arXiv research

The study explores Einstein Kropina metrics on Lie groups and homogeneous spaces.

The study finds homogeneous geodesics in homogeneous Kropina spaces.

Calculates S-curvature and mean Berwald curvature in homogeneous Finsler spaces.

Study flag curvature in homogeneous Finsler spaces with a specific metric.

In the present article we compute the flag curvature of a special type of invariant Kropina metrics on homogeneous spaces.

Using the navigation data (h,W) of a Kropina space, we characterize weakly-Berwald Kropina spaces and Berwald Kropina spaces by means of the Killing vector field W and the parallel vector field W, respectively. Moreover, the local 1-parameter group of Finslerian local isometries of the Kropina space coincides with the …

Study geodesics on strong Kropina spaces, global and local aspects.

Paper finds conditions for generalized Kropina spaces to have constant curvature.

The paper explores conditions for projective and dually flat metrics in Finsler spaces with Kropina changes of mth-root metrics.

Study minimal surfaces in Kropina 3D space, finding only planes as minimal translation surfaces.

The present paper studies globally defined Kropina metrics as solutions of the Zermelo's navigation problem. Moreover, we characterize the Kropina metrics of constant flag curvature showing that up to local isometry, there are only two model spaces of them: the Euclidean space and the odd-dimensional spheres.

In this paper conditions for a Kropina structure to be of constant flag curvature are obtained.

The concept of h-vector was introduced by H. Izumi in 1980. Recently we have obtained the Cartan connection for the Finsler space whose metric is given by Kropina change with an h-vector. In 1985, M. Matsumoto studied the theory of Finsler hypersurface. In this paper, we derive certain geometrical properties of a Finsl…

Study on metrizability and Ricci-flatness of Finsler spaces with Kropina metrics.

Study canonical curves and Kropina metrics in Lagrangian contact geometry.

Einstein-Kropina metrics extend Einstein condition to all signatures and classify Finsler gravity solutions.

In this paper, we discuss the Finsler spaces $(M^n,L)$ and $(M^n,\,^{*}L)$, where $^{*}L(x,y)$ is obtained from $L(x,y)$ by Kropina change $^{*}L(x,y)=\frac{L^2(x,y)}{b_i(x,y)\,y^i}$ and $b^{}_{i}(x,y)$ is an $h$-vector in $(M^n,L)$. We find the necessary and sufficient condition when the Cartan connection coefficients…

The paper studies Kropina metrics with a specific curvature property.

Study investigates metrizability of Finsler spaces with specific metrics.

The paper solves navigation problems on conic Kropina manifolds and establishes curvature relationships.

Study on geodesics in Kropina metrics with applications.

In this paper, a characteristic condition of Einstein Kropina metrics is given. By the characteristic condition, we prove that a non-Riemannian Kropina metric $F=\frac{α^2}β$ with constant Killing form $β$ on an n-dimensional manifold $M$, $n\geq 2$, is an Einstein metric if and only if $α$ is also an Einstein metric. …

In this paper, we consider Kropina change of $m$-th root Finsler metrics. We find necessary and sufficient condition under which the Kropina change of an $m$-th root Finsler metric be locally dually flat. Then we prove that the Kropina change of an $m$-th root Finsler metric is locally projectively flat if and only if …

In this paper, we consider a special class of singular Finsler metrics: $m$-Kropina metrics which are defined by a Riemannian metric and a $1$-form. We show that an $m$-Kropina metric ($m\ne -1$) of scalar flag curvature must be locally Minkowskian in dimension $n\ge 3$. We characterize by some PDEs a Kropina metric ($…

In this paper, a characteristic condition of the projectively flat Kropina metric is given. By it, we prove that a Kropina metric $F=α^2/β$ with constant curvature $K$ and $\|β\|_α=1$ is projectively flat if and only if $F$ is locally Minkowskian.

This paper examines a generalized Kropina metric and its geometric properties.

Study on generalized $m$-Kropina metrics in modified gravity and cosmology.

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we classify a class of singular $(α,β)$-metrics which are locally projectively flat with constant flag curvature in dimension $n= 2$ and $n \ge 3$ respectively. Further, we determine t…

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$ and characterize those which are respectively Douglasian and locally projectively flat in di…

Singular Finsler metrics, such as Kropina metrics and $m$-Kropina metrics, have a lot of applications in the real world. In this paper, we study a class of two-dimensional singular Finsler metrics defined by a Riemann metric $α$ and 1-form $β$, and we characterize those which are Douglasian or locally projectively flat…

In this paper, we find the necessary and sufficient conditions under which two classes of (q, a, b)-metrics are projectively related to a Kropina metric.

Chains in CR geometry are geodesics of a Kropina metric.

The paper studies minimal surfaces in deformed hyperbolic spaces and their properties.

The paper extends Ricci curvature in Finsler geometry and finds conditions for specific metrics.

Study proves existence of multiple geodesics in a specific metric space.

Study examines causal properties of Finsler spacetimes with cone Killing vectors.

Study weakly weighted Einstein-Finsler metrics, showing specific curvature properties and characterizing them.

For a standard Finsler metric F on a manifold M, its domain is the whole tangent bundle TM and its fundamental tensor g is positive-definite. However, in many cases (for example, the well-known Kropina and Matsumoto metrics), these two conditions are relaxed, obtaining then either a pseudo-Finsler metric (with arbitrar…

The paper classifies isoparametric hypersurfaces in conic Finsler spaces.

The paper shows that almost every path structure is not variational.

We give the explicit formulas of the flag curvatures of left invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag cur…

We generalize the Zermelo navigation problem and its solution on Riemannian manifolds $(M, h)$ admitting a space dependence of a ship's speed $0<|u(x)|_h\leq1$ in the presence of a perturbation $\tilde{W}$ determined by a strong velocity vector field satisfying $|\tilde{W}(x)|_h=|u(x)|_h$, with application of Finsler m…

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. Recently, we associated a new Riemannian metric to a codimension-one foliated Finsler space and proved integral formulae for general and for Randers spaces. In …

The paper studies special metrics on tangent Lie groups.

Paper proves conformal transformation of special $(α, β)$-metrics to Douglas spaces.

Introduces new types of homogeneous spaces and their properties.

Recently, wind Riemannian structures (WRS) have been introduced as a generalization of Randers and Kropina metrics. They are constructed from the natural data for Zermelo navigation problem, namely, a Riemannian metric $g_R$ and a vector field $W$ (the wind), where, now, the restriction of mild wind $g_R(W,W)<1$ is dro…

Survey of recent results on homogeneous finite-dimensional spaces.