EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 10, No. 5, 2017, 1099-1111 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global On Finsler s-manifolds Parisa Bahmandoust1, Dariush Latifi1,∗ 1 Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran Abstract. Finsler s−manifolds are a generalization of Riemannian s−manifolds. An important property of such manifolds is the homogeneity. In this paper we study Finsler s−manifolds. We first construct some example of Finsler s−manifolds which are neither Riemannian nor symmetric. Then we consider symmetric preserving diffeomorphism of Finsler s−manifolds. Finally we give some algebraic and existence theorem of these spaces. 2010 Mathematics Subject Classifications: 53C60, 53C30. Key Words and Phrases: Finsler s−manifold, Symmetric Finsler space, Generalized symmetric space, Homogeneous Finsler space. 1. Introduction Finsler manifold is a generalization of the Riemannian one, in the same as Riemannian manifold is for the Euclidean. A metric depends on the point and the direction. A Finsler metric on a manifold is a family of Minkowski norms on tangent spaces. Let (M,F ) be a Finsler space, where F is positively homogeneous of degree one. Then we have two ways to define the notion of an isometry of (M,F ). On the one hand, we call a diffeomorphism σ of M onto itself an isometry if F (dσx(y)) = F (y), for any x ∈ M and y ∈ TxM . On the other hand, we can also define an isometry of (M,F ) to be a one-to-one mapping of M onto itself which preserves the distance of each pair of points of M . It is well known that the two definitions are equivalent if the metric F is Riemannian. The equivalence of these two definitions in the general Finsler case is a result of S. Deng and Z. Hou [2]. Using these result, they proved that the group of isometries I(M,F ) of a Finsler space (M,F ) is a Lie transformation group of M and for any point x ∈ M , the isotropic subgroup Ix(M,F ) is a compact subgroup of I(M,F ). These results are important to study homogenous and symmetric Finsler spaces, for example [3, 4, 5, 12, 13, 14]. Symmetric spaces and generalized symmetric spaces have appeared to be very rich in content, stimulating the research in Lie groups, Mechanics, Physics, Gravity etc. The definition of symmetric Finsler space is a naturall generalization of E. Cartan’s definition of Riemannian symmetric spaces [8]. We call a Finsler space (M,F ) a symmetric ∗Corresponding author. Email addresses: bahmandoust.p@uma.ac.ir (P. Bahmandoust), latifi@uma.ac.ir (D. Latifi) http://www.ejpam.com 1099 c© 2017 EJPAM All rights reserved. P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1100 Finsler space if for any point p ∈M there exists an involutive isometry sp of (M,F ) such that p is an isolated fixed point of sp, [6, 5, 9, 12]. Affine and Riemannian s−manifold were first defined in [18] following the introduction of generalized Riemannian symmetric spaces in [19]. They form a more general class than the symmetric spaces [11]. An isometry of (M,F ) with an isolated fixed point x ∈M is called a symmetry of (M,F ) at x. A family {sx|x ∈ M} of symmetries of a connected Finsler space (M,F ) is called an s−structure of (M,F ), [7]. Σ−spaces and reduced Σ−spaces were first introduced by Loos as a generalization of reflection spaces and symmetric spaces [20]. He then proved that any Σ−space with compact Σ is a fibre bundle over a re- duced Σ−space. Basic properties of any reduced Σ−space M and affine and Riemannian Σ−space and Finsler Σ−space was given in [15, 21]. In this paper we are concerned with properties of Finsler spaces admitting such an s−structure. We construct some example of Finsler s−manifolds which are neither Rie- mannian nor symmetric. Then we study symmetry preserving diffeomorphism of Finsler s−manifolds and show that the group of symmetry preserving diffeomorphism is a transi- tive group. We then study some existence theorems and consider some geometric proper- ties of Finsler s−manifolds. 2. Preliminaries Let M be an n−dimensional smooth manifold without boundary and TM denote its tangent bundle. A Finsler structure on M is a map F : TM −→ [0,∞) which has the following properties [1]: (i) F is smooth on T̃M := TM\{0}. (ii) F (x, λy) = λF (x, y), for any x ∈M,y ∈ TxM and λ > 0. (iii) F 2 is strongly convex, i.e., gij(x, y) := 1 2 ∂2F 2 ∂yi∂yj (x, y) is positive definite for all (x, y) ∈ T̃M . Let V = vi∂/∂xi be a non-vanishing vector field on an open subset U ⊂ M . One can introduce a Riemannian metric gV and a linear connection ∇V on the tangent bundle over U as following [1] : gV (X,Y ) = XiY jgij(x, v), ∀X = Xi ∂ ∂xi , Y = Y i ∂ ∂xi , ∇V∂ ∂xi ∂ ∂xj = Γkij(x, v) ∂ ∂xk . From the torsion freeness and g−compatibility of Chern connection we have ∇VXY −∇VYX = [X,Y ], P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1101 XgV (Y, Z) = gV (∇VXY,Z) + gV (Y,∇VXZ) + 2CV (∇VXV, Y, Z), where CV is the Cartan tensor defined by CV (X,Y, Z) = XiY jZkCijk(x, v), Cijk(x, v) = 1 4 ∂3F 2(x, v) ∂yi∂yj∂yk , and it satisfies CV (V,X, Y ) = 0. Given a nonzero vector field V on a Finsler manifold (M,F ) with connection ∇V , one can consider the curvature tensor RV defined by RV (X,Y )Z = ∇VX∇VY Z −∇VY∇VXZ −∇V[X,Y ]Z. For a flag (V, σ) consisting of a nonzero tangent vector V ∈ TxM and a plane σ ⊂ TxM spanned by the tangent vectors V,W , the flag curvature is defined as K(V, σ) = K(V,W ) = gV (RV (V,W )W,V ) gV (V, V )gV (W,W )− gV (V,W )2 . In the Riemannian case the flag curvature is the sectional curvature of the plane σ and does not depend on V . A diffeomorphism, ϕ : M −→M , is an isometry on a Finsler manifold (M,F ) if it preserves the Finsler function: F (ϕ(x), dϕx(X)) = F (x,X) ∀x ∈M,X ∈ TxM. By the classical Dantzing-van der Waereden Theorem ([10]vol I, chapter I, Theorem 4.7 ) and the Montgomery- Zippin Theorem ([10],vol I, chapter I, Theorem 4.6), the group of isometries on a connected Finsler manifold form a Lie group. Strictly speaking, these theorems prove the statement for absolute homogeneous Finsler functions. For positive homogeneous Finsler functions consider the metric, d∗, defined by the function F ∗(X) = F (X) + F (−X) Then the G is a closed subgroup of G∗ defined for d∗. Thus both groups are Lie groups [22]. 3. Finsler s-manifolds Affine and Riemmannian s−manifolds were first defined in [18] following the introduc- tion of generalized Riemannian spaces in [19]. They form a more general class than the symmetric spaces of E. Cartan. Let (M, g) be a connected Riemannian manifold. A symmetry at x ∈ M is a isometry of (M, g) for which x is an isolated fixed point. A s−structure on (M, g) is a family {sx}x∈M P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1102 such that sx is a symmetry at x ∈ M , for each x ∈ M . An s−structure is called regular if for any two points x, y ∈M sx ◦ sy = sz ◦ sx, z = sx(y). If {sx}x∈M is regular, then the map s : M −→ I(M, g), x −→ sx is always C∞, here I(M, g) denotes the group of isometries of (M, g). An s−structure {sx}x∈M is called of order k if (sx)k = idM for all x ∈M and k is the minimal number with this property. It is well known that if (M, g) admits an s−structure, then it always admits an s−structure of finite order. Further if (M, g) admits a regular s−structure, then (M, g) admits a regular s−structure of finite order [11]. In particular if (M, g) admits an s−structure of order two then it is a usual Riemannian symmetric space. Let (M,F ) be a Finsler space, where F is positively homogeneous but not necessarily absolutely homogeneous. We introduce isometries of (M,F ) which form a Lie transfor- mation group of M as a result of S. Deng and Z. Hou [2] and moreover for any point x ∈ M , the isotropic subgroup Ix(M,F ) is a compact subgroup of I(M,F ), the group of isometries, which can be used to study homogeneous and symmetric Finsler spaces. The definition of symmetric Finsler space is a natural generalization of E. Cartan’s definition of Riemannian symmetric space [5], [6], [12]. We call a Finsler space (M,F ) a symmetric Finsler space if for any point p ∈M there exists an involutive isometry sp of (M,F ) such that p is an isolated fixed point of sp. If we drop the involution property in the definition of symmetric Finsler space keeping the property sx ◦ sy = sz ◦ sx, z = sx(y), we get a bigger class of Finsler manifolds as symmetric Finsler spaces. The definition of Finsler s−manifolds is a natural generalization of definition of Rie- mannian s−manifolds [7, 16]. Definition 1. Let (M,F ) be a connected Finsler space. An isometry on (M,F ) with an isolated fixed point x will be called a symmetry at x, and will usually be written as sx. Definition 2. A family {sx|x ∈M} of symmetries on a connected Finsler manifold (M,F ) is called an s−structure on (M,F ) An s−structure {sx|x ∈M} is called of order k (k ≥ 2) if (sx)k = id for all x ∈M and k is the least integer of this property. Obviously a Finsler space is symmetric if and only if it admits an s−structure of order 2. An s−structure {sx|x ∈ M} on (M,F ) is called regular if for every pair of points x, y ∈M sx ◦ sy = sz ◦ sx, z = sx(y). Definition 3. A Finsler s−manifold is a connected Finsler manifold (M,F ) admitting a regular s−structure and a Finsler space (M,F ) is said to be k−symmetric (k ≥ 2) if it admits a regular s−structure of order k. P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1103 Here we construct some Finsler s−manifolds which are non-Riemannian and non- symmetric. Example 1. Let k be a constant |k| < 1√ 3 . Consider the following Randers metric on R3, F (p1, p2, p3, y1, y2, y3) = √ y21 + y22 + y23 + k(y1 + y2 + y3) where p = (p1, p2, p3) ∈ R3 and (y1, y2, y3) ∈ TpR3. Define sp(x1, x2, x3) = (x3 − p3 + p1, x1 − p1 + p2, x2 − p2 + p3), for any p ∈ R3 we clearly see that sp is an isometry of F such that p is an isolated fixed point of sp. It is evident that s3p = id and s2p 6= id and sp 6= id and sp ◦ sq(x) = sz ◦ sp(x), z = sp(q). So {sp} is a regular 3-structure on (R3, F ) � Example 2. Let G be a compact connected Lie group. Consider the coset space (G × G)/G∗, where G∗ is the diagonal of G×G. (G×G)/G∗ is diffeomorphic to G via the map (g1, gg)G ∗ −→ g1g −1 2 G×G acts on G by (g1, g2)y = g1yg −1 2 . The isotropy group at the origin e ∈ G is G∗. Now define σ : G×G −→ G×G by σ(g1, g2) = (g2, g1), which is an involute automorphism. The fixed point set is (G×G)σ = G∗ and σ induces the map s : G −→ G, s(g) = g−1. Take a bi-invariant absolutely homogeneous Finsler metric F on G. Then F is invariant with respect to the action of G × G on G. It is also invariant with respect to s. Then (G,F ) is a symmetric Finsler space [17]. We now consider the more general case of Gk+1/G∗ where Gk+1 is the direct product of G with itself (k+1) times, and G∗ is the diagonal of Gk+1. We have Gk+1/G∗ ∼= Gk via π : Gk+1 −→ Gk, where π(g1, ..., gk+1) = (g1g −1 k+1, ..., gkg −1 k+1). Further define σ : Gk+1 −→ Gk+1 by σ(g1, ..., gk+1) = (gk+1, g1, ..., gk). P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1104 Then σ is an automorphism of order k + 1. It induces a map s : Gk −→ Gk defined by s(g1, ..., gk) = (g−1k , g1g −1 k , ..., gk−1g −1 k ). Let F be a bi-invariant Finsler metric on G. Then F generates a bi-invariant Finsler metric F k+1 on Gk+1 such that (Gk+1, F k+1) ∼= (G,F )× ...× (G,F ) Then F k+1 induces a Gk+1−invariant Finsler metric F [k] on Gk. The Finsler space (Gk, F [k]) is a (k+1)-symmetric Finsler space. Similar to the Riemannian case (Gk, F [k]) is not a symmetric space.� Example 3. Let (G1/H1, g1) , (G2/H2, g2) be two Riemannian s−manifolds with H1 and H2 compact and {τp} , {σq} be s−structures on G1/H1 , G2/H2, respectively of order k. Let M = G1/H1 × G2/H2 and o1 , o2 be the origins of G1/H1, G2/H2 respectively, and denote the origin of M by o = (o1, o2). Now for y = y1 + y2 ∈ ToM = To1(G1/H1) + To2(G2/H2), we define F (y) = √ g1(y1, y1) + g2(y2, y2) + s √ g1(y1, y1)s + g2(y2, y2)s where s is any integer ≥ 2. Then F (y) is a Minkowski norm on ToM which is invariant under H1 × H2. Hence it defines a G−invariant Finsler metric on M . It is easy to see that Finsler manifold (M,F ) is non-Riemannian s−manifold with regular s−structure {τp × σq}.� Given an s−structure {sx|x ∈ M} on (M,F ) we shall always denote by S the tensor field of type (1, 1) defined by Sx = (sx)∗ for all x ∈ M . Suppose there exists a nonzero vector X ∈ TxM such that SxX = X. Since sx is isometry, sx(expx(tX)), |t| < ε is a geodesic. Now expx(tX) and sx(expx(tX)) are two geodesics through x with the same initial vector X. Therefore, for any |t| < ε we have sx(expx(tX)) = expx(tX). But this contradicts to assumption that x is an isolated fixed point of sx. Therefore Sx has no non-zero invariant vector. Theorem 1. Let (M,F ) be a Finsler s−manifold. Then we have (a) For any x ∈M , Sx = (dsx)x has no invariant vector, (b) (M,F ) is homogeneous. That is, the group of isometries of (M,F ), I(M,F ), acts transitively on M . P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1105 (c) (M,F ) is forward complete. Proof: see [7].� Theorem 2. Let (M,F ) be a Finsler s−manifol with regular s−structure {sx}. Then there is a unique connection ∇̃ on M such that (i) ∇̃ is invariant under all sx (ii) ∇̃S = 0 Proof: The proof is similar to the Riemannian case [11].� If the Finsler space (M,F ) is of Berwald type, then ∇̃ is given by the formula ∇̃XY = ∇XY − (∇(I−S)−1XS)(S−1Y ) where ∇ is the Chern connection of (M,F ). Definition 4. Let (M,F ) be a generalized symmetric Finsler space, and let {sx} be the regular s−structure of (M,F ). Then a diffeomorphism φ : M −→ M is called symmetry preserving if φ(sx(y)) = sφ(x)φ(y) for all x, y ∈M . Obviously, all symmetries sx are symmetry preserving due to sx◦sy = sz◦sx, z = sx(y). We denote the group of symmetry preserving diffeomorphism by Aut({sx}). Let us denote by A(M) the Lie group of all affine transformations of M with respect to the connection ∇̃. Each symmetry preserving diffeomorphism is an affine transformation of (M, ∇̃), i.e. Aut(M, {sx}) ⊂ A(M). Lemma 1. An affine transformation φ ∈ A(M) is symmetry preserving if and only if it preserves the tensor field S. Consequently, Aut({sx}) is a closed subgroup of A(M) and hence a Lie transformation group of M . Proof: Let φ ∈ A(M) be symmetry preserving transformation then for each x ∈ M , maps φ ◦ sx , sφ(x) ◦ φ coincide, so (φ ◦ sx)∗x = (sφ(x) ◦ φ)∗x. Then φ preserves the tensor field S. On the other hand if φ ∈ A(M) preserves the tensor field S then for each x ∈M , (φ ◦ sx)∗x = (sφ(x) ◦ φ)∗x. Because φ ◦ sx and sφ(x) ◦ φ are affine transformations, so φ ◦ sx = sφ(x) ◦ φ that is φ is symmetry preserving map.� In the following we show that the group Aut({sx}) of all symmetry preserving diffeo- morphisms of (M,F ) is a transitive Lie transformation group. Theorem 3. The Lie transformation group Aut({sx}) act transitively on M . Proof: Let K ⊂ Aut({sx}) be the transformation group of M generated algebraicaly by all the symmetries sx, x ∈ M . Choose an origin o ∈ M . Let K(o) be the orbit of o with respect to K. Consider the map f(x) = sx(p) where p ∈ K(o) and x ∈ M . Clearly f(p) = p. For v ∈ TpM we have f∗p(v) = (Ip − Sp)v. Hence f∗p = (Ip − Sp) is a non-singular transformation and f maps a neighborhood U of p diffeomorphically onto P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1106 a neighborhood V of p. We get V ⊂ K(o) and the orbit K(o) is open. The union of all other orbits of K must be also open and hence K(o) is closed. Consequently K(o) = M . � Let V be a finite dimensional vector space and T : V −→ V an endomorphism. Then there is a unique decomposition V = V0T +V1T of V into T−invariant subspaces such that the restriction of T to V0T is nilpotent and the restriction of T to V1T is an automorphism. Definition 5. A regular homogeneous s−manifold is a triplet (G,H, σ), where G is a connected Lie group, H its closed subgroup and σ an automorphism of G such that (i) G◦σ ⊂ H ⊂ Gσ where Gσ is the subgroup consisting of the fixed points of σ in G and G◦σ denotes the identity component of Gσ. (ii) If T denotes the linear endomorphism Id− σ∗, then g0T = h. Clearly if (G,H, σ) is a regular homogeneous s−manifold, then g0T = h = ker T and g1T = Im(T ). Let G be a connected Lie group and H its closed subgroup. Consider the homogeneous manifold G/H. Here π : G −→ G/H will denote the canonical projection, and o = π(H) the origin of G/H. Let g and h be the Lie algebras of G and H respectively. Suppose that there is a subspace m ⊂ g such that g = h + m (direct sum of vector spaces) and Ad(h)m = m for every h ∈ H. Then the homogeneous space G/H is said to be reductive with respect to the decomposition g = h + m. Lemma 2. Let (G,H, σ) be a regular homogeneous s−manifold, then the homogeneous space G/H is reductive with respect to the decomposition g = h + g1T Proof: Let (G,H, σ) be a regular homogeneous s−manifold, since H ⊂ Gσ we obtain Ad(h) ◦ σ∗ = σ∗ ◦Ad(h) for each h ∈ H. Hence Ad(h) commutes with T on g and Ad(h)(g1T ) = (Ad(h) ◦ T )(g) = T (Ad(h)g) = T (g) = g1T . � Theorem 4. Let G be a connected Lie group, H its closed subgroup and σ an automor- phism of G such that (i) (Gσ)◦ ⊂ H ⊂ Gσ, (ii) σk = id, where k is the minimum number with this property, Then the triple (G,H, σ) is a regular homogeneous s−manifold of order k. P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1107 Proof: Let T = id− σ∗. We have to show that g0T = h. Clearly h = KerT and hence h ⊂ g0T . Suppose now that there is X ∈ g0T such that X is not in h. Without loss of generality we assume that TX 6= 0 , T 2X = 0. Then we get σ∗X = TX − σ2∗X = X − σ∗(X − σ∗X). Let Z = σ∗(X − σ∗X) = TX. So we have σ∗X = X − Z and σ∗Z = Z. Hence by the induction we get σ2∗X = X − 2Z . . . σk∗X = X − kZ Now Since σk∗X = X, we get Z = 0, a contradiction. This completes the proof. � Theorem 5. Let (G,H, σ) be a regular homogeneous s−manifold, π : G −→ G/H the canonical projection and let F be a G−invariant Finsler metric on G/H such that the transformation s of G/H determined by σ, i.e. s ◦ π = π ◦ σ is metric preserving at the origin eH of G/H. Then G/H is a Finslerian s−manifold and the symmetry sx is given by sx = g ◦ s ◦ g−1 g ∈ G, x = π(g) Proof: We will identify the elements of G with the corresponding transformations of M = G/H. Choose g ∈ G and x ∈M then x = π(g′) for some g′ ∈ G. Now, (s ◦ g ◦ s−1)(x) = (s ◦ g ◦ s−1 ◦ π)(g′) = (s ◦ g ◦ π)(σ−1(g′)) = (s ◦ π)(gσ−1(g′)) = (π ◦ σ)(gσ−1(g′)) = π(σ(g)g′) = σ(g)[π(g′)] = σ(g)(x). Hence we get s ◦ g ◦ s−1 = σ(g) g ∈ G (1) So for h ∈ H we obtain s ◦ h ◦ s−1 = h and hence h ◦ s ◦ h−1 = s. Consequently the transformation g ◦ s ◦ g−1 always depends only on π(g) and sπ(g) = g ◦ s ◦ g−1 g ∈ G defines a family {sx|x ∈ M} of diffeomorphisms of M . We can also easily that (x, y) −→ sx(y) is differentiable. Further for x ∈M , x = π(g) we have x = g(o) and hence sx(x) = (g ◦ s ◦ g−1)(x) = x, P. Bahmandoust, D. Latifi / Eur. J. Pure Appl. Math, 10 (5) (2017), 1099-1111 1108 because s(o) = o. Now for x, y ∈M put sx = g ◦ s ◦ g−1, sy = g′ ◦ s ◦ (g′)−1, where x = g(o) and y = g′(o). Then (g ◦ s ◦ g−1 ◦ g′ ◦ s−1)(o) = sx(g′(o)) = sx(y), on the other hand, (1) yields g◦s◦g−1◦g′◦s−1 = gσ(g−1g′). Thus, the map g◦s◦g−1◦g′◦s−1 coincides with the action of an element g′′ ∈ G, g′′(o) = sx(y). Now sx ◦ sy = g ◦ s ◦ g−1 ◦ g′ ◦ s ◦ (g′)−1 = g′′ ◦ s ◦ (g′′)−1 ◦ g ◦ s ◦ g−1 = ssx(y) ◦ sx. It remains to prove that sx∗ has no fixed vector except the null vector. If we identify g with TeG, then the projection π∗e : TeG −→ ToM induces an isomorphism of g1T onto ToM . From the relation π∗ ◦ σ∗ = s∗ ◦ π∗ we can see that π∗ ◦T = (Io− s∗o) ◦ π∗. Because T is an automorphism on g1T , Io − s∗o is an automorphism of ToM . From sπ(g) = g ◦ s ◦ g−1, g ∈ G, x = π(g), we obtain easily that Ip−Sp is an automorphism of TpM for each p ∈M . Thus {sx|x ∈M} is a regular s−structure on (M,F ).� Corollary 1. Let (G,H, σ) be a regular homogeneous s−manifold of order k, with the G−invariant Finsler metric F on G/H such that the transformation s of G/H determined by π ◦ σ = s ◦ π is metric preserving at the origin eH of G/H. Then G/H is a Finsler s−manifold of order k. Proof: It is a consequence of Theorem 4 and Theorem 5.� Theorem 6. Let (M,F ) be a Finsler s−manifold and o ∈M a fixed point. Let G be the identity component of the symmetry preserving group Aut({M, sx}) and Go the isotropy subgroup of G at o. Define a map σ : G −→ Aut(M, {sx}) by the formula σ(g) = so ◦ g ◦ s−1o g ∈ G. Then σ is an automorphism of G, and (G,Go, σ) is a regular homogeneous s−manifold. The symmetries sx are given by the formula sπ(g) = g ◦ so ◦ g−1, x = π(g) and M ' G/Go. Proof: By Theorem 3 Aut(M, {sx}) is transitive on M and M is connected. So G is also transitive on M . Obviously, the map σ given by σ(g) = so ◦ g ◦ s−1o g ∈ G REFERENCES 1109 is an isomorphism and σ(G) = G. Let Go be the isotropy group of G at o, then M ' G/Go. Let π : G −→ G/Go 'M be the canonical projection. Then for any g ∈ G we have (π ◦ σ)(g) = σ(g)(o) = (so ◦ g ◦ s−1o )(o) = so(g(o)) = (so ◦ π)(g). Hence on G we have π ◦ σ = so ◦ π. (2) Because g ∈ G is a symmetry preserving diffeomorphism, we have g ◦ sx = sg(x) ◦ g. In particular, for each h ∈ G we have σ(h) = so ◦ h ◦ s−1o = sh(o) ◦ h ◦ s−1o = h, so Go ⊂ Gσ. Let g, go, g σ denote the Lie algebra of G,Go, G σ respectively from (1) we have π∗ ◦ σ∗ = so ◦ π∗ on g = TeG. Let T = I − σ∗, hence on g we have π∗ ◦ T = (I − So) ◦ π∗. (3) Consider the decomposition g = g0T + g1T . Clearly go ⊆ gσ ⊆ g0T . Now we show that go = gσ = g0T . Suppose that there is a vector X ∈ g0T − go. Then X ′ = π∗(X) is a non-zero vector of To(M). On the other hand, T i(X) = 0 for some i, and from (2) we obtain (I − So)iX ′ = 0 for some i. Because (I − S0) is invertible, we get X ′ = 0, a contradiction. Hence go = gσ = g0T . Consequently (G,Go, σ) is a regular homogeneous s−manifold. Because g ∈ G is a symmetry preserving diffeomorphism we get sπ(g) = sg(o) = g ◦ so ◦ g−1 � References [1] D. Bao, S. S. Chern and Shen, An Introduction to Riemann-Finsler geometry, Springer-Verlag, New-York. 2000. [2] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific. J. Math. 207(1) (2002) 149-155. REFERENCES 1110 [3] S. Deng, Z. 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