Almost all Finsler metrics have infinite dimensional holonomy group

常曲率球对称射影 Finsler 度量的完整群

The holonomy of spherically symmetric projective Finsler metrics of constant curvature

本文证明常曲率球对称射影 Finsler 度量的完整群极大，并在单连通情形下同构于 $Diff_o({\mathbb S^{n-1}})$，首次显式刻画了非 Berwald 的 n 维 Finsler 流形完整群。

Metrizability of holonomy invariant projective deformation of sprays

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…

Holonomy of a class of bundles with fibre metrics

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.

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…

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…

Tangent Lie algebra of a diffeomorphism group and application to holonomy theory

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.

与芬斯勒空间相关的平均结构

Average structures associated with a Finsler space

该研究提出芬斯勒空间的平均方法，将芬斯勒几何对象关联为仿射几何对象，推导了平均联络的相关性质，推广了Gauss-Bonnet定理并证明了对称空间性质的遗传性。

Entropy of foliations with leafwise Finsler structure

We extend the notion of the geometric entropy of foliation to foliated manifolds equipped with leafwise Finsler structure. We study the relation between the geometric entropy and the topological entropy of the holonomy pseudogroup. The case of foliated manifold with leafwise Randers structure. In this case the estimates for one dimensional foliation defined by a vector field in term of topological entropy of a flow are presented.

Two new families of Finsler connections on even-dimensional manifolds

Let $F^{2n}=(M,M',F^{\ast})$ be an even-dimensional pseudo-Finsler manifold. We construct an almost hypercomplex structure on any chart domain of a certain atlas of $M'$ by using a considered non-linear connection. Then by using the almost hypercomplex structure we define two new families of Finsler connections. Also we show that for any Finsler connection $\nabla$ there exists a linear connection $D$ such that the local almost hypercomplex structure is parallel with respect to it.

关于局部对偶平坦和射影平坦的几乎有理Finsler度量

On Locally Dually and Projectively Flat Almost Rational Finsler Metrics

研究几乎有理Finsler度量的局部对偶平坦和射影平坦的充要条件，并刻画同时满足两者的度量，提出局部Minkowski性猜想。

关于几乎有理Finsler度量

On almost rational Finsler metrics

本文提出几乎有理Finsler度量概念，给出其成为Riemannian度量的充要条件，研究相关几何量的有理性，证明若干曲率条件下的刚性结果，并引入扩展m次根度量作为新例子。

A holonomy invariant anisotropic surface energy in a Riemannian manifold

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…

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.

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。

Some remarks on Finsler manifolds with constant flag curvature

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…

On submaximal dimension of the group of almost isometries of Finsler metrics

We show that the second greatest possible dimension of the group of (local) almost isometries of a Finsler metric is $\frac{n^2 -n}{2} +1$ for $n= dim(M)\ne 4 $ and $\frac{n^2 -n}{2} +2 =8$ for $n=4$. If a Finsler metric has the group of almost isometries of dimension greater than $\frac{n^2 -n}{2} +1$, then the Finsler metric is Randers, i.e., $F(x,y)= \sqrt{g_x(y,y)} + τ(y)$. Moreover, if $n\ne 4$, the Riemannian metric $g$ has constant sectional curvature and, if in addition $n\ne 2$, the 1-form $τ$ is closed, so (locally) the metric admits the group of local isometries of the maximal dimen…

几乎Finsler流形的特征张量

Characteristic tensors for almost Finsler manifolds

本文引入了几乎Finsler流形和部分Finsler流形，扩展了标准Finsler流形定义，允许非平凡切口包含在同源缩放下固定的点，并且允许基本张量具有非正特征值。通过双部空间给出了较为简单的几乎Finsler流形和部分Finsler流形的例子，并探讨了其物理应用。

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…

On A Class of Finsler Metrics of Einstein-Reversibility

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…

Complex connections with trivial holonomy

Given an almost complex manifold (M, J), we study complex connections with trivial holonomy and such that the corresponding torsion is either of type (2,0) or of type (1,1) with respect to J. Such connections arise naturally when considering Lie groups, and quotients by discrete subgroups, equipped with bi-invariant and abelian complex structures.

Almost regular Poisson manifolds and their holonomy groupoids

We look at Poisson geometry taking the viewpoint of singular foliations, understood as suitable submodules generated by Hamiltonian vector fields rather than partitions into (symplectic) leaves. The class of Poisson structures which behave best from this point of view, are those whose submodule generated by Hamiltonian vector fields arises from a smooth holonomy groupoid. We call them almost regular Poisson structures and determine them completely. They include regular Poisson and log symplectic manifolds, as well as several other Poisson structures whose symplectic foliation presents singular…

Berwald metrics constructed by Chevalley's polynomials

Berwald metrics are particular Finsler metrics which still have linear Berwald connections. Their complete classification is established in an earlier work, [Sz1], of this author. The main tools in these classification are the Simons-Berger holonomy theorem and the Weyl-group theory. It turnes out that any Berwald metric is a perturbed-Cartesian product of Riemannian, Minkowski, and such non-Riemannian metrics which can be constructed on irreducible symmetric manifolds of $rank > 1$. The existence of these metrics are well established by the above theories. The present paper has several new fe…

孤立高阶不动点附近抛物几何的和乐

Holonomy of Parabolic Geometries Near Isolated Higher-Order Fixed Points

该研究针对具特定迷向自同构的Cartan几何，证明存在平凡和乐的开子集，推导了和乐相关的实用结果，并刻画了几类具高阶不动点的几何结构。

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 …

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.

Almost Kaehler Ricci Flows and Einstein and Lagrange-Finsler Structures on Lie Algebroids

In this work we investigate Ricci flows of almost Kaehler structures on Lie algebroids when the fundamental geometric objects are completely determined by (semi) Riemannian metrics, or effective) regular generating Lagrange/ Finsler, functions. There are constructed canonical almost symplectic connections for which the geometric flows can be represented as gradient ones and characterized by nonholonomic deformations of Grigory Perelman's functionals. The first goal of this paper is to define such thermodynamical type values and derive almost Kähler - Ricci geometric evolution equations. The se…

Quasi-Kähler manifolds with trivial Chern Holonomy

In this paper we study almost complex manifolds admitting a quasi-Kähler Chern-flat metric (Chern-flat means that the holonomy of the Chern connection is trivial). We prove that in the compact case such manifolds are all nilmanifolds. Some partial classification results are established and we prove that a quasi-Kähler Chern-flat structure can be tamed by a symplectic form if and only if the ambient space is isomorphic to a flat torus.

Holonomy of the Obata connection on SU(3)

A hypercomplex structure on a smooth manifold is a triple of integrable almost complex structures satisfying quaternionic relations. The Obata connection is the unique torsion-free connection that preserves each of the complex structures. The holonomy group of the Obata connection is contained in $GL(n, \mathbb{H})$. There is a well-known construction of hypercomplex structures on Lie groups due to Joyce. In this paper we show that the holonomy of the Obata connection on SU(3) coincides with $GL(2, \mathbb{H})$.

On a Class of Weakly Weighted Einstein Metrics

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.

On Non-Positively Curved Homogeneous Finsler Metrics

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…

Finsler几何中Killing向量场的十二种刻画

Twelve characterisations of Killing vector fields in Finsler geometry

本文在Finsler几何中利用Frölicher-Nijenhuis形式体系，建立了Killing向量场的十二种等价刻画，并给出局部方程形式，揭示其多方面的几何本质。

An almost flat manifold with a cyclic or quaternionic holonomy group bounds

A long-standing conjecture of Farrell and Zdravkovska and independently S.~T.~Yau states that every almost flat manifold is the boundary of a compact manifold. This paper gives a simple proof of this conjecture when the holonomy group is cyclic or quaternionic. The proof is based on the interaction between flat bundles and involutions.

A Teichmuller space for negatively curved surfaces

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 …

黄金芬斯勒几何：局部性质与全局变形

Golden Finsler Geometry: Local Properties and Global Deformations

研究有限维光滑流形上黄金芬斯勒结构，从局部和全局角度研究。局部计算相关张量等，研究射影平坦性；全局定义黄金芬斯勒变化，研究其几何性质及结构变换，最终证明\(\widetilde{F}\)与\(F\)不能射影相关。

On isotropic Berwald scalar curvature

In this short paper, we establish a closer relation between the Berwald scalar curvature and the $S$-curvature. In fact, we prove that a Finsler metric has isotropic Berwald scalar curvature if and only if it has weakly isotropic $S$-curvature. For Finsler metrics of scalar flag curvature and of weakly isotropic $S$-curvature, they have almost isotropic $S$-curvature if and only if the flag curvature is weakly isotropic.

Highly symmetric 2-plane fields on 5-manifolds and 5-dimensional Heisenberg group holonomy

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…

Banach等距猜想中的稳定性与近单色Finsler曲面

Stability in the Banach isometric conjecture and nearly monochromatic Finsler surfaces

本研究改进经典拓扑论证方法，得到Banach等距猜想k=2情形的稳定版本，同时证明非环面/克莱因瓶的近单色Finsler 2维曲面近似黎曼曲面，并用Banach-Mazur距离量化稳定性。

具有极小测地线的Finsler圆盘的最小面积

Minimal area of Finsler disks with minimizing geodesics

该研究证明内部测地线均为长度极小的Finsler圆盘的Holmes--Thompson面积下界，构造紧例并发现非旋转对称度量可达等号，通过随机测地线离散化度量证明定理，还得到相关积分几何公式在特定Finsler流形上成立的辅助结果。

SU(4)-holonomy via the left-invariant hypo and Hitchin flow

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M,g) with holonomy in Spin(7) starting with a cocalibrated G_2-structure on a seven-dimensional manifold. As Sp(2)\subseteq SU(4)\subseteq Spin(7), one may also obtain Calabi-Yau fourfolds or hyperKähler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics with Hol(g)\subseteq SU(4) but Hol(g)\neq Sp(2). We investigate when we actually get Hol(g)=SU(4) and obtain so many new explicit examp…

Spin(9) and almost complex structures on 16-dimensional manifolds

For a Spin(9)-structure on a Riemannian manifold M^16 we write explicitly the matrix psi of its Kähler 2-forms and the canonical 8-form Phi. We then prove that Phi coincides up to a constant with the fourth coefficient of the characteristic polynomial of psi. This is inspired by lower dimensional situations, related to Hopf fibrations and to Spin(7). As applications, formulas are deduced for Pontrjagin classes and integrals of Phi and Phi^2 in the special case of holonomy Spin(9).

关于带有特殊π-形式的广义Matsumoto度量

On Generalized Matsumoto Metrics with a Special $π$-form

本文研究了广义Matsumoto度量的构造及其几何性质，探讨了其与原度量的关系及不变性条件。

Fractional Almost Kahler - Lagrange Geometry

The goal of this paper is to encode equivalently the fractional Lagrange dynamics as a nonholonomic almost Kahler geometry. We use the fractional Caputo derivative generalized for nontrivial nonlinear connections (N-connections) originally introduced in Finsler geometry, with further developments in Lagrange and Hamilton geometry and, in our approach, with fractional derivatives. For fundamental geometric objects induced canonically by regular Lagrange functions, we construct compatible almost symplectic forms and linear connections completely determined by a "prime" Lagrange (in particular, F…

具有可约特征holonomy的异质G₂系统

The heterotic G$_2$ system with reducible characteristic holonomy

本文提出了一种在退化3-(α,δ)-Sasaki流形上构造异质G₂系统精确解的方法，通过分析G₂连接获得新解。

Systems of symplectic forms on four-manifolds

We study almost Hermitian 4-manifolds with holonomy algebra, for the canonical Hermitian connection, of dimension at most one. We show how Riemannian 4-manifolds admitting five orthonormal symplectic forms fit therein and classify them. In this set-up we also fully describe almost Kaehler 4-manifolds.

对称空间上芬斯勒度量的理想泊松-沃罗诺伊镶嵌

Ideal Poisson--Voronoi tessellations for Finsler metrics on symmetric spaces

本文研究高阶对称空间上芬斯勒度量的理想泊松-沃罗诺伊镶嵌，证明多面体芬斯勒度量对应的镶嵌可呈现前人发现的两种行为，并引入具有良好性质的理想塞尔伯格-泊松-沃罗诺伊镶嵌。

New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres

There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 6-manifolds play a distinguished role both in the general structure theory and also because of their connection with singular spaces with holonomy group the compact exceptional Lie group G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 if and only if M is a nearly Kaehler 6-manifold. A central problem in the field has been the absence of any complete …

On Landsberg general $(α,β)$-metrics with a conformal 1-form

In this paper, we study almost regular Landsberg general $(α,β)$-metrics in Finsler geometry. The corresponding equivalent equations are given. By solving the equations, we give the classification of Landsberg general $(α,β)$-metrics under the conditon that $β$ is closed and conformal to $α$. Under this condition, we prove that regular Landsberg general $(α,β)$-metrics must be Berwaldian when the dimension is greater than two. For the almost regular case, the classification also is given and some new non-Berwaldian Landsberg metrics are found.

调和Spin(7)流之分析

The analysis of the harmonic-spin(7) flow

本文研究八维黎曼流形上Spin(7)-结构的调和性概念，并分析其调和解流的解析性质，以比较等距Spin(7)-结构。

Nearly Kähler Geometry and $(2,3,5)$-Distributions via Projective Holonomy

We show that any dimension $6$ nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly Kähler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly Kähler and nearly para-Kähler parts separated by a hypersurface which canon…

Toric G_2 and Spin(7) holonomy spaces from gravitational instantons and other examples

Non-compact G_2 holonomy metrics that arise from a T^2 bundle over a hyper-Kahler space are discussed. These are one parameter deformations of the metrics studied by Gibbons, Lu, Pope and Stelle in hep-th/0108191. Seven-dimensional spaces with G_2 holonomy fibered over the Taub-Nut and the Eguchi-Hanson gravitational instantons are found, together with other examples. By considering the Apostolov-Salamon theorem math.DG/0303197, we construct a new example that, still being a T^2 bundle over hyper-Kahler, represents a non trivial two parameter deformation of the metrics studied in hep-th/010819…

Finsler Branes and Quantum Gravity Phenomenology with Lorentz Symmetry Violations

A consistent theory of quantum gravity (QG) at Planck scale almost sure contains manifestations of Lorentz local symmetry violations (LV) which may be detected at observable scales. This can be effectively described and classified by models with nonlinear dispersions and related Finsler metrics and fundamental geometric objects (nonlinear and linear connections) depending on velocity/ momentum variables. We prove that the trapping brane mechanism provides an accurate description of gravitational and matter field phenomena with LV over a wide range of distance scales and recovering in a systema…

一种并发广义Kropina变换

A Concurrent Generalized Kropina Change

本文研究具特定π-形式的广义Kropina度量，从容许并发π-向量场的芬斯勒流形出发考察φ-并发广义Kropina变换，分析其几何对象、给出实例，证明喷雾非射影相关等性质及变换保持几乎有理性质。

Quaternionic Kaehler and Spin(7) metrics arising from quaternionic contact Einstein structures

We construct left invariant quaternionic contact (qc) structures on Lie groups with zero and non-zero torsion and with non-vanishing quaternionic contact conformal curvature tensor, thus showing the existence of non-flat quaternionic contact manifolds. We prove that the product of the real line with a seven dimensional manifold, equipped with a certain qc structure, has a quaternionic Kaehler metric as well as a metric with holonomy contained in Spin(7). As a consequence we determine explicit quaternionic Kaehler metrics and Spin(7)-holonomy metrics which seem to be new. Moreover, we give expl…

The Relation Between the Associate Almost Complex Structure to $HM'$ and $(HM',S,T)$-Cartan Connections

In the present paper, the $(HM',S,T)$-Cartan connections on pseudo-Finsler manifolds, introduced by A. Bejancu and H.R. Farran, are obtained by the natural almost complex structure arising from the nonlinear connection $HM'$. We prove that the natural almost complex linear connection associated to a $(HM',S,T)$-Cartan connection is a metric linear connection with respect to the Sasaki metric $G$. Finally we give some conditions for $(M', J, G)$ to be a Kähler manifold.