Geodesics in Randers spaces of constant curvature

子Randers度量

Sub-Randers metrics

提出子Randers度量，通过添加单形式到子黎曼度量定义，利用庞特里亚金最大值原理导出正规测地线方程，证明Zermelo导航生成正规测地线，并推广Hopf-Rinow定理。

Curvatures of left invariant Randers metric on the five-dimensional Heisenberg group

A left invariant Z-Randers metric on the five-dimensional Heisenberg group is a left invariant Randers metric with deformation vector from the center of the Heisenberg algebra. In this note we prove that for every left invariant Z-Randers metric on the five-dimensional Heisenberg group there exist flags of strictly negative and there exist flags of strictly positive curvatures.

On the flag curvature of invariant Randers metrics

In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric…

Flag Curvature of Invariant Randers Metrics on Homogeneous Manifolds

In this article we review the recent results about the flag curvature of invariant Randers metrics on homogeneous manifolds and by using a counter example we show that the formula which obtained for the flag curvature of these metrics is incorrect. Then we give an explicit formula for the flag curvature of invariant Randers metrics on the naturally reductive homogeneous manifolds $(G/H,g)$, where the Randers metric induced by the invariant Riemannian metric $g$ and an invariant vector field $\tilde{X}$ which is parallel with $g$.

The holonomy group of projectively flat Randers two-manifolds of constant curvature

In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of th…

The geometry of a Randers rotational surface

We study the behavior of geodesics on a Randers surface of revolution. The main tool is the extension of Clairaut relation from Riemannian case to the Randers case. Moreover, we show that our Randers surface of revolution can be embedded in a Minkowski space as hypersurface.

Vanishing S-curvature of Randers spaces

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

On the Projective Algebra of Randers Metrics of Constant Flag Curvature

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=α+β)$ of a Randers space is characterized as a certain Lie subalgebra of the projective algebra $p(M,α)$. Certain subgroups of the projective group $P(M,F)$ and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the…

On the Randers metrics on two-step homogeneous nilmanifolds of dimension five

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left invariant Randers metric of Berwald type has three dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.

On Ricci-Tensor of Randers Metrics

In this paper, we study Randers metrics and find a condition on Ricci tensor of these metrics to be Berwaldian. This generalize Shen's Theorem which says: every R-°at complete Randers metric is locally Minkowskian. Then we find a necessary and sufficient condition on Ricci tensor under which a Randers metric of scalar °ag curvature is of zero °ag curvature.

Randers Metrics of Berwald type on 4-dimensional hypercomplex Lie groups

In the present paper we study Randers metics of Berwald type on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulas for computing flag curvature of these metrics. By this study we construct two 4-dimensional Berwald spaces, one of them has non-negative flag curvature and the other one has non-positive flag curvature.

利用同伦理论研究仿射控制系统解以获得 Randers-Kropina 度量的多条连接测地线

Multiple connecting geodesics of a Randers-Kropina metric via homotopy theory for solutions of an affine control system

本文借助 Lusternik-Schnirelman 理论及受控漂移仿射控制系统解空间的同伦结果，证明不可缩流形上任意两点间存在无穷多条 Randers-Kropina 测地线。

Homogeneous Finsler sphere with constant flag curvature

We prove that a homogeneous Finsler sphere with constant flag curvature $K\equiv1$ and a prime closed geodesic of length $2π$ must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Finsler sphere with $K\equiv1$ must be Randers. Then we discuss the behavior of geodesics on a homogeneous Finsler sphere with $K\equiv1$. We prove that many geodesic properties for homogeneous Randers spheres with $K\equiv1$ can be generalized to the non-Randers case.

Classification of Randers metric of isotropic projective Ricci curvature

In this paper, the concept of isotropic projective Ricci curvature has been investigated. By classification of Randers metric of isotropic projective Ricci curvature, it is shown that Randers metric of projective Ricci curvature is reversible if and only if it is of square projective Ricci curvature.

Integral formulae for codimension-one foliated Randers spaces

Integral formulae for foliated Riemannian manifolds provide obstructions for existence of foliations or compact leaves of them with given geometric properties. This paper continues our recent study and presents new integral formulae and their applications for codimension-one foliated Randers spaces. The goal is a generalization of Reeb's formula (that the total mean curvature of the leaves is zero) and its companion (that twice total second mean curvature of the leaves equals to the total Ricci curvature in the normal direction). We also extend results by Brito, Langevin and Rosenberg (that to…

Nontrivial minimal surfaces in a hyperbolic Randers space

The contribution of this paper is two-fold. The first one is to derive a simple formula of the mean curvature form for a hypersurface in the Randers space with a Killing field, by considering the Busemann-Hausdorff measure and Holmes-Thompson measure simultaneously. The second one is to obtain the explicit local expressions of two types of nontrivial rotational BH-minimal surfaces in a Randers domain of constant flag curvature $K=-1$, which are the first examples of BH-minimal surfaces in the hyperbolic Randers space.

Wind Riemannian spaceforms and Randers metrics of constant flag curvature

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 dropped. Here, the models of WRS spaceforms of constant flag curvature are determined. Indeed, the celebrated classification of Randers metrics of constant flag curvature by Bao, Robles and Shen, extended to the Kropina case in the works by Yoshikawa, Okubo and Sabau, can be used to…

A Riemannian approach to Randers geodesics

In certain circumstances tools of Riemannian geometry are sufficient to address questions arising in the more general Finslerian context. We show that one such instance presents itself in the characterisation of geodesics in Randers spaces of constant flag curvature. To achieve a simple, Riemannian derivation of this special family of curves, we exploit the connection between Randers spaces and the Zermelo problem of time-optimal navigation in the presence of background fields. The characterisation of geodesics is then proven by generalising an intuitive argument developed recently for the sol…

The flag curvature of a submanifold of a Randers-Minkowski space in terms of Zermelo data

The main result of this paper is an expression of the flag curvature of a submanifold of a Randers-Minkowski space $({\mathscr V},F)$ in terms of invariants related to its Zermelo data $(h,W)$. More precisely, these invariants are the sectional curvature and the second fundamental form of the positive definite scalar product $h$ and some projections of the wind $W$. This expression allows for a promising characterization of submanifolds with scalar flag curvature in terms of Riemannian quantities, which, when a hypersurface is considered, seems quite approachable. As a consequence, we prove th…

Stationary-Complete Spacetimes with non-standard splittings and pre-Randers metrics

Using the relativistic Fermat's principle, we establish a bridge between stationary-complete manifolds which satisfy the observer-manifold condition and pre-Randers metrics, namely, Randers metrics without any restriction on the one-form. As a consequence, we give a description of the causal ladder of such spacetimes in terms of the elements associated with the pre-Randers metric: its geodesics and the associated distance. We obtain, as applications of this interplay, the description of conformal maps of Killing submersions, and existence and multiplicity results for geodesics of pre-Randers m…

The Isoperimetric Problem in Randers Poincaré Disc

It is known that a simply connected Riemann surface satisfies the isoperimetric equality if and only if it has constant Gaussian curvature. In this article, we show that Randers Poincaré disc satisfies the isoperimetric equality with respect to different volume forms. These metrics do not necessarily have constant (negative) flag curvature. Thus we show that the Osserman's result \cite{RO} in the Riemannian case can not be extended to the corresponding Finslerian case.

A class of Finsler surfaces whose geodesics are circles

We determine all Finsler metrics of Randers type for which the Riemannian part is a scalar multiple of the Euclidean metric, on an open subset of the Euclidean plane, whose geodesics are circles. We show that the Riemannian part must be of constant Gaussian curvature, and that for every such Riemannian metric there is a class of Randers metrics satisfying the condition, determined up to the addition of a total derivative, depending on a single parameter. As one of several applications we exhibit a Finsler metric whose geodesics are the oriented horocycles in the Poincare disk, in each of the t…

Isoparameteric hypersurfaces in a Randers sphere of constant flag curvature

In this paper, I study the isoparametric hypersurfaces in a Randers sphere $(S^n,F)$ of constant flag curvature, with the navigation datum $(h,W)$. I prove that an isoparametric hypersurface $M$ for the standard round sphere $(S^n,h)$ which is tangent to $W$ remains isoparametric for $(S^n,F)$ after the navigation process. This observation provides a special class of isoparametric hypersurfaces in $(S^n,F)$, which can be equivalently described as the regular level sets of isoparametric functions $f$ satisfying $-f$ is transnormal. I provide a classification for these special isoparametric hype…

Left invariant Randers metrics on the 3-dimensional Heisenberg group

In the previous paper [MR2430243] we computed some geometric quantities such as curvature and flag curvature for a general left invariant Finsler metric on a two-step nilpotent group. In the present paper we give a more complete description of the Chern--Rund connection defined by a left invariant Randers metric on the 3 dimensional Heisenberg group.

Isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces

In this paper, we study the relationship between isoparametric hypersurfaces and hypersurfaces with constant principal curvatures in Finsler spaces. We give some examples of isoparametric hypersurfaces with (non)constant principal curvatures on Randers manifolds with nonconstant flag curvatures. Furthermore, we construct an example of a conformally flat Randers manifold which admits a family of nonisoparametric hyperplanes with constant principal curvatures.

Ricci-quadratic homogeneous Randers spaces

A Finsler space is called Ricci-quadratic if its Ricci curvature $Ric(x,y)$ is quadratic in $y$. It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold $M$. In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of Berwald type.

Geodesic sprays and frozen metrics in rheonomic Lagrange manifolds

We define systems of pre-extremals for the energy functional of regular rheonomic Lagrange manifolds and show how they induce well-defined Hamilton orthogonal nets. Such nets have applications in the modelling of e.g. wildfire spread under time- and space-dependent conditions. The time function inherited from such a Hamilton net induces in turn a time-independent Finsler metric - we call it the associated frozen metric. It is simply obtained by inserting the time function from the net into the given Lagrangean. The energy pre-extremals then become ordinary Finsler geodesics of the frozen metri…

Weighted Projective Ricci Curvature in Finsler Geometry

In this paper, we introduce the weighted projective Ricci curvature as an extension of projective Ricci curvature introduced by Z. Shen. We characterize the class of Randers metrics of weighted projective Ricci flat curvature. We find the necessary and sufficient condition under which a Kropina metric has weighted projective Ricci flat curvature. Finally, we show that every projectively flat metric with isotropic weighted projective Ricci and isotropic S-curvature is a Kropina metric or Randers metric.

Finsler空间上的几乎Ricci孤立子

Almost Ricci solitons on Finsler spaces

本文在Finsler测度空间上引入并研究（梯度）几乎Ricci孤立子，给出其等价刻画，并分类Randers情形，构造了Finsler几何中首批非平凡的梯度Ricci孤立子例子。

Convex regions of stationary spacetimes and Randers spaces. Applications to lensing and asymptotic flatness

By using Stationary-to-Randers correspondence (SRC), a characterization of light and time-convexity of the boundary of a region of a standard stationary (n+1)-spacetime is obtained, in terms of the convexity of the boundary of a domain in a Finsler n or (n+1)-space of Randers type. The latter convexity is analyzed in depth and, as a consequence, the causal simplicity and the existence of causal geodesics confined in the region and connecting a point to a stationary line are characterized. Applications to asymptotically flat spacetimes include the light-convexity of stationary hypersurfaces whi…

四维超复李群上 Douglas 型非 Berwald Randers 度量

Non-Berwaldian Randers metrics of Douglas type on four-dimensional hypercomplex Lie groups

本文分类了四维超复李群上由不变超 Hermitian 度量产生的 Douglas 型非 Berwald Randers 度量，并给出旗曲率公式，证明某些方向上其与截面曲率同号。

Geodesic orbit spheres and constant curvature in Finsler geometry

In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\equiv1$ on homogeneous spheres other than $Sp(n)/Sp(n-1)$.

关于非零旗帜曲率的贝瓦尔德空间

On Berwald Spaces with non-Zero Flag Curvature

该研究证明非零旗帜曲率的贝瓦尔德流形是黎曼流形，并扩展了刚性定理，同时探讨了各向同性贝瓦尔德流形的性质及推广了贝瓦尔德-兰德斯度量的结果。

On the Ricci curvature of homogeneous Finsler spaces with $(α,β)$-metrics

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In this paper, we consider homogeneous Finsler spaces with square metric and Randers change of square metric. First, we derive the explicit formulae for Ricci curvature of homogeneous Finsler spaces with these metrics. Next, we find a necessary and sufficient condition under which a homogeneous Finsler space with either of these metrics is of vanishing $S$-curvature. The formulae for Ricci curvature of homogeneous Finsler spaces with square metric an…

Homogeneous geodesics in homogeneous Finsler spaces

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogeneous geodesics on compact semi-simple Lie group is established. We introduce the notion of naturally reductive homogeneous Finsler space. As a special case, we study homogeneous geodesics in homogeneous Randers spaces. Finally, we study some curvature properties of homogeneous ge…

Recent results on complex Cartan spaces

In this paper, we first provide an updated survey of the geometry of complex Cartan spaces. New characterizations for some particular classes of complex Cartan spaces are pointed out, e.g. Landsberg-Cartan, strongly Berwald-Cartan and others. We introduce the Cartan-Randers spaces which offer examples of Berwald-Cartan and strongly Berwald-Cartan spaces. Then, we investigate the complex geodesic curves of a complex Cartan space, using the image by Legendre transformation ($\mathcal{L}-$ duality) of complex geodesic curves of a complex Finsler space. Assuming the weakly Kähler condition for a c…

Finsler 测地线与电磁轨迹的多重性：基于局部形变

Multiplicity of Finsler geodesics and electromagnetic trajectories via localized deformations

本文通过局部共形形变构造能量势垒，证明 Finsler 流形上存在任意多条连接两点的测地线，并推广到 Randers 度量、电磁轨迹及光线，得到多重性结果。

Complete Finsler spaces of constant negative Ricci curvature

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

Killing frames and S-curvature of homogeneous Finsler spaces

In this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.

On $S$-Curvature of Homogeneous Finsler spaces with $(α, β)$-metrics

The study of curvature properties of homogeneous Finsler spaces with $(α, β)$-metrics is one of the central problems in Riemann-Finsler geometry. In the present paper, the existence of invariant vector fields on homogeneous Finsler spaces with square $(α, β)$-metric and Randers changed square $(α, β)$-metric is proved. Further, an explicit formula for $S$-curvature of these $(α, β)$-metrics is established. Finally, using the formula of $S$-curvature, the mean Berwald curvature of afore said $(α, β)$-metrics is calculated.

关于齐次Kropina空间中齐次测地线的存在性

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

本文研究奇异齐次Kropina空间中齐次测地线的存在性，证明任意齐次Kropina空间通过任意点至少存在一条齐次测地线，并给出Douglas型情形的充要条件及三维李群例子。

Finsler Metrics of Constant Positive Curvature on the Lie Group $s^3$

Guided by the Hopf fibration, we single out a family (indexed by a positive constant K) of right invariant Riemannian metrics on the Lie group $S^3$. Using the Yasuda-Shimada theorem as an inspiration, we determine for each K>1 a privileged right invariant Killing field of constant length. Each such Riemannian metric pairs with the corresponding Killing field to produce a y-global and {\it explicit} Randers metric on $S^3$. Using the machinery of spray curvature and Berwald's formula for it, we prove directly that the said Randers metric has constant positive flag curvature K, as predicted by …

The curvatures of spherically symmetric Finsler metrics in $R^n$

In this paper, we classify the spherically symmetric Berwald metrics in $\mathbb{R}^n$. For the spherically symmetric Landsberg metrics, we prove that there do not exist any non-Berwald metrics among the regular case. The partial differential equation systems which can respectively characterize the spherically symmetric Finsler metrics with constant flag curvature and Einstein metrics of this type is also obtained. Utilizing these equations, we find an effective way to construct the non-projective, non-Randers Finsler metrics with constant flag curvature and many explicit examples are given by…

Finsler spaces with infinite dimensional holonomy group

Our paper is devoted to the study of the holonomy groups of Finsler surfaces using the methods of infinite dimensional Lie theory. The notion of infinitesimal holonomy algebra will be introduced, by the smallest Lie algebra of vector fields on an indicatrix, containing the curvature vector fields and their horizontal covariant derivatives with respect to the Berwald connection. We obtain that the topological closure of the holonomy group contains the exponential image of any tangent Lie algebra of the holonomy group. A class of Randers surfaces is determined, for which the infinitesimal holono…

具有常数量曲率和零高斯曲率的$\mathbb{S}^5$中极小超曲面是全测地的

Minimal hypersurfaces in $\mathbb{S}^5$ with constant scalar curvature and zero Gauss curvature are totally geodesic

本文证明五维球面中常数量曲率且零高斯曲率的闭极小超曲面必为全测地超曲面。

Finsler spheres with constant flag curvature and finite orbits of prime closed geodesics

In this paper, we consider a Finsler sphere $(M,F)=(S^n,F)$ with the dimension $n>1$ and the flag curvature $K\equiv 1$. The action of the connected isometry group $G=I_o(M,F)$ on $M$, together with the action of $T=S^1$ shifting the parameter $t\in \mathbb{R}/\mathbb{Z}$ of the closed curve $c(t)$, define an action of $\hat{G}=G\times T$ on the free loop space $λM$ of $M$. In particular, for each closed geodesic, we have a $\hat{G}$-orbit of closed geodesics. We assume the Finsler sphere $(M,F)$ described above has only finite orbits of prime closed geodesics. Our main theorem claims, if the …

Constant mean curvature surfaces of Delaunay type along a closed geodesic

In this paper, we construct Delaunay type constant mean curvature surfaces along a nondegenerate closed geodesic in a 3-dimensional Riemannian manifold.

Submersions and curves of constant geodesic curvature

Considering Riemannian submersions, we find necessary and sufficient conditions for when sub-Riemannian normal geodesics project to curves of constant first geodesic curvature or constant first and vanishing second geodesic curvatures. We describe a canonical extension of the sub-Riemannian metric and study geometric properties of the obtained Riemannian manifold. This work contains several examples illustrating the results.

Flat Moebius strips of given isotopy types in R^3 whose centerlines are geodesics or lines of curvature

We construct real analytic flat Moebius strips of arbitrary isotopy types, whose centerlines are geodesics or lines of curvature.

Geodesics and curvature of semidirect product groups

Geodesics and curvature of semidirect product groups with right invariant metrics are determined. In the special case of an isometric semidirect product, the curvature is shown to be the sum of the curvature of the two groups. A series of examples, like the magnetic extension of a group, are then considered.

The geodesic total curvature of spherical curves

The geodesic total curvature of rectifiable spherical curves is analyzed. We extend to the case of high dimension spheres the explicit formula that holds true for curves supported into the 2-sphere. For this purpose, we take advantage of some new integral-geometric formulas concerning both the Euclidean and geodesic total curvature of spherical curves.

Average curvatures of Weil-Petersson geodesics in Teichmuller space

We compute curvatures of a three-manifold formed by a Weil-Petersson geodesic in Teichmuller space.

Existence of curves with constant geodesic curvature in a Riemannian 2-sphere

We prove the existence of immersed closed curves of constant geodesic curvature in an arbitrary Riemannian 2-sphere for almost every prescribed curvature. To achieve this, we develop a min-max scheme for a weighted length functional.

A metric interpretation of the geodesic curvature in the Heisenberg group

In this paper we study the notion of geodesic curvature of smooth horizontal curves parametrized by arc lenght in the Heisenberg group, that is the simplest sub-Riemannian structure. Our goal is to give a metric interpretation of this notion of geodesic curvature as the first corrective term in the Taylor expansion of the distance between two close points of the curve.