arXiv research

A geodesic orbit manifold is a complete Riemannian manifold all of whose geodesics are orbits of one-parameter groups of isometries. We give both a geometric and an algebraic characterization of geodesic orbit manifolds that are diffeomorphic to $\mathbf{R}^n$. Along the way, we establish various structural properties …

Study geodesic orbit metrics in quaternionic Stiefel manifolds.

Geodesic orbit and weakly symmetric properties in spray geometry.

Geodesic orbit metrics on real flag manifolds identified.

Study Einstein Lie groups and geodesic orbit manifolds, finding some are not geodesic orbit.

Study geodesic orbit metrics on specific homogeneous spaces.

Geodesic orbit property studied for Lorentz manifolds.

Classifies geodesic orbit spaces with abelian isotropy subgroups.

Study geodesic orbit Lorentz nilmanifolds, proving structural properties.

Study on homogeneous geodesics in sub-Riemannian geometry.

This survey explores compact geodesic orbit manifolds and their properties.

Researchers extend geodesic orbit properties to pseudo-Riemannian nilmanifolds of specific signature.

New families of non-singular geodesic orbit nilmanifolds discovered.

Proves a quantitative closing lemma for negatively curved manifolds.

This paper classifies geodesic orbit metrics on compact Lie group $G_2$.

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

Geodesics spiral around Reeb orbits in 3D contact manifolds.

Study geodesics on $SL(n, \mathbb{R})$ adjoint orbits.

We study cohomogeneity one Riemannian manifolds and we establish some simple criterium to test when a singular orbit is totally geodesic. As an application, we classify compact, positively curved Riemannian manifolds which are acted on isometrically by a non semisimple Lie group with an hypersurface orbit.

Geodesic orbit spaces and their families are studied in pseudo-Riemannian manifolds.

Study isotropic embeddings of Lie group orbits and their application to magnetic geodesic flows.

The study of solutions with fixed energy of certain classes of Lagrangian (or Hamiltonian) systems is reduced, via the classical Maupertuis--Jacobi variational principle, to the study of geodesics in Riemannian manifolds. We are interested in investigating the problem of existence of brake orbits and homoclinic orbits,…

The paper finds infinitely many magnetic geodesics on non-compact manifolds.

Paper finds periodic orbits for convex Lagrangian systems on noncompact manifolds.

We prove the equidistribution of (weighted) periodic orbits of the geodesic ow on noncompact negatively curved manifolds toward equilibrium states in the narrow topology, i.e. in the dual of bounded continuous functions. We deduce an exact asymptotic counting for periodic orbits (weighted or not), which was previously …

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 this paper we prove that the compact Lie group $G_2$ admits a left-invariant Einstein metric that is not geodesic orbit. In order to prove the required assertion, we develop some special tools for geodesic orbit Riemannian manifolds. It should be noted that a suitable metric is discovered in a recent paper by I. Chr…

A geodesic orbit manifold (GO manifold) is a Riemannian manifold (M,g) with the property that any geodesic in M is an orbit of a one-parameter subgroup of a group G of isometries of (M,g). The metric g is then called a G-GO metric in M. For an arbitrary compact homogeneous manifold M=G/H, we simplify the general proble…

We survey some results on the existence (and non-existence) of periodic Reeb orbits on contact manifolds, both in the open and closed case. We place these statements in the context of Finsler geometry by including a proof of the folklore theorem that the Finsler geodesic flow can be interpreted as a Reeb flow. As a mil…

The goal of this paper is to clarify connections between Killing fields of constant length on a Rimannian geodesic orbit manifold $(M,g)$ and the structure of its full isometry group. The Lie algebra of the full isometry group of $(M,g)$ is identified with the Lie algebra of Killing fields $\mathfrak{g}$ on $(M,g)$. We…

Proves existence of many non-$\mathbb R$-covered Anosov flows on hyperbolic 3-manifolds.

Geodesic graphs for special Finsler metrics on spheres are studied.

The paper classifies Sasaki-Einstein orbits in compact Hermitian symmetric spaces.

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\mathrm{PSL}_2(\mathbb{Z})\backslash\mathrm{PSL}_2(\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus has a well-defined volume. We present strong nu…

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…

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. In particular, we discuss some important totally geodesic submanifolds that inherit the property to be geodesic orbit. For a given geodesic…

The paper classifies geodesic orbit spaces with simple isotropy groups.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

The study constructs families of nilpotent Lie groups with geodesic orbit metrics.

Researchers classify geodesic orbit spaces for compact Lie groups of rank two.

We investigate the rudiments of Riemannian geometry on orbit spaces $M/G$ for isometric proper actions of Lie groups on Riemannian manifolds. Minimal geodesic arcs are length minimising curves in the metric space $M/G$ and they can hit strata which are more singular only at the end points. This is phrased as convexity …

We produce infinitely many examples of Anosov flows in closed 3-manifolds where the set of periodic orbits is partitioned into two infinite subsets. In one subset every closed orbit is freely homotopic to infinitely other closed orbits of the flow. In the other subset every closed orbit is freely homotopic to only one …

Homoclinic orbits found in geodesic flows on surfaces.

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, …

Geodesic orbit metrics proven on specific homogeneous spaces.

The paper adapts results for Reeb flows and Hamiltonian flows, showing all orbits are closed have identical periods.

The paper is devoted to the study of geodesic orbit Riemannian spaces that could be characterize by the property that any geodesic is an orbit of a 1-parameter group of isometries. The main result is the classification of compact simply connected geodesic orbit Riemannian spaces $G/H$ with two irreducible submodules in…

Study periodic geodesics on contact 3D manifolds, proving existence and precise properties.