Search

In this paper, we construct a Funk–Finsler structure in various models of the hyperbolic plane. In particular, in the unit disc of the Klein model, it turns out to be a Randers metric, which is a non-Berwald Douglas metric. Further, using Finsler isometries we obtain the Funk–Finsler structures in other models of the hyperbolic plane. Finally, we also investigate the geometry of this Funk–Finsler metric by explicitly computing the S-curvature, Riemann curvature, flag curvature, and the Ricci curvature in the Klein unit disc. [ABSTRACT FROM AUTHOR]

Purpose – In this paper, the authors prove that the Douglas space of second kind with a generalised form of special (α, β)-metric F, is conformally invariant. Design/methodology/approach – For, the authors have used the notion of conformal transformation and Douglas space. Findings – The authors found some results to show that the Douglas space of second kind with certain (α, β)-metrics such as Randers metric, first approximate Matsumoto metric along with some special (α, β)-metrics, is invariant under a conformal change. Originality/value – The authors introduced Douglas space of second kind and established conditions under which it can be transformed to a Douglas space of second kind.

In this article, we study the geodesic orbit Randers spaces of the form ( G / H , F ) {(G/H,F)} , such that G is one of the compact classical Lie groups SO ⁢ ( n ) {{\mathrm{S}}{\mathrm{O}}(n)} , SU ⁢ ( n ) {{\mathrm{S}}{\mathrm{U}}(n)} , Sp ⁢ ( n ) {{\mathrm{S}}{\mathrm{p}}(n)} , and H is a diagonally embedded product H 1 × ⋯ × H s {H_{1}\times\cdots\times H_{s}} , where H i {H_{i}} is of the same type as G. Such spaces include spheres, Stiefel manifolds, Grassmann manifolds, and flag manifolds. The present work is a contribution to the study of geodesic orbit Randers spaces ( G / H , F ) {(G/H,F)} with H semisimple. We construct new examples of non-Riemannian Randers g.o. metrics in homogeneous bundles over generalized Stiefel manifolds which are not naturally reductive. Also, we obtain the specific expressions of these Randers g.o. metrics. [ABSTRACT FROM AUTHOR]

In this paper, we study a new Finslerian quantity T ^ defined by the T-curvature and the angular metric tensor. We show that the T ^ -curvature not only gives a measure of the failure of a Finsler metric to be of scalar flag curvature but also has a vanishing trace. We find that the T ^ -curvature is closely related to the Riemann curvature, the Matsumoto torsion and the Θ-curvature. We solve Z. Shen's open problem in terms of the T ^ -curvature. Finally, we give a global rigidity result for Finsler metrics of negative Ricci curvature on a compact manifold via the T ^ -curvature, generalizing a theorem previously only known in the case of negatively curved Finsler metrics of scalar flag curvature. [ABSTRACT FROM AUTHOR]

The Randers metrics are popular metrics similar to the Riemannian metrics, frequently used in physical and geometric studies. The weak Einstein-Finsler metrics are a natural generalization of the Einstein-Finsler metrics. Our proof shows that if $(M,F)$ is a simply-connected and compact Randers manifold and $F$ is a weak Einstein-Douglas metric, then every special projective vector field is Killing on $(M,F)$. Furthermore, we demonstrate that if a connected and compact manifold $M$ of dimension $n \geq 3$ admits a weak Einstein-Randers metric with Zermelo navigation data $(h,W)$, then either the $S$-curvature of $(M,F)$ vanishes, or $(M,h)$ is isometric to a Euclidean sphere ${\mathbb{S}^n}(\sqrt{k})$, with a radius of $1/\sqrt{k}$, for some positive integer $k$.

The geodesic model based on the eikonal partial differential equation (PDE) has served as a fundamental tool for the applications of image segmentation and boundary detection in the past two decades. However, the existing approaches commonly only exploit the image edge-based features for computing minimal geodesic paths, potentially limiting their performance in complicated segmentation situations. In this paper, we introduce a new variational image segmentation model based on the minimal geodesic path framework and the eikonal PDE, where the region-based appearance term that defines then regional homogeneity features can be taken into account for estimating the associated minimal geodesic paths. This is done by constructing a Randers geodesic metric interpretation of the region-based active contour energy functional. As a result, the minimization of the active contour energy functional is transformed into finding the solution to the Randers eikonal PDE. We also suggest a practical interactive image segmentation strategy, where the target boundary can be delineated by the concatenation of several piecewise geodesic paths. We invoke the Finsler variant of the fast marching method to estimate the geodesic distance map, yielding an efficient implementation of the proposed region-based Randers geodesic model for image segmentation. Experimental results on both synthetic and real images exhibit that our model indeed achieves encouraging segmentation performance. [ABSTRACT FROM AUTHOR]

The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projective Ricci at isotropic S-curvature Randers metrics from a navigation data point of view and conclude that these metrics are weak Einsteinian. [ABSTRACT FROM AUTHOR]

In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown. [ABSTRACT FROM AUTHOR]

Finsler metrics being Ricci-quadratic is a non-Riemannian condition since the Ricci curvature (tensor) is always Ricci-quadratic for Riemannian metrics. In this paper, we introduce the notion of strongly Ricci-quadratic Finsler metrics. We classify strongly Ricci-quadratic Randers metrics expressed in a navigation form. [ABSTRACT FROM AUTHOR]

In the present paper, we study the Randers metric on two-spheres of revolution in order to obtain new families of Finsler of Randers type metrics with simple cut locus. We determine the geodesics behavior, conjugate and cut loci of some families of Finsler metrics of Randers type whose navigation data is not a Killing field and without sectional or flag curvature restrictions. Several examples of Randers metrics whose cut locus is simple are shown.

In this paper, we discuss several Ricci curvature tensors and their relationship with the Ricci curvature and some non-Riemannian quantities. By these Ricci curvature tensors, we shall have a better understanding on the non-Riemannian quantities. [ABSTRACT FROM AUTHOR]

In this article, we have shown that two important pairs of (α, β)-metrics, namely, generalized (α, β)-metric F = μ1β + μ2β + μ3 β2 β, (where μ1, μ2 and μ3 are constants) with Kropina metric and generalized (α, β)-metric with Randers metric, have same Douglas tensors if and only if both are Douglas metrics. Furthermore, we study the projective relation between two important (α, β)-metrics with dimension n ≥ 0, where β and β? are one-forms, while β and β? denote Riemannian metrics. [ABSTRACT FROM AUTHOR]

We consider a geometrical approach to the optimisation problems motivated by transportation system management. Here, we provide a comprehensive account of geometric programming based on the elementary Finsler geometry in R n . Then, we present a Finslerian dynamical model for the nonlinear complementarity problem of traffic equilibrium that can be applied to a variety of equilibrium problems. [ABSTRACT FROM AUTHOR]

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. For this aim, we study the Landsberg curvature of three-dimensional homogeneous Finsler manifolds. First, we express the second Matsumoto torsion of three-dimensional Finsler manifolds, explicitly. Then, we show that the mean Landsberg curvature of three-dimensional homogeneous Finsler manifolds satisfy an ODE. Finally, we prove that every homogeneous 3-dimensional L-reducible Finsler manifold has constant relatively isotropic mean Landsberg curvature if and only if it is a Landsberg metric or a Randers metric of Berwald-type. [ABSTRACT FROM AUTHOR]

We consider a geodesic problem in a manifold endowed with a Randers–Kropina metric. This is a type of a singular Finsler metric arising both in the description of the lightlike vectors of a spacetime endowed with a causal Killing vector field and in the Zermelo’s navigation problem with a wind represented by a vector field having norm not greater than one. By using Lusternik–Schnirelman theory, we prove existence of infinitely many geodesics between two given points when the manifold is not contractible. Due to the type of non-holonomic constraints that the velocity vectors must satisfy, this is achieved thanks to some recent results about the homotopy type of the set of solutions of an affine control system associated with a totally non-integrable distribution. [ABSTRACT FROM AUTHOR]

Due to global warming, wildfires became more severe and frequent than in the past, and more complex mathematics is needed to provide the model of propagation and predict the behavior of the fire. In this work, for a wildfire spreading in some flat land of zero slope, by using Randers metric the equations of wavefronts and wave rays are provided which lead to presenting a more precise and reliable model of propagation. By considering some hypothetical fire in Boa Nova National Park as an example, a simulation is implemented.

In the present work, we study the optimal control paths in the Zermelo navigation problem from the geometric and differential equations point of view rather than the optimal control point of view, where the latter has been carried out in our recent work. Here, we obtain the precise form of the system of ODE where the solutions are optimal trajectories of Zermelo’s navigation problem. Having a precise equation allows optimizing a cost function more accurately and efficiently. The advantage of these equations is to approximate optimal trajectories in the general case by the first order approximation of external fields w. The latter could be solved numerically since we have retrieved simpler equations for these paths.

In order to extend the sphere theorem for Finsler metrics, the concept of reversibility introduced by H-B. Rademacher for a compact Finsler manifold. In this paper, we extend this notion to the general Finsler manifolds. Then we find an upper bound for the reversibility of some important spherically symmetric Finsler metrics. Furthermore, we introduce the concept of sub-reversibility for a general Finsler manifold and obtain a non-zero lower bound for this new quantity. [ABSTRACT FROM AUTHOR]

Every Landsberg metric and every Landsbeg metric is a weakly Landsberg metric, but the converse is not true generally. Let (M; F) be a 3-dimensional Finsler manifold. In this paper, we find a condition under which the notions of weakly Landsberg metric and Landsberg metric are equivalent. [ABSTRACT FROM AUTHOR]

In this article, we are going to discuss the geometry of the navigation problem on a Finsler manifold. We will give proofs for several important local and global results.

On a real vector space $V$, a Randers norm $\hat{F}$ is defined by $\hat{F}=\hat{\alpha}+\hat{\beta}$, where $\hat{\alpha}$ is a Euclidean norm and $\hat{\beta}$ is a covector. We show that the unit sphere $\Sigma$ in the Randers space $(V,\hat{F})$ has positive flag curvature, if and only if $|\hat{\beta}|_{\hat{\alpha}}< (5-\sqrt{17})/2 \approx 0.43845$, thus answering a problem proposed by Prof. Zhongmin Shen. Moreover, we prove that the flag curvature of $\Sigma$ has a universal lower bound $-4$.

• Determining the equations of wavefronts and wave rays by Randers geodesics. • Providing the model of wildfire propagation by Huygens' principle. • Determining the paths through them the fire reaches a special place. • Determining the paths through which the fire spreads faster. Here, by using the validity of the Huygens' envelope principle for wavefronts in Randers spaces, we present some models for wildfire propagation in a flat land under the presence of wind. The wind could be constant or space-dependent. In the models, the height of the propagation is considered, trajectories of fire particles are tracked, and the equations that give the wavefront at each time are provided. Furthermore, we determine the paths of great importance in the process of wildfire management, called strategic paths. These paths can be used to determine the locations of barriers that we make to block the fire and stop it from propagation toward some special area; or to determine the paths through which the fire surrounds, for instance, the transmission lines of the power system. Finally, we provide several interesting examples of spreading the wildfire with different conditions. We implement the examples in the MATLAB environment to solve and provide the models of fire propagations. [ABSTRACT FROM AUTHOR]

Let $G$ be a Lie group equipped with a left-invariant Randers metric $F$. Suppose that $F^v$ and $F^c$ denote the vertical and complete lift of $F$ on $TG$, respectively. We give the necessary and sufficient conditions under which $F^v$ and $F^c$ are generalized Douglas-Weyl metrics. Then, we characterize all 2-step nilpotent Lie groups $G$ such that their tangent Lie groups $(TG, F^c)$ are generalized Douglas-Weyl Randers metrics.

Randers distances are an asymmetric generalization of Riemannian distances, and arise in optimal control problems subject to a drift term, among other applications. We show that Randers eikonal equation can be approximated by a logarithmic transformation of an anisotropic second order linear equation, generalizing Varadhan's formula for Riemannian manifolds. Based on this observation, we establish the convergence of a numerical method for computing Randers distances, from point sources or from a domain's boundary, on Cartesian grids of dimension 2 and 3, which is consistent at order 2/3, and uses tools from low-dimensional algorithmic geometry for best efficiency. We also propose a numerical method for optimal transport problems whose cost is a Randers distance, exploiting the linear structure of our discretization and generalizing previous works in the Riemannian case. Numerical experiments illustrate our results. [ABSTRACT FROM AUTHOR]

In this paper, we study homogeneous geodesics in generalized symmetric spaces equipped with an invariant Randers metric. we give a five dimensional example and we obtain homogeneous geodesics of this space in some special cases. [ABSTRACT FROM AUTHOR]

The aim of the paper is to brief out an (a, ß)-metric and its applications to Psychology. Further, we provide some interesting results on geometry and psychology. [ABSTRACT FROM AUTHOR]

The class of weakly stretch Finsler metrics contains the class of stretch metric. Randers metrics are important Finsler metrics which are defined as the sum of a Riemann metric and a 1-form. In this paper, we prove that every Randers metric with closed and conformal one-form is a weakly stretch metric if and only if it is a Berwald metric. [ABSTRACT FROM AUTHOR]

In this paper we study homogeneous geodesics in five-dimensional generalized symmetric spaces equipped with a left invariant Randers metric and we obtain homogeneous geodesics of this space in some special cases. [ABSTRACT FROM AUTHOR]

The notion of the Ricci curvature is defined for sprays on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. In this paper, we introduce the notion of projectively Ricci-flat sprays. We establish a global rigidity result for projectively Ricci-flat sprays with nonnegative Ricci curvature. Then we study and characterize projectively Ricci-flat Randers metrics. [ABSTRACT FROM AUTHOR]

We characterize generalized Douglas-Weyl Randers metrics in terms of their Zermelo navigation data. Then, we study the Randers metrics induced by some important classes of almost contact metrics. Furthermore, we construct a family of generalized Douglas-Weyl Randers metrics which are not R-quadratic. We show that the Randers metric induced by a Kenmotsu manifold is a Douglas metric which is not of isotropic S-curvature. We show that the Randers metric induced by a Kenmotsu or Sasakian manifold is not Einsteinian. By using D-homothetic deformation of a Kenmotsu or Sasakian manifold, we construct a family of generalized Douglas-Weyl Randers metrics and show that the Lie group of projective transformations does not act transitively on the set of generalized Douglas-Weyl Randers metrics. [ABSTRACT FROM AUTHOR]

One of the most important problems in Finsler geometry is to classify Finsler metrics of scalar flag curvature. In this paper, we study the classification problem of Randers metrics of scalar flag curvature. Under the condition that β is a Killing 1-form, we obtain some important necessary conditions for Randers metrics to be of scalar flag curvature. [ABSTRACT FROM AUTHOR]

The class of generalized P-reducible manifolds (briefly GP-reducible manifolds) was first introduced by Tayebi and his collaborates [1]. This class of Finsler manifolds contains the classes of P-reducible manifolds, C-reducible manifolds and Landsberg manifolds. We prove that every compact GP-reducible manifold with positive or negative character is a Randers manifold. The norm of Cartan torsion plays an important role for studying immersion theory in Finsler geometry. We find the relation between the norm of Cartan torsion, mean Cartan torsion, Landsberg and mean Landsberg curvatures of the class of GP-reducible manifolds. Finally, we prove that every GP-reducible manifold admitting a concurrent vector field reduces to a weakly Landsberg manifold.

In this study, we give the linear condition for the Bertrand curve of a given curve in Finslerian 3- space. Using the Izumiya and Takeuchi's paper as an inspiration, Bertrand curves of Finslerian spherical curves are defined according to Randers metric which is a special Finslerian metric. Also, we obtain the Finsler helix derived from the Finslerian circle according to Randers metric and their visualizations are given by using the Mathematica program. [ABSTRACT FROM AUTHOR]

Present paper deals with the introduction of two special Finsler spaces due to deformations of some special Finsler spaces with the help of (α, β) - metrics. Also, the nonholonomic Finsler frames for product of Matsumoto metric and Kropina metric and also for product of Randers metric and Kropina metric are also obtained. [ABSTRACT FROM AUTHOR]

Here, we find some necessary conditions for a projective vector field on a Randers metric to preserve the non-Riemannian quantities and H. They are known in the contexts as the C-projective and H-projective vector fields. We find all projective vector fields of the Funk type metrics on the Euclidean unit ball Bn(1). [ABSTRACT FROM AUTHOR]

The well-known Cheng–Shen conjecture says that every W -quadratic Randers metric on a closed manifold is a Berwald metric. The class of W -quadratic Randers metrics contains the class of generalized Douglas–Weyl Randers metrics. In this paper, we give a classification of left-invariant Randers metrics of generalized Douglas–Weyl type on three-dimensional Lie groups. Based on our classification theorem, we find a counter-example for the Cheng–Shen conjecture. [ABSTRACT FROM AUTHOR]

In this paper, we study a property of Riemannian and Ricci curvatures under which it reproduces itself, namely, recurrent Finsler metrics. We prove that if (M , F) is a recurrent Finsler manifold of non-zero isotropic flag curvature, then F is a Landsberg metric. It follows that Every positively complete 2-dimensional Randers metric is recurrent if and only if it is a Riemannian or locally Minkowskian metric. Next, we study two positive (or negative) projectively related Ricci parallel Finsler metrics on a compact manifold. We show that the projective equivalence is trivial and then the Riemannian curvatures are equal. In the same vein, we explore the class of Ricci-recurrent Randers metrics with closed and conformal form, and show that the related Riemannian metric is Ricci-recurrent if and only if the Randers metric is a Berwald metric. Finally, we find the necessary and sufficient condition under which a Kropina metric be Ricci-recurrent, provided that its one-form is closed and conformal. [ABSTRACT FROM AUTHOR]

In this paper, we study one of the oldest open problems in Finsler geometry which was introduced by Matsumoto-Shimada in 1977 about the existence of a concrete L -reducible Finsler metric that is not C -reducible. To spot such a Finsler metric, we study the class of spherically symmetric Finsler metrics. We prove two rigidity theorems for spherically symmetric Finsler metrics. First, we prove that every spherically symmetric Finsler metric is semi- C -reducible. Second, we show that every non-Riemannian spherically symmetric Finsler metric is a generalized L -reducible metric. Finally, under a particular condition, we prove that every non-Riemannian L -reducible spherically symmetric Finsler metric on a manifold of dimension n ≥ 3 must be a Randers metric. This result provides a negative answer to Matsumoto-Shimada's problem in the class of spherically symmetric Finsler metrics. [ABSTRACT FROM AUTHOR]

In the present paper, we study the Finsler deformation of some special (α, β)-metric. We determine a nonholonomic Finsler frame for Finsler space with (α, β)-metric F= αeβ/α and Randers change of Matsumoto metric. [ABSTRACT FROM AUTHOR]

We study the cut and conjugate locus of a point in a complete Finsler manifold. Our study focuses on their intersection. We propose the Finsler version of Klingenberg's lemma and determine the structure of the cut and conjugate locus in a Blaschke Finsler manifold. In order to have some examples, we study how cut and conjugate loci change under a projective Randers change. We see the relation between invariance of cut loci and exactness of a closed 1-form in its projective Randers changes. [ABSTRACT FROM AUTHOR]

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the sectional curvature of the hyper-Hermitian metrics have the same sign. [ABSTRACT FROM AUTHOR]

In the year 1998, M. Matsumoto [11] has studied the conditions for some Finsler spaces with (α, β)-metric to be of Douglas type. The purpose of the present paper is devoted to find the various conditions for a Finsler space Fn with a special (α, β)-metric to be a Douglas space for assumptions α² = 0(modβ), in the entire paper. [ABSTRACT FROM AUTHOR]

We introduce a new non-Riemannian quantity named mean stretch curvature. A Finsler metric with vanishing mean stretch curvature is called weakly stretch metric. We prove that every complete P-reducible weakly stretch metric with bounded Cartan torsion is a Landsberg metric. Then, we classify complete weakly stretch surfaces and show that every complete weakly stretch surface is Riemannian or Landsbergian. This provides a natural extension of Szabó's rigidity theorem on Berwald surfaces. [ABSTRACT FROM AUTHOR]

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five equipped with a left invariant Randers metric. We first study geodesic vectors and investigates the set of all homogeneous geodesics. Then we obtain some conditions for left invariant Randers metric to be of Douglas type. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a generalized asymmetric fronts propagation model based on the geodesic distance maps and the Eikonal partial differential equations. One of the key ingredients for the computation of the geodesic distance map is the geodesic metric, which can govern the action of the geodesic distance level set propagation. We consider a Finsler metric with the Randers form, through which the asymmetry and anisotropy enhancements can be taken into account to prevent the fronts leaking problem during the fronts propagation. These enhancements can be derived from the image edge-dependent vector field such as the gradient vector flow. The numerical implementations are carried out by the Finsler variant of the fast marching method, leading to very efficient interactive segmentation schemes. We apply the proposed Finsler fronts propagation model to image segmentation applications. Specifically, the foreground and background segmentation is implemented by the Voronoi index map. In addition, for the application of tubularity segmentation, we exploit the level set lines of the geodesic distance map associated with the proposed Finsler metric providing that a thresholding value is given. [ABSTRACT FROM AUTHOR]