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 holonomy algebra coincides with the curvature algebra. We prove that for all projectively flat Randers surfaces of non-zero constant flag curvature the infinitesimal holonomy algebra has infinite dimension and hence the holonomy group cannot be a Lie group of finite dimension. Finally, in the case of the Funk metric we prove that the infinitesimal holonomy algebra is a dense subalgebra of the Lie algebra of the full diffeomorphism group and hence the topological closure of the holonomy group is the orientation preserving diffeomorphism group of the circle.
We show that the set of Finsler metrics on a manifold contains an open everywhere dense subset of Finsler metrics with infinite-dimensional holonomy groups.
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.
The aim of this paper is to show that holonomy properties of Finsler manifolds can be very different from those of Riemannian manifolds. We prove that the holonomy group of a positive definite non-Riemannian Finsler manifold of non-zero constant curvature with dimension >2 cannot be a compact Lie group. Hence this holonomy group does not occur as the holonomy group of any Riemannian manifold. In addition, we provide an example of left invariant Finsler metric on the Heisenberg group, so that its holonomy group is not a (finite dimensional) Lie group. These results give a positive answer to the…
In this paper we prove that the holonomy group of a simply connected locally projectively flat Finsler manifold of constant curvature is a finite dimensional Lie group if and only if it is flat or it is Riemannian.
In this paper we show that the topological closure of the holonomy group of a certain class of projectively flat Finsler 2-manifolds of constant curvature is maximal, that is isomorphic to the connected component of the diffeomorphism group of the circle. This class of 2-manifolds contains the standard Funk plane of constant negative curvature and the Bryant-Shen-spheres of constant positive curvature. The result provides the first examples describing completely infinite dimensional Finslerian holonomy structures.
Our goal in this paper is to make an attempt to find the largest Lie algebra of vector fields on the indicatrix such that all its elements are tangent to the holonomy group of a Finsler manifold. First, we introduce the notion of the curvature algebra, generated by curvature vector fields, then we define the infinitesimal holonomy algebra 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. At the end we introduce the notion of the holonomy algebra of a Finsler ma…
In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $λ\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal c…
This paper is concerned with the holonomy of a class of spaces which includes Landsberg spaces of Finsler geometry. The methods used are those of Lie groupoids and algebroids as developed by Mackenzie. We prove a version of the Ambrose-Singer Theorem for such spaces. The paper ends with a discussion of how the results may be extended to Finsler spaces and homogeneous nonlinear connections in general.
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 introduce the notion of tangent space TG of a (not necessary smooth) subgroup G of the diffeomorphism group Diff(M) of a compact manifold M. We prove that TG is a Lie subalgebra of the Lie algebra of smooth vector fields on M. The construction can be generalized to subgroups of any (finite or infinite dimensional) Lie groups. The tangent Lie algebra TG introduced this way is a generalization of the classical Lie algebra in the smooth cases. As a working example, we discuss in detail the tangent structure of the holonomy group and fibered holonomy group of Finsler manifolds.
In this paper we investigate the following question: under what conditions can a second-order homogeneous ordinary differential equation (spray) be the geodesic equation of a Finsler space. We show that the Euler-Lagrange partial differential system on the energy function can be reduced to a first order system on this same function. In this way we are able to give effective necessary and sufficient conditions for the local existence of a such Finsler metric in terms of the holonomy algebra generated by horizontal vector-fields. We also consider the Landsberg metrizability problem and prove sim…
We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and has applications in the characterisation of isometric embeddings into Heisenberg groups.
In the present paper, the notion of $\mathbb{R}$-complex Finsler space with Infinite Series ($α, β$)- metric $\dfrac{β^2}{β- α}$ is defined. The Fundamental metric fields $g_{ij}$, $g_{i\bar{j}}$, their determinants and the inverse of these tensor fields are obtained. Also some properties of these spaces are studied.
In this paper, we investigate a holonomy invariant elliptic anisotropic surface energy for hypersurfaces in a complete Riemannian manifold, where "holonomy invariant" means that the elliptic parametric Lagrangian (i.e., a Finsler metric) of the Riemannian manifold used to define the anisotropic surface energy is constant along each holonomy subbundle of the tangent bundle of the Riemannian manifold. First we obtain the first variational formula for this anisotropic surface energy. Next we shall introduce the notions of an anisotropic convex hypersurface, an anisotropic equifocal hypersurface a…
This paper studies the relation between two notions of holonomy on a conformal manifold. The first is the conformal holonomy, defined to be the holonomy of the normal tractor connection. The second is the holonomy of the Fefferman-Graham ambient metric of the conformal manifold. It is shown that the infinitesimal conformal holonomy and the infinitesimal ambient holonomy always agree up to the order that the ambient metric is defined.
In this article, we give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of Lie groups defined by T.Robart [13], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (not necessarily complete) Finsler n-manifold of constant positive flag curvature out of a hypersurface in suitably general position in complex projective n-space. <p> The third remark is that there is a description of the Finsler metrics of constant curvature on the 2-sphere in t…
In this paper, we study geodesics and geodesic vectors for homogeneous exponential Finsler space and homogeneous infinite series Finsler space. Further, we find necessary and sufficient condition for a non-zero vector in these homogeneous spaces to be a geodesic vector.
In this paper, first we prove the existence of invariant vector field on a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce an explicit formula for the the $S$-curvature of homogeneous Finsler space with these metrics. Using this formula, we further derive the formula for mean Berwald curvature of the homogeneous Finsler space with the above mentioned metrics.
In this paper, first we derive an explicit formula for the flag curvature of a homogeneous Finsler space with infinite series $(α, β)$-metric and exponential metric. Next, we deduce it for naturally reductive homogeneous Finsler space with the above mentioned metrics.
We show that the infinite-dimensional space of Zoll Finsler metrics on the projective plane strongly deformation retracts to the canonical round metric. In particular, this space of Zoll Finsler metrics is connected. Moreover, the strong deformation retraction arises from a deformation of the geodesic flow of every Zoll Finsler projective plane to the geodesic flow of the round metric through a family of smooth free circle actions induced by the curvature flow of the canonical round projective plane. This construction provides a description of the geodesics of the Zoll Finsler metrics along th…
For compact manifolds with infinite fundamental group we present sufficient topological or metric conditions ensuring the existence of two geometrically distinct closed geodesics. We also show how results about generic Riemannian metrics can be carried over to Finsler metrics.
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…
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.
A Riemannian metric is of constant curvature if and only if it is locally projectively flat. There are infinitely many locally projectively flat Finsler metrics of constant curvature, that are special solutions to the Hilbert's Fourth Problem. In this paper, we use the technique in the paper titled "Finsler metrics with K=0 and S=0" (math.DG/0109060) to construct infinitely many Finsler metrics on the 2-sphere with constant curvature K=1 and infinitely many Finsler metrics on the 2-disk with constant curvature K = -1. These metrics are not projectively flat. So far, the classification of Finsl…
In this paper we discuss some examples and general properties of holonomy groups of $\nabla^φ$ introduced by Wylie and the author, the connection corresponding to the $N=1$ Bakry-Émery Ricci curvature, and also Wylie's $\bar{\mathrm{sec}}_f$. In particular we classify all possible holonomy groups in dimension 2 and also provide two infinite families: $SL_n(\mathbb{R})$ and $SO^+(p,q)$.
A torsion-free G_2 structure admitting an infinitesimal isometry is shown to give rise to a 4-manifold equipped with a complex symplectic structure and a 1-parameter family of functions and 2-forms linked by second order equations. Reversing the process in various special cases leads to the construction of explicit metrics with holonomy equal to G_2.
Finsler space is differentiable manifold for which Minkowski space is the fiber of the tangent bundle. To understand structure of the reference frame in Finsler space, we need to understand the structure of orthonormal basis in Minkowski space. In this paper, I considered the definition of orthonormal basis in Minkowski space, the structure of metric tensor relative to orthonormal basis, procedure of orthogonalization. Linear transformation of Minkowski space which preserves the scalar product is called a motion. Linear transformation which maps the orthonormal basis into an infinitely close o…
First we present a short overview of the long history of projectively flat Finsler spaces. We give a simple and quite elementary proof of the already known condition for the projective flatness, and we give a criterion for the projective flatness of a special Lagrange space (Theorem 1). After this we obtain a second-order PDE system, whose solvability is necessary and sufficient for a Finsler space to be projectively flat (Theorem 2). We also derive a condition in order that an infinitesimal transformation takes geodesics of a Finsler space into geodesics. This yields a Killing type vector fie…
We first describe the action of the fundamental group of a closed surface of variable negative curvature on the oriented geodesics in its universal covering in terms of a naturally-defined flat connection whose holonomy lies in the group of Hamiltonian diffeomorphisms of S^1 x R. Consideration of the holonomy necessitates an extension from Riemannian to Finsler metrics. The second part of the paper follows the Higgs bundle approach to flat connections adapted to this infinite dimensional group and focuses on a family of metrics, relying on a construction of O.Biquard, which is parametrized by …
We prove the shifting theorems of the critical groups of critical points and critical orbits for the energy functionals of Finsler metrics on Hilbert manifolds of $H^1$-curves, and two splitting lemmas for the functionals on Banach manifolds of $C^1$-curves. Two results on critical groups of iterated closed geodesics are also proved; their corresponding versions on Riemannian manifolds are based on the usual splitting lemma by Gromoll and Meyer (1969). Our approach consists in deforming the square of the Finsler metric in a Lagrangian which is smooth also on the zero section and then in using …
We prove that sub-Riemannian manifolds are infinitesimally Hilbertian (i.e., the associated Sobolev space is Hilbert) when equipped with an arbitrary Radon measure. The result follows from an embedding of metric derivations into the space of square-integrable sections of the horizontal bundle, which we obtain on all weighted sub-Finsler manifolds. As an intermediate tool, of independent interest, we show that any sub-Finsler distance can be monotonically approximated from below by Finsler ones. All the results are obtained in the general setting of possibly rank-varying structures.
In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a technique to construct non-projectively flat Finsler metrics with zero curvature in each dimension. The technique can be used to construct many non-projectively flat Finsler metrics of constant curvature.
In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using linearly parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. Concerning the nonlinearly ones, we propose two que…
This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here we prove T-duality and Hori formulae for winding q-loop spaces, which are infinite dimensional subspaces of loop spaces.
Igarashi introduce the concept of $(α, β)$-metric in Cartan space $\ell^{n}$ analogously to one in Finsler space and obtained the basic important geometric properties and also investigate the special class of the space with $(α, β)$-metric in $\ell^{n}$ in terms of $ 'invariants' $. In the present paper we determine the $ 'invariants' $ in two different cases of deformed infinite series metric which characterize the special classes of Cartan spaces $\ell^{n}$.
We establish the splitting lemmas (or generalized Morse lemmas) for the energy functionals of Finsler metrics on the natural Hilbert manifolds of $H^1$-curves around a critical point or a critical $\R^1$ orbit of a Finsler isometry invariant closed geodesic. They are the desired generalization on Finsler manifolds of the corresponding Gromoll-Meyer's splitting lemmas on Riemannian manifolds (\cite{GM1, GM2}). As an application we extend to Finsler manifolds a result by Grove and Tanaka \cite{GroTa78, Tan82} about the existence of infinitely many, geometrically distinct, isometry invariant clos…
We describe the infinitesimal moduli space of pairs $(Y, V)$ where $Y$ is a manifold with $G_2$ holonomy, and $V$ is a vector bundle on $Y$ with an instanton connection. These structures arise in connection to the moduli space of heterotic string compactifications on compact and non-compact seven dimensional spaces, e.g. domain walls. Employing the canonical $G_2$ cohomology developed by Reyes-Carrión and Fernández and Ugarte, we show that the moduli space decomposes into the sum of the bundle moduli $H^1_{\check{d}_A}(Y,\mathrm{End}(V))$ plus the moduli of the $G_2$ structure preserving the i…
The main result of this paper is the following: any `weighted' Riemannian manifold $(M,g,μ)$ - i.e. endowed with a generic non-negative Radon measure $μ$ - is `infinitesimally Hilbertian', which means that its associated Sobolev space $W^{1,2}(M,g,μ)$ is a Hilbert space. We actually prove a stronger result: the abstract tangent module (à la Gigli) associated to any weighted reversible Finsler manifold $(M,F,μ)$ can be isometrically embedded into the space of all measurable sections of the tangent bundle of $M$ that are $2$-integrable with respect to $μ$.
Nurowski showed that any generic 2-plane field $D$ on a 5-manifold $M$ determines a natural conformal structure $c_D$ on $M$; these conformal structures are exactly those (on oriented $M$) whose normal conformal holonomy is contained in the (split, real) simple Lie group $G_2$. Graham and Willse showed that for real-analytic $D$ the same holds for the holonomy of the real-analytic Fefferman-Graham ambient metric of $c_D$, and that both holonomy groups are equal to $G_2$ for almost all $D$. We investigate here independently interesting plane fields for which the associated holonomy groups are a…
In his 1910 "Five Variables" paper, Cartan solved the equivalence problem for the geometry of $(2, 3, 5)$ distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type $\textrm{G}_2$. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least $6$ (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution $\bf E$. We produce a closed form for the Fefferman-…
While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses: the Walker and Kundt metrics. In this paper we have chosen an example from each of the two subcases of the Ricci-flat VSI Walker metrics respectively. To investigate the difference between the metrics we determine the existence of a null, geodesic, expansion-free, shear-free and…
Let $M=S^n/ Γ$ and $h$ be a nontrivial element of finite order $p$ in $π_1(M)$, where the integer $n, p\geq2$, $Γ$ is a finite abelian group which acts freely and isometrically on the $n$-sphere and therefore $M$ is diffeomorphic to a compact space form. In this paper, we prove that for every irreversible Finsler compact space form $(M,F)$ with reversibility $λ$ and flag curvature $K$ satisfying \[ \frac{4p^2}{(p+1)^2} \big(\fracλ{λ+1} \big)^2 < K \leq 1,\;\;λ< \frac{p+1}{p-1}, \] there exist at least $n-1$ non-contractible closed geodesics of class $[h]$. In addition, if the metric $F$ is bum…
The connection between Yang--Mills gauge fields on $4$-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if …
