Horizontal Forms of Chern Type on Complex Finsler Bundles

Traveling along horizontal broken geodesics of a homogenous Finsler submersion

In this paper, we discuss how to travel along horizontal broken geodesics of a homogenous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $\mathcal{A}_{q}(\mathcal{C})$ of the set of analytic vector fields $\mathcal{C}$ determined by the family of horizontal unit geodesic vector fields $\mathcal{C}$ to the fibers $\mathcal{F}=\{ρ^{-1}(c)\}$ of a homogenous analytic Finsler submersion $ρ: M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examp…

Corrigendum to "Chern connection of a pseudo-Finsler metric as a family of affine connections"

In this note, we give the correct statements of [2,Proposition 3.3 and Theorem 3.4] and a formula of the Chern curvature in terms of the curvature tensor $R^V$ of the affine connection $\nabla^V$ and the Chern tensor $P$.

次芬斯勒海森堡群 $\mathbb{H}^1$ 中具有低正则性的面积极小水平图

Area-minimizing horizontal graphs with low-regularity in the sub-Finsler Heisenberg group $\mathbb{H}^1$

研究次芬斯勒海森堡群$\mathbb{H}^1$中低正则性面积极小水平图，构造以水平线或共点水平半直线有限并为奇异集的欧氏局部利普希茨整面积极小水平图，还给出多类面积极小锥例子。

Chern connection of a pseudo-Finsler metric as a family of affine connections

We consider the Chern connection of a (conic) pseudo-Finsler manifold $(M,L)$ as a linear connection $\nabla^V$ on any open subset $Ω\subset M$ associated to any vector field $V$ on $Ω$ which is non-zero everywhere. This connection is torsion-free and almost metric compatible with respect to the fundamental tensor $g$. Then we show some properties of the curvature tensor $R^V$ associated to $\nabla^V$ and in particular we prove that the Jacobi operator of $R^V$ along a geodesic coincides with the one given by the Chern curvature.

Chern forms of holomorphic Finsler vector bundles and some applications

In this paper, we present two kinds of total Chern forms $c(E,G)$ and $\mathcal{C}(E,G)$ as well as a total Segre form $s(E,G)$ of a holomorphic Finsler vector bundle $π:(E,G)\to M$ expressed by the Finsler metric $G$, which answers a question of J. Faran (\cite{Faran}) to some extent. As some applications, we show that the signed Segre forms $(-1)^ks_k(E,G)$ are positive $(k,k)$-forms on $M$ when $G$ is of positive Kobayashi curvature; we prove, under an extra assumption, that a Finsler-Einstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Fi…

Sasakian Finsler structures on pulled-back bundle

Under a pulled-back approach given in [1] and firstly presented in [2], we introduce, in this paper, the concepts of almost contact and normal almost contact Finsler structures on the pulled-back bundle. Properties of structures partly Sasakians are studied. Using the hh-curvature tensor of Chern connection given in [2], we obtain some characterizations of horizontally Finslerian K-contact structures via the horizontal Ricci tensor and the flag curvature.

Almost Paracontact Finsler Structures on Vector Bundle

In this paper, we define almost paracontact and normal almost paracontact Finsler structures on a vector bundle and find some conditions for integrability of these structures. We define paracontact metric, para- Sasakian and K-paracontact Finsler structures and study some properties of these structures. For a K-paracontact Finsler structure, we find the vertical and horizontal flag curvatures. Then, we define vertical ?-flag curvature and prove that every locally symmetric para-Sasakian Finsler structure has negative vertical ?-flag curvature. Finally, we define the horizontal and vertical Ric…

Models of Finsler Geometry on Lie algebroids

Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are studied on prolongation of Lie algebroids. Considering pullback bundle, covariant derivatives are scrutinized based on anchor map. Several Finsler geometry models on Lie algebroid structures, will organized via recent arguments. Finally, it will be overhaulled some special Finsl…

On Doubly Warped Product Finsler Manifolds

In this paper, we introduce horizontal and vertical warped product Finsler manifold. We prove that every C-reducible or proper Berwaldian doubly warped product Finsler manifold is Riemannian. Then, we find the relation between Riemmanian curvatures of doubly warped product Finsler manifold and its components, and consider the cases that this manifold is flat or has scalar flag curvature. We define the doubly warped Sasaki-Matsumoto metric for warped product manifolds and find a condition under which the horizontal and vertical tangent bundles are totally geodesic. Also, we obtain some conditio…

Finsler 几何上的调和向量场与 Hodge Laplacian 算子

Harmonic vector fields and the Hodge Laplacian operator on Finsler geometry

本文在 Finsler 流形上定义了水平微分、散度与 $p$-调和形式，证明了 Hodge 型定理，提出了调和向量场的新定义，并导出了基于调和 Ricci 标量的 Bochner-Yano 型分类定理。

On two-dimensional complex Finsler manifolds

In this paper, we investigate the two-dimensional complex Finsler manifolds. The tools of this study are the complex Berwald frames and the Chern-Finsler connection with respect to these frames.

Adiabatic limit and connections in Finsler Geometry

In this paper, we identify the Bott connection on the natural foliation of the projective sphere bundle of a Finsler manifold to the Chern connection of this manifold. As a consequence, the symmetrization of the Bott connection turns out to be the Cartan connection of the Finsler manifold. Following Liu-Zhang \cite{LiuZ}, the Cartan connection can also be obtained through an adiabatic limit process. Furthermore, a Chern-Simons type form is defined and its conformal properties are discussed.

Finsler structures on holomorphic Lie algebroids

Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid E and we introduce Finsler structures, partial and Chern-Finsler connections on it. First, we recall some basic notions on holomorphic Lie algebroids. Then, using an idea from E. Martinez, we introduce the concept of complexified prolongation of such an algebroid. Also, we study nonlinear and linear connections on the tangent bundle of E and on the prolongation of E and we inves…

Finsler几何的度量方法

A metrical approach to Finsler geometry

本文提出Finsler几何的度量表述，从度量与Finsler联络的相容性公理推导出度量的Hessian性质、非线性联络及Chern或Cartan联络，适用于场论和Finsler引力研究。

Symplectic Connections Induced by the Chern Connection

Let $(M,ω)$ be a symplectic manifold and $F$ be a Finsler structure on $M$. In the present paper we define a lift of the symplectic two-form $ω$ on the manifold $TM\backslash 0$, and find the conditions that the Chern connection of the Finsler structure $F$ preserves this lift of $ω$. In this situation if $M$ admits a nowhere zero vector field then we have a non-empty family of Fedosov structures on $M$.

复Finsler度量的刻画

Characterizations of complex Finsler Metrics

本文研究复Finsler流形上的典范联络与Chern-Finsler联络的曲率关系，给出平衡复Finsler度量的刻画及成为Kähler-Finsler度量的充要条件，并探讨其共形变换。

Existence and uniqueness of Chern connection in the Klein-Grifone approach

The Klein-Grifone approach to global Finsler geometry is adopted. A global existence and uniqueness theorem for Chern connection is formulated and proved. The torsion and curvature tensors of Chern connection are derived. Some properties and the Bianchi identities for this connection are investigated. A concise comparison between Berwald, Cartan and Chern connections is presented.

Tangent Lie algebras to the holonomy group of a Finsler manifold

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…

Geometrical structures on the prolongation of a pre-Lie algebroid on fibered manifolds and application to Partial Finsler geometry on foliated anchored bundle

A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The principal purpose of this paper is to show that how all these results only depend of the foliated structure and do not depend of the pre-Lie algebroid structure that we can put on a foliated anchored bundle. As application, we obtain a Finsler connection on a foliated anchored bundle …

Remarks on the Schwarzian Derivatives and the Invariant Quantization by means of a Finsler Function

Let $(M,F)$ be a Finsler manifold. We construct a 1-cocycle on $\Diff(M)$ with values in the space of differential operators acting on sections of some bundles, by means of the Finsler function $F.$ As an operator, it has several expressions: in terms of the Chern, Berwald, Cartan or Hashiguchi connection, although its cohomology class does not depend on them. This cocycle is closely related to the conformal Schwarzian derivatives introduced in our previous work. The second main result of this paper is to discuss some properties of the conformally invariant quantization map by means of a Sazak…

Laplace operators on holomorphic Lie algebroids

The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace operators considered in the paper are also locally expressed using the Chern-Finsler connection of the algebroid.

On Concircularly Recurrent Finsler Manifolds

Two special Finsler spaces have been introduced and investigated, namely $R^h$-recurrent Finsler space and consircularly recurrent Finsler space. The defining properties of these spaces are formulated in terms of the first curvature tensor of Cartan connection. The following three results constitute the main object of the present paper: 1. A concircularly flat Finsler manifold is necessarily of constant curvature (Theorem A); 2. Every $R^h$-recurrent Finsler manifold is concirculaly recurrent with the same recurrence form (Theorem B); 3. Every horizontally integrable concircularly recurrent Fi…

Critical Remarks on Finsler Modifications of Gravity and Cosmology by Zhe Chang and Xin Li

I do not agree with the authors of papers arXiv:0806.2184 and arXiv:0901.1023v1 (published in Phys. Lett., respectively, B668 (2008) 453 and B676 (2009) 173). They consider that \textit{"In Finsler manifold, there exists a unique linear connection - the Chern connection ... It is torsion freeness and metric compatibility ... "}. There are well known results (for example, presented in monographs by H. Rund and R. Miron and M. Anastasiei) that in Finsler geometry there exist an infinite number of linear connections defined by the same metric structure and that the Chern and Berwald connections \…

A horizontal Chern-Gauss-Bonnet formula on totally geodesic foliations

Under suitable conditions, we show that the Euler characteristic of a foliated Riemannian manifold can be computed only from curvature invariants which are transverse to the leaves. Our proof uses the hypoelliptic sub-Laplacian on forms recently introduced by two of the authors.

经典域上全纯不变强伪凸复Finsler度量的几何

Geometry of holomorphic invariant strongly pseudoconvex complex Finsler metrics on the classical domains

本文在I-IV型不可约经典域上通过对Bergman度量形变构造了一类全纯不变强伪凸复Finsler度量，证明了其完备性及与Bergman度量相似的负曲率性质。

Non-tame Morse-Smale flows and odd Chern-Weil theory

Using a certain well-posed ODE problem introduced by Shilnikov in the sixties, G. Minervini proved in his PhD thesis [17], among other things, the Harvey-Lawson Diagonal Theorem but without the restrictive tameness condition for Morse flows. Here we combine the same techniques with the ideas of Latschev in order to construct local resolutions for the flow of the graph of a section of a fiber bundle. This is endowed with a vertical vector field which is horizontally constant and Morse-Smale in every fiber. The resolution allows the removal of the tameness hypothesis from the homotopy formula in…

Curve Flows in Lagrange-Finsler Geometry, Bi-Hamiltonian Structures and Solitons

Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces $G/SO(n)$ provided with an N-connection structure and an adap…

Concurrent $π$-vector fields and energy beta-change

The present paper deals with an \emph{intrinsic} investigation of the notion of a concurrent $π$-vector field on the pullback bundle of a Finsler manifold $(M,L)$. The effect of the existence of a concurrent $π$-vector field on some important special Finsler spaces is studied. An intrinsic investigation of a particular $β$-change, namely the energy $β$-change ($\widetilde{L}^{2}(x,y)=L^{2}(x,y)+ B^{2}(x,y) with \ B:=g(\barζ,\barη)$; $\barζ $ being a concurrent $π$-vector field), is established. The relation between the two Barthel connections $Γ$ and $\widetildeΓ$, corresponding to this change…

Connections in holomorphic Lie algebroids

The main purpose of this note is the study of the total space of a holomorphic Lie algebroid $E$. The paper is structured in three parts. In the first section we briefly introduce basic notions on holomorphic Lie algebroids. The local expressions are written and the complexified holomorphic bundle is introduced. The second section is a little broader and includes two approaches to study the geometry of complex manifold $E.$ The first part contains the study of the tangent bundle $T_{C}E=T^{\prime }E\oplus T^{\prime \prime }E$ and its link, via tangent anchor map, with the complexified tangent …