arXiv research

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

Geodesic graphs for special Finsler metrics on spheres are studied.

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…

Geodesic orbit metrics proven on specific homogeneous spaces.

Study geodesic orbit metrics in quaternionic Stiefel manifolds.

The paper classifies geodesic orbit spaces with simple isotropy groups.

Classifies geodesic orbit spaces with abelian isotropy subgroups.

Study geodesic orbit metrics on specific homogeneous spaces.

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…

Geodesic orbit property studied for Lorentz manifolds.

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

A Finsler space $(M,F)$ is called a geodesic orbit space if any geodesic of constant speed is the orbit of a one-parameter subgroup of isometries of $(M, F)$. In this paper, we study Finsler metrics on Euclidean spaces which are geodesic orbit metrics. We will show that, in this case $(M, F)$ is a fiber bundle over a s…

Study on homogeneous geodesics in sub-Riemannian geometry.

We prove some rigidity results on geodesic orbit Finsler spaces with non-positive curvature. In particular, we show that a geodesic Finsler space with strictly negative flag curvature must be a non-compact Riemannian symmetric space of rank one.

Counting geodesics on compact symmetric spaces using orbit dimensions and topological data.

Geodesic orbit metrics on real flag manifolds identified.

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

Geodesics spiral around Reeb orbits in 3D contact manifolds.

The study bounds the number of closed geodesics in a specific orbit closure of surfaces.

We study the geodesic orbit property for nilpotent Lie groups $N$ when endowed with a pseudo-Riemannian left-invariant metric. We consider this property with respect to different groups acting by isometries. When $N$ acts on itself by left-translations we show that it is a geodesic orbit space if and only if the metric…

Study geodesic orbit Lorentz nilmanifolds, proving structural properties.

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

This survey explores compact geodesic orbit manifolds and their properties.

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

The paper extends two-step homogeneous geodesics to homogeneous Finsler spaces.

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 …

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

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 …

The paper studies geodesic orbit properties in Finsler spaces.

Study on geodesics of Finsler metrics derived from Riemannian metrics.

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…

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

We study geodesics in generalized Wallach spaces which are expressed as orbits of products of three exponential terms. These are homogeneous spaces $M=G/K$ whose isotropy representation decomposes into a direct sum of three submodules $\frak{m}=\frak{m}_1\oplus\frak{m}_2\oplus\frak{m}_3$, satisfying the relations $[\fr…

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

Geodesic orbit and weakly symmetric properties in spray geometry.

Homoclinic orbits found in geodesic flows on surfaces.

We show that strongly contracting geodesics in Outer space project to parameterized quasigeodesics in the free factor complex. This result provides a converse to a theorem of Bestvina--Feighn, and is used to give conditions for when a subgroup of $\mathrm{Out}(\mathbb{F})$ has a quasi-isometric orbit map into the free …

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…

Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky g…

New families of non-singular geodesic orbit nilmanifolds discovered.

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, …

Study geodesic orbit property on pseudo-Riemannian H-type nilmanifolds.

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…

Study counts and equidistributes geodesic orbits on curved spaces.

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

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.

We study the geodesics on an invariant surface of a three dimensional Riemannian manifold. The main results are: the characterization of geodesic orbits; a Clairaut's relation and its geometric interpretation in some remarkable three dimensional spaces; the local description of the geodesics; the explicit description o…

For positive integers $p$ and $q$ let $G:=\textrm{PSO}(p,q)$ be the projective indefinite special-orthogonal group of signature $(p,q)$. We study counting problems in the Riemannian symmetric space $X_G$ of $G$ and in the pseudo-Riemannian hyperbolic space $\mathbb{H}^{p,q-1}$. Let $S\subset X_G$ be a totally geodesic …