G/K}=(T_{G/K},π_{G/K},G/K, \textrm{R}^{m})$ be the tangent bundle of special examples of odd dimension solvable Lie groups equipped with left invariant Riemannian metric. In this paper we…

In this paper we discuss general properties of geodesic surfaces that are locally biLipschitz homogeneous. In particular, we prove that they are locally doubling and that there exists a special doubling measure analogous to the Haar measure for locally compact groups.

An affine manifold is said to be geodesically complete if all affine geodesics extend for all time. It is said to be affine Killing complete if the integral curves for any affine Killing vector field extend for all time. We use the solution space of the quasi-Einstein equation to examine these concepts in the setting o…

A homogeneous Riemannian manifold $(M=G/K, g)$ is called a space with homogeneous geodesics or a $G$-g.o. space if every geodesic $γ(t)$ of $M$ is an orbit of a one-parameter subgroup of $G$, that is $γ(t) = \exp(tX)\cdot o$, for some non zero vector $X$ in the Lie algebra of $G$. We give an exposition on the subject, …

Characterizes equigeodesics on specific homogeneous spaces.

New upper bound for geodesic complexity derived from cut locus decompositions.

New compact ECS manifolds with rank 2 discovered, differing from previous rank 1 examples.

Study shows nontrivial intersections of subgroups on homogeneous spaces.

When a closed Finsler manifold admits continuous isometric actions, estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics. To generalize the works of H. Duan, Y. Long, H.B. Rademacher, W. Wang and others on the existence of two…

We study the family of $α$-connections of Amari-Chentsov on the homogeneous space $\mathcal{D}(M)/\mathcal{D}_μ(M)$ of diffeomorphisms modulo volume-preserving diffeomorphims of a compact manifold $M$. We show that in some cases their geodesic equations yield completely integrable Hamiltonian systems.