The index of a geodesic in a Randers space and some remarks about the lack of regularity of the energy functional of a Finsler metric
Erasmo Caponio
备注：
Contribution to the proceedings of "Workshop on Finsler geometry and its applications", Debrecen, 24--29 May, 2009. 8 pages, AMSLaTex. v2 minor revision: typos fixed, references updated
In some recent papers, the relations existing between the metric properties of Randers spaces and the conformal geometry of stationary Lorentzian manifolds were discovered and investigated. In this note, we focus on the equality between the index of a geodesic in a Randers space and that of its lightlike lift in the associated conformal stationary spacetime. Moreover we make some remarks about regularity of the energy functional of a Finsler metric on the infinite dimensional manifold of $H^1$ curves between two points, in connection with infinite dimensional techniques in Morse Theory.
In this paper, we consider a Finsler space with a Randers change of Quartic metric F = $\sqrt[4]{α^4 + β^4} + β$. The conditions for this space to be with reversible geodesics are obtained. Further, we study some geometrical properties of F with reversible geodesics and prove that the Finsler metric F induces a generalized weighted quasi-distance $d_F$ on M.
In this paper, we consider Randers change of some special $ (α, β)- $ metrics. First we find the fundamental metric tensor and Cartan tensor of these Randers changed $ (α, β)- $metrics. Next, we establish a general formula for inverse of fundamental metric tensors of these metrics. Finally, we find the necessary and sufficient conditions under which the Randers change of these $ (α, β)- $ metrics are projectively and locally dually flat.
If a non-reversible Finsler norm is the sum of a reversible Finsler norm and a closed 1-form, then one can uniquely recover the 1-form up to potential fields from the boundary distance data. We also show a boundary rigidity result for Randers metrics where the reversible Finsler norm is induced by a Riemannian metric which is boundary rigid. Our theorems generalize Riemannian boundary rigidity results to some non-reversible Finsler manifolds. We provide an application to seismology where the seismic wave propagates in a moving medium.
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.
In the year 1984 Shibata investigated the theory of a change which is called a $ β$-change of a Finsler metric. On the other hand in 1985 a systematic study of geometry of hypersurfaces in Finsler spaces was given by Matsumoto. In the present paper is to devoted to the study of a condition for a Randers conformal change to be projective and find out when a totally geodesic hypersurface $ F^{n-1} $ remains to be a totally geodesic hypersurface $ F^{n-1} $ under the projective Randers conformal change. Further obatined the condition under which a Finslerian hypersurfaces given by the projective …
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.
A linear connection on a Finsler manifold is called compatible to the metric if its parallel transports preserve the Finslerian length of tangent vectors. Generalized Berwald manifolds are Finsler manifolds equipped with a compatible linear connection. Since the compatibility to the Finslerian metric does not imply the unicity of the linear connection in general, the first step of checking the existence of compatible linear connections on a Finsler manifold is to choose the best one to look for. A reasonable choice is introduced in \cite{V14} called the extremal compatible linear connection, w…
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two points $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we give a complete classification of connected simply connected Clifford-Wolf homogeneous Randers spaces.
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…
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.
A Clifford-Wolf translation of a connected Finsler space is an isometry which moves each point the sam distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous if for any two point $x_1, x_2\in M$ there is a Clifford-Wolf translation $ρ$ such that $ρ(x_1)=x_2$. In this paper, we study Clifford-Wolf translations of left invariant Randers metrics on compact Lie groups. The mian result is that a left invariant Randers metric on a connected compact simple Lie group is Clifford-Wolf homogeneous if and only if the indicatrix of the metric is a round sphere with respect to a bi-invarian…
In complex Finsler geometry, an open problem is: does there exist a weakly Kähler Finsler metric which is not Kähler? In this paper, we give an affirmative answer to this open problem. More precisely, we construct a family of the weakly Kähler Finsler metrics which are non-Kähler. The examples belong to the unitary invariant complex Randers metrics. Furthermore, a uniformization theorem of the unitary invariant complex Randers metrics with constant holomorphic curvature is proved under the weakly Kähler condition.
In this paper, it is proved that any conformal vector field is homothetic on a locally projectively flat $(α,β)$-space of non-Randers type in dimension $n\ge 3$, and the local solutions of such a vector field are determined. While on a locally projectively flat Randers space, examples showthat the conformal vector fields are not necessarily homothetic.
We obtain some results in both Lorentz and Finsler geometries, by using a correspondence between the conformal structure (Causality) of standard stationary spacetimes on $M=\R\times S$ and Randers metrics on $S$. In particular, for stationary spacetimes, we give a simple characterization of when they are causally continuous or globally hyperbolic (including in the latter case, when $S$ is a Cauchy hypersurface), in terms of an associated Randers metric. Consequences for the computability of Cauchy developments are also derived. Causality suggests that the role of completeness in many results o…
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.
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…
We consider remodeling the planar search patterns, in the presence of the river-type perturbation represented by the weak vector field, basing on the time-optimal paths as Finslerian solutions to the Zermelo navigation problem via Randers metric.
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…
We investigate the relation between weighted quasi-metric Spaces and Finsler Spaces. We show that the induced metric of a Randers space with reversible geodesics is a weighted quasi-metric space.
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.
We show that in dimension 2 every Finsler metric with at least 3-dimensional Lie algebra of projective vector fields is locally projectively equivalent to a Randers metric. We give a short list of such Finsler metrics which is complete up to coordinate change and projective equivalence.
In this paper, we investigate the spherically symmetric Finsler metrics with isotropic S-curvature and obtain a characterized equation. As an application, we prove that these metrics with Douglas type must be Randers metrics or Berwald metrics. This result leads to two classification theorems.
We proof that in dimension two, a Finsler metric is Douglas and generalized Berwald, if and only if it is Berwald or a Randers metric $α+ β$, where $β$ is closed and is of constant length with respect to $α$.
Finsler metrics of scalar flag curvature play an important role to show the complexity and richness of general Finsler metrics. In this paper, on an $n$-dimensional manifold $M$ we study the Finsler metric $F=F(x,y)$ of scalar flag curvature ${\bf K} = {\bf K}(x,y)$ and discover some equations ${\bf K}$ should be satisfied. As an application, we mainly study the metric $F$ of weakly isotropic flag curvature ${\bf K} = \frac{3 θ}{F} + σ$, where $θ=θ_i(x) y^i \neq 0$ is a $1$-form and $σ=σ(x)$ is a scalar function. We prove that in this case, $F$ must be a Randers metric when $dim(M) \geq 3$. Fu…
In this paper, I will show how to use beta-deformations to deal with dual flatness of Randers metrics. beta-deformations is a new method in Riemann-Finsler geometry, it is introduced by the author(see arxiv:1209.0845). Later on I will provide more applications of the new kind of deformations in Finsler geometry.
Here, it is introduced a concept of convolution metric in Finslerian Geometry. This convolution metric is a kind of function obtained by a given mathematical operation between two Finslerian metrics. Some basic properties of the Finslerian convolution metrics are studied. Then it is characterized Finslerian convolution metrics which are of type Riemannian, Minkowskian as well as Randers. Furthermore, some examples of the Finslerian convolutions are given.
In this paper we introduce in study the projectively related complex Finsler metrics. We prove the complex versions of the Rapcsák's theorem and characterize the weakly Kähler and generalized Berwald projectively related complex Finsler metrics. The complex version of Hilbert's Fourth Problem is also pointed out. As an application, the projectiveness of a complex Randers metric is described.
In this paper we argue that when gauge invariance is taken into consideration, there is no consistent geometric framework of Finsler class that can accommodate Randers type spaces. In this context, an alternative non-Finslerian framework for Randers spacetimes compatible with gauge invariance is introduced.
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.
In this paper, we prove two rigidity results for non-positively curved homogeneous Finsler metrics. Our first main result yields an extension of Hu-Deng's well-known result proven for the Randers metrics. Indeed, we prove that every connected homogeneous Finsler space with non-positive flag curvature and isotropic S-curvature is Riemannian or locally Minkowskian. We extend the Szabó's rigidity theorem for Berwald surfaces and show that homogeneous isotropic Berwald metrics with non-positive flag curvature are Riemannian or locally Minkowskian. We prove that a homogeneous $(α, β)$-metrics has i…
A smooth curve on $G/H$ is called a Riemannian equigeodesic if it is a homogeneous geodesic for all $G$-invariant Riemannian metrics on $G/H$. With the $G$-invariant Riemannian metric replaced by other classes of $G$-invariant metrics, we can similarly define Finsler equigeodesic, Randers equigeodesic, $(α,β)$ equigeodesic, etc. In this paper, we study Randers and $(α,β)$ equigeodesics. For a compact homogeneous manifold, we prove Randers and $(α,β)$ equigeodesics are equivalent, and find a criterion for them. Using this criterion we can classify the equigeodesics on many compact homogeneous m…
The geometry and analysis on Finsler manifolds is a very important part of Finsler geometry. In this article, we introduce some important and fundamental topics in global Finsler geometry and discuss the related properties and the relationships in them. In particular, we optimize and improve the various definitions of Lie derivatives on Finsler manifolds. We also characterize the gradient vector fields and obtain a gradient estimate for any smooth function on a Randers manifold.
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…
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.
In this paper, we give global expressions of geodesics and isoparametric functions on a Randers sphere by navigation. We obtain isoparametric families and focal submanifolds in (S^{n}; F; dμ_{BH}) by Cartan-Münzner polynomials. Further more, we construct some examples of closed and non-closed geodesics, isoparametric functions, isoparametric families and focal submanifolds.
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 $Sp(n)/Sp(n-1)$.
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.
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.
An isometry of a Finsler space is called Clifford-Wolf translation (CW-translation) if it moves all points the same distance. A Finsler space $(M, F)$ is called Clifford-Wolf homogeneous (CW-homogeneous) if for any $x, y\in M$ there is a CW-translation $σ$ such that $σ(x)=y$. We prove that if $F$ is a homogeneous Finsler metric on the sphere $S^n$ such that $(S^n, F)$ is CW-homogeneous, then $F$ must be a Randers metric. This gives a complete classification of CW-homogeneous Finsler metrics on spheres.
Recently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group can not be a compact Lie group, if the Finsler manifold of dimension $> 2$ has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant-Shen manifolds of non-zero constant flag curvature have 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…
In this paper, we study Clifford-Wolf translations of Finsler spaces. We first give a characterization of Clifford-Wolf translations of Finsler spaces in terms of Killing vector fields. In particular, we show that there is a natural correspondence between Clifford-Wolf translations and the Killing vector fields of constant length. In the special case of homogeneous Randers spaces, we give some explicit sufficient and necessary conditions for an isometry to be a Clifford-Wolf translation. Finally, we construct some explicit examples to explain some of the results of this paper.
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…
In this paper, we first give two fundamental principles under a technique to characterize conformal vector fields of $(α,β)$ spaces to be homothetic and determine the local structure of those homothetic fields. Then we use the principles to study conformal vector fields of some classes of $(α,β)$ spaces under certain curvature conditions. Besides, we construct a family of non-homothetic conformal vector fields on a family of locally projectively Randers spaces.
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…
Weighted Ricci curvatures with various weights of the S-curvature have played an important role in Finsler geometry, especially in the control of volumes of Finsler manifolds. In this paper, we study general weighted Ricci curvatures, and characterize Randers metrics of almost isotropic weighted Ricci curvatures.
We study extrinsic geometry of a codimension-one foliation ${\cal F}$ of a closed Finsler space $(M,F)$, in particular, of a Randers space $(M,α+β)$. Using a unit vector field $ν$ orthogonal (in the Finsler sense) to the leaves of ${\cal F}$ we define a new Riemannian metric $g$ on $M$, which for Randers case depends nicely on $(α,β)$. For that $g$ we derive several geometric invariants of ${\cal F}$ (e.g. the Riemann curvature and the shape operator) in terms of $F$, then under natural assumptions on $β$ which simplify derivations, we express them in terms of corresponding invariants arising …
In this paper, we introduce the notion of Einstein-reversibility for Finsler met- rics. We study a class of p-power Finsler metrics determined by a Riemann metric and 1-form which are of Einstein-reversibility. It shows that such a class of Finsler metrics of Einstein-reversibility are always Einstein metrics. In particular, we show that all p-power metrics but Randers metrics, square metrics and 2-dimensional square-root metrics, are always Ricci-flat-parallel. Further, the local structure is almost determined for 2-dimensional square-root metrics which are Einsteinian (equivalently, of isotr…
Given a Finsler manifold $(M,F)$, it is proved that the first eigenvalue of the Finslerian $p$-Laplacian is bounded above by a constant depending on $\ p$, the dimension of $M$, the Busemann-Hausdorff volume and the reversibility constant of $(M,F)$. For a Randers manifold $(M,F:=\sqrt{g}+β)$, where $g$ is a Riemannian metric on $M$ and $β$ an appropriate $1$-form on $M$, it is shown that the first eigenvalue $λ_{1,p}(M,F)$ of the Finslerian $p$-Laplacian defined by the Finsler metric $F$ is controled by the first eigenvalue $λ_{1,p}(M,g)$ of the Riemannian $p$-Laplacian defined on $(M,g)$. Fi…
In this paper we introduce the concept of singular Finsler foliation, which generalizes the concepts of Finsler actions, Finsler submersions and (regular) Finsler foliations. We show that if $\mathcal{F}$ is a singular Finsler foliation on a Randers manifold $(M,Z)$ with Zermelo data $(\mathtt{h},W),$ then $\mathcal{F}$ is a singular Riemannian foliation on the Riemannian manifold $(M,\mathtt{h} )$. As a direct consequence we infer that the regular leaves are equifocal submanifolds (a generalization of isoparametric submanifolds) when the wind $W$ is an infinitesimal homothety of $\mathtt{h}$ …
An $(α,β)$-manifold $(M,F)$ is a Finsler manifold with the Finsler metric $F$ being defined by a Riemannian metric $α$ and $1$-form $β$ on the manifold $M$. In this paper, we classify $n$-dimensional $(α,β)$-manifolds (non-Randers type) which are positively complete and locally projectively flat. We show that the non-trivial class is that $M$ is homeomorphic to the $n$-sphere $S^n$ and $(S^n,F)$ is projectively related to a standard spherical Riemannian manifold, and then we obtain some special geometric properties on the geodesics and scalar flag curvature of $F$ on $S^n$, especially when $F$…
