arXiv research

Classifies special homogeneous curves with polynomial equations.

We show how to specify preferred parameterisations on a homogeneous curve in an arbitrary homogeneous space. We apply these results to limit the natural parameters on distinguished curves in parabolic geometries.

In this paper, we study normal homogeneous Finsler spaces. We first define the notion of a normal homogeneous Finsler space, using the method of isometric submersion of Finsler metrics. Then we study the geometric properties. In particular, we establish a technique to reduce the classification of normal homogeneous Fin…

The paper extends Heintze-Kobayashi-Wolf theory to negatively curved homogeneous Finsler manifolds.

In this paper, we explore the similarity between normal homogeneity and $δ$-homogeneity in Finsler geometry. They are both non-negatively curved Finsler spaces. We show that any connected $δ$-homogeneous Finsler space is $G$-$δ$-homo-geneous, for some suitably chosen connected quasi-compact $G$. So $δ$-homogeneous Fins…

Ricci flow preserves positive sectional curvature on homogeneous spheres

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…

Of concern is the study of the space of curves in homogeneous spaces. Motivated by applications in shape analysis we identify two curves if they only differ by their parametrization and/or a rigid motion. For curves in Euclidean space the Square-Root-Velocity-Function (SRVF) allows to define and efficiently compute a d…

The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and t…

In this paper, we use the flag curvature formula for homogeneous Finsler spaces in our previous work to classify odd dimensional smooth coset spaces admitting positively curved reversible homogeneous Finsler metrics. We will show that the most features of L. Bérard-Bergery's classification results for odd dimensional p…

The paper proves rigidity results for non-positively curved homogeneous Finsler metrics.

Let $G$ be a compact Lie group acting isometrically on a compact Riemannian manifold $M$ with nonempty fixed point set $M^G$. We say that $M$ is fixed-point homogeneous if $G$ acts transitively on a normal sphere to some component of $M^G$. Fixed-point homogeneous manifolds with positive sectional curvature have been c…

Characterizes equigeodesics on specific homogeneous spaces.

In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) Condition), which means that in each tangent plane, we can find a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact …

The $L^p$-cohomology in degree 1 of Riemannian homogeneous spaces is computed. It turns out that reduced cohomology does not vanish exactly for spaces quasiisometric to negatively curved homogeneous spaces.

Researchers classify special curved spheres in a complex space.

The pseudoconvex and disprisoning conditions for geodesics of linear connections are extended to the solution curves of general homogeneous sprays. The main result is that pseudoconvexity and disprisonment are jointly stable in the fine topology on the space of all homogeneous sprays of any degree of homogeneity.

Homogeneous magnetic trajectories in a special linear group proven.

Study on triharmonic curves in 3D spaces, proving their existence and classification.

Rollings of reductive homogeneous spaces are studied using intrinsic curves.

We define a Riemannian structure as a pre-homogeneous geometric structure with curvature R. We show that R=0 if and only if the underlying metric has constant curvature. We define pre-homogeneous geometric structures and pose some problems.

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a …

A classical result of Milman roughly states that every Lipschitz function on $\mathbb{S}^n$ is almost constant on a sufficiently high-dimensional sphere $\mathbb{S}^m\subset \mathbb{S}^n$. In this paper we extend the result by proving that any Lipschitz function on a positively curved homogeneous space is almost consta…

Linear inequality found for certain curved spaces.

We examine homogeneous metrics on spheres and determine which ones have positive sectional curvature. The answer is subtle and surprisingly difficult to prove. In some cases we also determine their pinching constants. This completes the classification of all homogeneous metrics with positive curvature (apart from one s…

We establish metrics of positive $2^\mathrm{nd}$-intermediate Ricci curvature, i.e. $\mathrm{Ric}_2>0$, on products of positively curved homogeneous spaces. Using these examples, we demonstrate that the Hopf conjectures, Petersen-Wilhelm conjecture, Berger fixed point theorem, and Hsiang-Kleiner theorem for positively …

The paper classifies compact homogeneous Finsler manifolds with positive flag curvature.

This paper constructs a continuous decomposition of the Sierpiński curve into acyclic continua one of which is an arc. This decomposition is then used to construct another continuous decomposition of the Sierpiński curve. The resulting decomposition space is homeomorphic to the continuum obtained from taking the Sierpi…

Classifies special quartic curves up to equivalence.

We express invariants of Finsler manifolds in a geometrical way by means of using moving planes and their associated Jacobi curves, which are curves in a fixed homogeneous Grassmann manifold. Some applications are given.

In this paper we study the shape space of curves with values in a homogeneous space $M = G/K$, where $G$ is a Lie group and $K$ is a compact Lie subgroup. We generalize the square root velocity framework to obtain a reparametrization invariant metric on the space of curves in $M$. By identifying curves in $M$ with thei…

This is a survey paper on our recent works concerning the classification of positively curved homogeneous Finsler spaces, and some related topics. At the final part, we present some open problems in this field.

The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…

Let $G$ be a connected Lie group acting locally simply transitively on a manifold $M$. By connecting curves in $M$ we mean the orbits of one-parameter subgroups of $G$. To block a pair of points $m_1,m_2\in M$ is to find a finite set $B\subset M\setminus{m_1,m_2}$ such that every connecting curve joining $m_1$ and $m_2…

In this paper we completely classify the homogeneous two-spheres, especially, the minimal homogeneous ones in the quaternionic projective space $\textbf{HP}^n$. According to our classification, more minimal constant curved two-spheres in $\textbf{HP}^n$ are obtained than Ohnita conjectured in the paper "Homogeneous har…

The paper studies Randers and $(α,β)$ equigeodesics on compact homogeneous manifolds.

We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(…

Totally geodesic dual leaves on curved manifolds are also curved.

We propose definitions of homogeneity and projective equivalence for systems of ordinary differential equations of order greater than two, which allow us to generalize the concept of a spray (for systems of order two). We show that the Euler-Lagrange fields of parametric Lagrangians of order greater than one which are …

Let $G$ be the identity component of the isometry group for an arbitrary curved two-point homogeneous space $M$. We consider algebras of $G$-invariant differential operators on bundles of unit spheres over $M$. The generators of this algebra and the corresponding relations for them are found. The connection of these ge…

The paper extends distributions by singular curves, revealing structural equivalences.

We determine an explicit expression for the Ricci tensor of a K-manifold, that is of a compact Kaehler manifold M with vanishing first Betti number, on which a semisimple group G of biholomorphic isometries acts with an orbit of codimension one. We also prove that the Kaehler form and the Ricci form of M are uniquely d…

Solves equivalence problem for curves in G(2) flag varieties.

This is a slightly altered version of the authors thesis from 2014. In the first main part we show that the quotient space of a compact, simply connected and nonnegatively curved Riemannian 4-manifold by an effective, isometric circle-action admits an approximation in Gromov-Hausdorff topology by smooth, positively cur…

A Finsler geometry may be understood as a homogeneous variational problem, where the Finsler function is the Lagrangian. The extremals in Finsler geometry are curves, but in more general variational problems we might consider extremal submanifolds of dimension $m$. In this minicourse we discuss these problems from a ge…

We investigate contact magnetic curves in the real special linear group of degree 2. They are geodesics of the Hopf tubes over the projection curve. We prove that periodic contact magnetic curves in SL(2,R) can be quantized in the set of rational numbers. Finally, we study contact homogeneous magnetic trajectories in S…

Invariant covariant derivatives on homogeneous spaces are characterized.

Study on triharmonic curves in Sol space with constant curvature and torsion.