Ju TAN(譚舉)School of Microelectronics and Data Science，Anhui University of Technology,Maanshan 243032,China E-mail : tanju200r@163.com

Ming XU(許明)+School of Mathematical Sciences，Capital Normal University，Beijing 100048，China E-mail: mgmgmgxu@163.com

In this paper, we discuss natural reductiveness in Finsler geometry, which has been given from different view points, by Latifi in [22], and by Deng and Hou in [9].

The equivalence between these two definitions was proven in [36].It should be remarked that, unlike the situation in Riemannian geometry, the Finslerian natural reductiveness cannot be implied from the normal homogeneity, and in some sense, it is even stronger, because naturally reductive Finsler metrics are Berwaldian.

The first goal of this paper is to characterize the natural reductiveness of non-Riemannian(α1,α2) metric.The (α1,α2) metrics are defined in [12].We believe that these generalize Randers metrics [26] and (α,β)-metrics [23], and they share some similar properties [1, 12, 34,35].The f-product in [5] is a special case of the Berwald (α1,α2) metric; it is defined as a Finsler metric F = f(α1,α2) on M1×M2, where each αiis a Riemannian metric on Mi.A general(α1,α2) metric can be similarly presented as F =f(α1,α2), but each αiis only defined on a (possibly non-integrable) distribution Viin T M with T M =V1+V2.

From a geometric point of view, we prove the following explicit and simple description:

Theorem 1.1Any non-Riemannian naturally reductive homogeneous (α1,α2) metric is locally isometric to an f-product between two naturally reductive Riemannian metrics.

For connected and simply connected manifolds,the local f-product description in Theorem 1.1 is global and the inverse of Theorem 1.1 is valid (see Corollary 3.3).With some minor changes, the proof of Theorem 1.1 might argue in favour of a similar description for the non-Riemannian naturally reductive (α,β) manifold.

Notice that naturally reductive Finsler metrics are Berwald[9].These results are consistent with to Szab′o’s description for Berwald metrics, i.e., they are some local “product” among Riemannian manifolds and non-Riemannian affine symmetric spaces[29].Here the speciality of the (α1,α2) metric type specifies how the “product” is built and constrains the product factor and the factor number.

Switching to the algebraic point of view,we prove that if the non-Riemannian homogeneous(α1,α2) metric F is defined using the homogeneous Riemannian metric α, then the natural reductiveness of F is in fact inherited from α (see Corollary 3.4).This byproduct generalizes the corresponding result for (α,β) metrics (see [25, Theorem 3.7]).Meanwhile, we see that a non-Riemannian homogeneous (α1,α2) manifold (M,F) may achieve its natural reductiveness for diffferent homogeneous space representations M = G/H.The assertion that M is locally an f-product does not imply that G is also locally a product (see Example 3.6).

The second task of this paper is to discuss the curvature properties of a homogeneous(α1,α2) manifold.The S-curvature was found by Shen when he proved the volume comparison theorem in Finsler geometry [27].The Hessian of the S-curvature S(x,y) for its y-entries essentially provides the mean Berwald curvature (E-curvature in short).E-curvature helps us study the celebrated Landsberg Conjecture [17].In general Finsler geometry, there are many properties related to the S-curvature(for example, the vanishing, constant,isotropic and weakly isotropic S-curvature conditions), and there are many others related to the E-curvature(for example, the vanishing, constant and isotropic E-curvature conditions).These properties are closely related,and some are equivalent in homogeneous Finsler geometry[33].If we specify the homogeneous metric to be of Randers or(α,β) type, even more equivalences can be proven([6, 11, 30]).We transport these observation to homogeneous (α1,α2) metrics and prove the following theorem:

Theorem 1.2For a homogeneous (α1,α2) metric, the following are equivalent:

(1) it has weakly isotropic S-curvature;

(2) it has vanishing S-curvature;

(3) it has isotropic E-curvature;

(4) it has vanishing E-curvature.

Flag curvature generalizes sectional curvature in Riemannian geometry [2], but its calculation in general and homogeneous Finsler geometry is much harder.However, for a naturally reductive Finsler metric, the vanishing of the spray vector field greatly simplifies the homogeneous flag curvature formula [16].For a naturally reductive (α1,α2) metric of the form F =αφ(α2/α), we present an explicit flag curvature formula (see Theorem 4.4).

The arrangement of the rest of this paper is as follows: in Section 2, we summarize some preliminaries in general and homogeneous Finsler geometry.In Section 3, we prove Theorem 1.1.In Section 4,we prove Theorem 1.2 and calculate the flag curvature formula for a naturally reductive (α1,α2) metric.

2 Preliminaries

In this section, we summarize some fundamental knowledge on general Finsler geometry from [2, 28], and recall the basic settings of homogeneous Finsler geometry in [7].The notion of the (α1,α2) metric is from [12].

2.1 Minkowski Norms and Finsler Metrics

Definition 2.1LetVbe a finite dimensional real vector space.A Minkowski norm onVis a continuous function F :V→[0,+∞) satisfying the following properties:

(1) regularity: F is positive and smooth onV{0};

(2) positive 1-homogeneity: F(λy)=λF(y) for any y ∈Vand λ ≥0;

(3) strong convexity: for any y ∈V{0}, the fundamental tensor

Definition 2.2A Finsler metric on a smooth manifold M is a continuous function F :T M →[0,+∞) such that

(1) F is smooth on the slit tangent bundle T M{0};

(2) the restriction of F to each tangent space TxM is a Minkowski norm.

We also call the pair (M,F) a Finsler manifold or a Finsler space.

For example, a Finsler metric is Riemannian when the fundamental tensor (gij(x,y)) depends on x ∈M only, for any standard local coordinate x = (xi) ∈M and y = yi?xi∈TxM,or, equivalently, the Cartan tensor

vanishes everywhere.

A Randers metric has the form F = α+β, where α is a Riemannian metric and β is a one-form with a pointwise α-norm smaller than 1.An(α,β)metric has the form F =αφ(β/α),where φ(s) is a positive smooth function of one variable.

A Minkowski norm or a Finsler metric is said to be reversible if opposite (tangent) vectors have the same length.Obviously, all (α1,α2) metrics (including Riemannian metrics) are reversible, and most (α,β) metrics (including Randers metrics) are not.The (α,β) and (α1,α2)metrics can achieve the maximal non-Euclidean linear symmetry degree in each tangent space,i.e., O(n -1) and O(n1)×O(n2), respectively, where n = n1+n2is the dimension of the manifold.Thus the calculation and formulae for these metrics have relatively low complexity.

2.2 E-curvature, S-curvature and Flag Curvature

For a Finsler manifold (M,F), the geodesic spray is the smooth tangent vector field G on T M