Generalized Projectively Symmetric Spaces

Abstract

We study generalized projectively symmetric spaces. We first study some geometric properties of projectively symmetric spaces and prove that any such space is projectively homogeneous and under certain conditions the projective curvature tensor vanishes. Then we prove that given any regular projective s-space (𝑀, ∇), there exists a projectively related connection ∇, such that (𝑀, ∇) is an affine s-manifold.

1. Introduction

Affine and Riemannian s-manifolds were first defined in [1] following the introduction of generalized Riemannian symmetric spaces in [2]. They form a more general class than the symmetric spaces of E. Cartan. More details about generalized symmetric spaces can be found in the monograph [3]. Let 𝑀 be a connected manifold with an affine connection ∇, and let 𝐴(𝑀,∇) be the Lie transformation group of all affine transformation of 𝑀. An affine transformation 𝑠_𝑥 will be called an affine symmetry at a point 𝑥 if 𝑥 is an isolated fixed point of 𝑠_𝑥. An affine manifold (𝑀,∇) will be called an affine s-manifold if there is a differentiable mapping 𝑠∶𝑀→𝐴(𝑀,∇), such that for each 𝑥∈𝑀, 𝑠_𝑥 is an affine symmetry at 𝑥.

In [4] Podestà introduced the notion of a projectively symmetric space in the following sense. Let (𝑀,∇) be a connected 𝐶^∞ manifold with an affine torsion free connection ∇ on its tangent bundle; (𝑀,∇) is said to be projectively symmetric if for every point 𝑥 of 𝑀 there is an involutive projective transformation 𝑠_𝑥 of 𝑀 fixing 𝑥 and whose differential at 𝑥 is −Id. The assignment of a symmetry 𝑠_𝑥 at each point 𝑥 of 𝑀 can be viewed as a map 𝑠∶𝑀→𝑃(𝑀,∇), and 𝑃(𝑀,∇) can be topologised, so that it is a Lie transformation group. In the above definition, however, no further assumption on 𝑠 is made; even continuity is not assumed.

In this paper we define and state prerequisite results on projective structures and define projective symmetric spaces due toPodestà. Then we generalize them to define projective s-manifolds as manifolds together with more general symmetries and consider the cases where they are essential or inessential. A projective s-manifold is called inessential if it is projectively equivalent to an affine s-manifold and essential otherwise. We prove that these spaces are naturally homogeneous, and moreover under certain conditions the projective curvature tensor vanishes. Later we define regular projective s-manifolds and prove that they are inessential.

2. Preliminaries

Let 𝑀 be a connected real 𝐶^∞ manifold whose tangent bundle 𝑇𝑀 is endowed with an affine torsion free connection ∇. We recall that a diffeomorphism 𝑠 of 𝑀 is said to be projective transformation if 𝑠 maps geodesics into geodesics when the parametrization is disregarded [5]; equivalently 𝑠 is projective if the pull back 𝑠^∗∇ of the connection is projectively related to ∇, that is, if there exists a global 1-form 𝜋 on 𝑀, such that 𝑠^∗∇_𝑋𝑌=∇_𝑋𝑌+𝜋(𝑋)𝑌+𝜋(𝑌)𝑋∀𝑋,𝑌∈𝜒(𝑀). (1) If the form 𝜋 vanishes identically on 𝑀, then 𝑠 is said to be an affine transformation.

Definition 1. (𝑀,∇) is said to be projectively symmetric if for every point 𝑥 in 𝑀 there exists a projective transformation 𝑠_𝑥 with the following properties: (a)𝑠_𝑥(𝑥)=𝑥 and 𝑥 is an isolated fixed point of 𝑠_𝑥. (b)𝑠_𝑥 is involutive.

It is easy to see that conditions (a) and (b) imply (𝑑𝑠_𝑥)_𝑥=−Id. Moreover we recall that a projective transformation is determined if we fix its value at a point, its differential, and its second jet at this point [6], hence a symmetry at 𝑥 in 𝑀 is not uniquely determined in general by the condition (a) and (b).

Affine symmetric spaces are affine homogeneous, but in general projectively symmetric spaces are not projective homogeneous; for more detail and examples see [4, 7], but if following Ledger and Obata define the case of a differentiable distribution of projective symmetries in an affine manifold, then this happens.

Let (𝑀,𝑔) be a connected Riemannian manifold. An isometry 𝑠_𝑥 of (𝑀,𝑔) for which 𝑥∈𝑀 is an isolated fixed point will be called a Riemannian symmetry of 𝑀 at 𝑥. Clearly, if 𝑠_𝑥 is a symmetry of (𝑀,𝑔) at 𝑥, then the tangent map 𝑆_𝑥=(𝑑𝑠_𝑥)_𝑥 is an orthogonal transformation of 𝑇_𝑥𝑀 having no fixed vectors (with the exception of 0). An s-structure on (𝑀,𝑔) is a family {𝑠_𝑥∣𝑥∈𝑀} of symmetries of (𝑀,𝑔).

A Riemannian s-manifold is a Riemannian manifold (𝑀,𝑔) together with a map 𝑠∶𝑀→𝐼(𝑀,𝑔), such that for each 𝑥∈𝑀 the image 𝑠_𝑥 is a Riemannian symmetry at 𝑥.

For any affine manifold (𝑀,∇) let 𝐴(𝑀,∇) denote the Lie group of all affine transformation of (𝑀,∇). An affine transformation 𝑠_𝑥∈𝐴(𝑀,∇) for which 𝑥∈𝑀 is an isolated fixed point will be called an affine symmetry at 𝑥. An affine s-manifold is an affine manifold (𝑀,∇) together with a differentiable mapping 𝑠∶𝑀→𝐴(𝑀,∇), such that for each 𝑥∈𝑀, the image 𝑠_𝑥 is an affine symmetry at 𝑥.

Let 𝑀 be an affine s-manifold. Since 𝑠∶𝑀→𝐴(𝑀,∇) is assumed to be differentiable, the tensor field 𝑆 of type (1,1) defined by 𝑆_𝑥=(𝑑𝑠_𝑥)_𝑥 for each 𝑥∈𝑀 is differentiable. The tensor field 𝑆 is defined similarly for a Riemannian s-manifold, although it may not be smooth. For either affine or Riemannian s-manifolds we call 𝑆 the symmetry tensor field.

Following [3] an s-structure {𝑠_𝑥} is called regular if for every pair of points 𝑥,𝑦∈𝑀 as follows: 𝑠_𝑥∘𝑠_𝑦=𝑠_𝑧∘𝑠_𝑥,where𝑧=𝑠_𝑥𝑦. (2)

3. Projective s-Space

Let 𝑀 be a connected manifold with an affine connection ∇, and let 𝑃(𝑀,∇) bethe group of all projective transformations of 𝑀.

Definition 2. A projective transformation 𝑠_𝑥 will be called a projective symmetry or simply a symmetry at the point 𝑥, if 𝑥 is an isolated fixed point of 𝑠_𝑥 and (𝑑𝑠_𝑥)_𝑥=𝑆 does not leave any nonzero vector fixed.

Definition 3. A connected affine manifold (𝑀,∇) will be called a projective s-manifold or simply ps-manifold if for each 𝑥∈𝑀 there is a projective symmetry 𝑠_𝑥, such that the mapping 𝑠∶𝑀→𝑃(𝑀,∇), 𝑥↦𝑠_𝑥 is smooth.

A symmetry 𝑠_𝑥 will be called a symmetry of order 𝑘 at 𝑥, if there exist a positive integer 𝑘, such that 𝑠^𝑘_𝑥=Id, and 𝑀 will be called ps-manifold of order 𝑘, if 𝑘 is the least positive number such that each symmetry is of order 𝑘. Evidently every ps-manifold of order 2 is a projective symmetric space.

Lemma 4. Let 𝐺 be a topological transformation group acting on a connected topological space 𝑀, if for each point 𝑥 in 𝑀, the 𝐺-orbit of 𝑥 contains a neighborhood of 𝑥, then 𝐺 is transitive on 𝑀.

Proof. Since 𝐺 is transitive on each orbit, for each 𝑥 the 𝐺-orbit 𝐺(𝑥) of 𝑥 is open by our assumption. The complement 𝐶(𝑥) of 𝐺(𝑥) in 𝑀 is also open, being a union of orbits. Thus 𝐺(𝑥) is open and closed. It is nonempty and therefore coincides with the connected space 𝑀, thus 𝐺 is transitive.

Theorem 5. If 𝑀 is a 𝑝𝑠-manifold, then 𝑃(𝑀,∇) is transitive on 𝑀.

Proof. We fix a point 𝑥₀∈𝑀 and consider the 𝐶^∞ map ℎ∶𝑀→𝑀 given by ℎ(𝑥)=𝑠_𝑥(𝑥₀); since 𝑠_𝑥(𝑥)=𝑥 for every 𝑥 in 𝑀, the differential (𝑑ℎ)_𝑥0 of ℎ at the point 𝑥₀ is given by (𝑑ℎ)_𝑥0=𝐼−𝑆_𝑥0, where 𝑆_𝑥0 is the differential of 𝑠_𝑥0 at 𝑥₀. (𝑑ℎ)_𝑥0 is nonsingular because no eigenvalue of 𝑆_𝑥0 is equal to 1. Hence ℎ is a diffeomorphism on some neighborhood 𝑊 of 𝑥₀ in 𝑀, and ℎ(𝑊) is a neighborhood of 𝑥₀ contained in the 𝑃(𝑀,∇)-orbit 𝑃(𝑀,∇)_𝑥0 of 𝑥₀, therefore from the above lemma 𝑃(𝑀,∇) is transitive.

Definition 6. Let 𝑀 be a ps-manifold;since 𝑠∶𝑀→𝑃(𝑀,∇) is assumed to be differentiable, the tensor field 𝑆 of type (1,1) defined by 𝑆_𝑥=(𝑑𝑠_𝑥)_𝑥 is differentiable, we call 𝑆 the symmetry tensor field.

Lemma 7. If 𝑠_𝑥 is a projective symmetry of (𝑀,∇) then there exists a connection ∇ projectively equivalent with ∇ which is 𝑠_𝑥-invariant.

Proof. Since 𝑠_𝑥 is a projective symmetry of (𝑀,∇) then there is a 1-form 𝛼 on 𝑀, such that 𝑠^∗_𝑥∇_𝑋𝑌=∇_𝑋𝑌+𝛼(𝑋)𝑌+𝛼(𝑌)𝑋. (3) We are looking for a connection ∇ with the following properties: ∇_𝑋𝑌=∇_𝑋𝑌+𝜋(𝑋)𝑌+𝜋(𝑌)𝑋. (4) As ∇ should be 𝑠_𝑥-invariant we need 𝑠^∗_𝑥∇_𝑋𝑌=∇_𝑋𝑌 (5) that is, 𝑠_𝑥 is an affine transformation of (𝑀,∇). We have 𝑠^∗_𝑥∇_𝑋𝑌=𝑠_𝑥⁻¹_∗∇_{𝑠𝑥∗𝑋}𝑠_𝑥∗𝑌=𝑠_𝑥⁻¹_∗∇_{𝑠𝑥∗𝑋}𝑠_𝑥∗𝑌+𝜋𝑠_𝑥∗𝑋𝑠_𝑥∗𝑌+𝜋𝑠_𝑥∗𝑌𝑠_𝑥∗𝑋=𝑠^∗_𝑥∇_𝑋𝑌+𝜋𝑠_𝑥∗𝑋𝑌+𝜋𝑠_𝑥∗𝑌𝑋. (6) It follows from (3) 𝑠^∗_𝑥∇_𝑋𝑌=∇_𝑋(𝑋)(𝑌)𝑋𝑠𝑌+𝛼𝑌+𝛼+𝜋_𝑥∗𝑋𝑌+𝜋𝑠_𝑥∗𝑌𝑋. (7) From (5) we have ∇_𝑋𝑌+𝛼(𝑋)𝑌+𝛼(𝑌)𝑋+𝜋𝑠_𝑥∗𝑋𝑌+𝜋𝑠_𝑥∗𝑌𝑋=∇_𝑋𝑌+𝜋(𝑋)𝑌+𝜋(𝑌)𝑋, (8) thus it is enough to have for every vector field 𝑍 as follows: 𝛼(𝑍)+𝜋𝑠_𝑥∗𝑍=𝜋(𝑍) (9) which is equivalent to 𝜋𝑍−𝑠_𝑥∗𝑍=𝛼(𝑍) (10) or simply 𝜋∘𝐼−𝑠_𝑥∗𝑍=𝛼(𝑍) (11) since 𝑠_𝑥 is symmetry, then 𝐼−𝑠_𝑥 is invertible; hence we obtain 𝜋=𝛼∘𝐼−𝑠_𝑥∗⁻¹ (12) thus if we choose 𝜋 as (12), then (4) and (5) are true, and ∇ is the required connection.

So it would be convenient to introduce the following definition for connection ∇ and 1-form 𝜋=𝛼∘(𝐼−𝑠_𝑥∗)⁻¹.

Definition 8. Let (𝑀,∇) be a ps-manifold, and let 𝑠_𝑥 be the projective symmetry at the point 𝑥. Then we call the associate connection ∇ the fundamental connection of 𝑠_𝑥. Also the 1-form 𝜋=𝛼∘𝐼−𝑠_𝑥∗⁻¹ (13) will be called the fundamental 1-form of 𝑠_𝑥, where 𝛼 is the 1-form on 𝑀, such that 𝑠^∗∇_𝑋𝑌=∇_𝑋𝑌+𝜋(𝑋)𝑌+𝜋(𝑌)𝑋∀𝑋,𝑌∈𝜒(𝑀). (14)

Definition 9. The projective curvature tensor of (𝑀,∇) is defined as follows [5, 8]: 𝑊^𝑖_𝑗𝑘𝑙=Π^𝑖_𝑗𝑘𝑙−1𝛿𝑛−1^𝑖𝑘Π_𝑗𝑙−𝛿^𝑖_𝑙Π_𝑗𝑘, (15) where Π^𝑖_𝑗𝑘=Γ^𝑖_𝑗𝑘−2𝛿𝑛+1^𝑖(𝑗Γ^𝑙_𝑘)𝑙,Π^𝑖_𝑗𝑘𝑙=𝜕_𝑘Π^𝑖_𝑗𝑙−𝜕_𝑙Π^𝑖_𝑗𝑘+Π^ℎ_𝑗𝑙Π^𝑖_ℎ𝑘−Π^ℎ_𝑗𝑘Π^𝑖_ℎ𝑙,Π_𝑗𝑘=Π^ℎ_𝑗ℎ𝑘. (16) The projective curvature tensor 𝑊 is invariant with respect to projective transformations [5, 8].

Theorem 10. In a ps-manifold (𝑀,∇), let 𝑠_𝑥 be a symmetry, and let ∇ be the fundamental connection of 𝑠_𝑥, if ∇𝑆=0; that is, ((∇_𝑋𝑆)(𝑌)=𝜋(𝑌)𝑆(𝑋)−𝜋(𝑆(𝑌))𝑋), then (∇𝑊)_𝑥=0.

Proof. Let 𝑠∶𝑀→𝑃(𝑀,∇) be the 𝑝𝑠-structure and ∇𝑆=0. Let 𝑋,𝑌,𝑍∈𝑇_𝑥𝑀 be tangent vectors, and let 𝜔∈𝑇^∗_𝑥𝑀 be a covector at 𝑥∈𝑀. By parallel translation along each geodesics through 𝑥, 𝑋,𝑌,𝑍, and 𝜔 can be extended to local vector fields 𝑋,𝑌,𝑍, and 𝜔 with vanishing covariant derivative with respect to ∇ at 𝑥. Because 𝑆 is parallel, the local vector fields 𝑆𝑋,𝑆𝑌,𝑆𝑍, and 𝑆^∗𝜔 have also vanishing covariant derivative at 𝑥. (Here 𝑆^∗ denotes the transpose map to 𝑆.) As 𝑊 is invariant with respect to the projective transformation 𝑠_𝑝,𝑝∈𝑀, we have 𝑊𝑆^∗𝜔,𝑋,𝑌,𝑍=𝑊𝜔,𝑆𝑋,𝑆𝑌,𝑆𝑍. (17) Now we show that ∇𝑊(𝑆^∗𝜔,𝑋,𝑌,𝑍,𝑈) and ∇𝑊(𝜔,𝑆𝑋,𝑆𝑌,𝑆𝑍,𝑆𝑈) are equal at 𝑥. These are equal if and only if (𝑆^∗𝜔)(∇_𝑈𝑊(𝑋,𝑌,𝑍)) and 𝜔(∇_𝑆𝑈𝑊(𝑆𝑋,𝑆𝑌,𝑆𝑍)) are equal, which follows from the assumption on ∇. That is ∇𝑊𝑆^∗_𝑥𝜔,𝑋,𝑌,𝑍,𝑈=∇𝑊𝜔,𝑆_𝑥𝑋,𝑆_𝑥𝑌,𝑆_𝑥𝑍,𝑆_𝑥𝑈 (18) or ∇𝑊(𝜔,𝑋,𝑌,𝑍,𝑈)=∇𝑊𝑆^∗−1𝜔,𝑆𝑋,𝑆𝑌,𝑆𝑍,𝑆𝑈. (19) Differentiating covariantly (17) with respect to ∇ in the direction of 𝑆𝑈 at 𝑥 and using (19) we get ∇𝑊(𝜔,𝑋,𝑌,𝑍,𝑆𝑈)=∇𝑊(𝜔,𝑋,𝑌,𝑍,𝑈), (20) thus ∇𝑊_𝑥(𝜔,𝑋,𝑌,𝑍,(𝐼−𝑆)𝑈)=0, (21) for all 𝑋,𝑌,𝑍,𝑈∈𝑇_𝑥𝑀 and 𝜔∈𝑇^∗_𝑥𝑀, and because (𝐼−𝑆)_𝑥 is a nonsingular transformation we obtain ∇𝑊_𝑥=0. (22)

Theorem 11. Let (𝑀,∇) be a 𝑝𝑠-manifold of dimension 𝑛>2; if there exist two different projective symmetries 𝜎₁,𝜎₂ at a point 𝑞 of 𝑀, such that 𝜎_1∗𝑞=𝜎_2∗𝑞 and ∇𝑆=0, where ∇ is the fundamental connection corresponding to 𝜎₁, then the projective curvature tensor 𝑊; vanishes that is, 𝑀 is projectively flat.

Proof. By a similar method used in Proposition  1.1 of [7] the proof follows from Lemma 7 and Theorem 10.

Corollary 12. If (𝑀,∇) is a ps-manifold of order 2, and two different projective symmetry 𝜎₁,𝜎₂ can be defined at a point 𝑞, then 𝑀 is projectively flat.

Proof. It is evident from the fact that 𝜎_1∗𝑞=𝜎_2∗𝑞.

Proposition 13. Let (𝑀,∇) be ps-manifold, such that at every point 𝑥 of 𝑀 the projective symmetry is uniquely determined. Then the linear isotropy representation 𝜌∶𝑃(𝑀,∇)_𝑥→𝐺𝐿(𝑛,𝑅) is faithful for every 𝑥∈𝑀.

Proof. Since 𝑠_𝑥 and 𝑠_𝑥⁻¹ both are projective symmetry at 𝑥, then we have 𝑠_𝑥=𝑠_𝑥⁻¹; that is, 𝑠²_𝑥=Id, thus (𝑀,∇) is a ps-manifold of order 2. Now, our assertion follows from Theorem  1.1 of [7].

4. Regular Projective s-Space

Definition 14. A 𝑝𝑠-manifold (𝑀,∇) is called regular 𝑝𝑠-manifold or simply 𝑟𝑝𝑠-manifold if for all 𝑝,𝑞∈𝑀,   𝑠_𝑝∘𝑠_𝑞=𝑠_𝑧∘𝑠_𝑝, where 𝑧=𝑠_𝑝(𝑞).

Lemma 15. Let (𝑀,∇) be a regular ps-manifold, then the (1,1) tensor field 𝑆 is invariant under all symmetries 𝑠_𝑥; that is 𝑑𝑠_𝑥(𝑆𝑋)=𝑆𝑑𝑠_𝑥𝑋, (23) for all 𝑋∈𝜒(𝑀).

Proof. Since 𝑀 is regular ps-manifold, then for all 𝑋∈𝑇_𝑦𝑀 we have 𝑑(𝑠_𝑥∘𝑠_𝑦)𝑋=𝑑(𝑠_𝑧∘𝑠_𝑥)𝑋 and so 𝑑𝑠_𝑥(𝑆𝑋)=𝑆_𝑧(𝑑𝑠_𝑥𝑋). Thus 𝑆 is 𝑠_𝑥 invariant for all 𝑠_𝑥.

Lemma 16. Let (𝑀,∇) be a connected ps-manifold, such that at every point 𝑥 of 𝑀 the projective symmetry is uniquely determined, then (𝑀,∇) is rps-manifold.

Proof. Suppose 𝑝,𝑞∈𝑀 and 𝑧=𝑠_𝑝(𝑞); then from the uniqueness of the projective symmetry, we have 𝑠_𝑝∘𝑠_𝑞=𝑠_𝑧∘𝑠_𝑝, so (𝑀,∇) is regular ps-manifold.

Remark 17. A general question is to find condition under which, given a ps-manifold (𝑀,∇), there exists a projectively related connection ∇, such that (𝑀,∇) is an affine s-manifold; we shall call such spaces inessential ps-manifold and essential otherwise.

Definition 18. A ps-manifold (𝑀,∇) is called inessential ps-manifold if there exists a projectively related connection ∇ such that (𝑀,∇) is an affine s-manifold.

Let us denote by Φ_ℎ the 1-form corresponding to an element ℎ of 𝑃(𝑀,∇). We want to see when (𝑀,∇) is inessential, in order to show that (𝑀,∇) is inessential, we must find a connection ∇ which is projectively related to ∇ and is invariant under all symmetries. Let 𝑠_𝑞 be a symmetry at 𝑞, we must find a one-form 𝜋, such that ∇_𝑋𝑌=∇_𝑋𝑌+𝜋(𝑋)𝑌+𝜋(𝑌)𝑋. (24) As 𝑠_𝑞 is a projective transformation for (𝑀,∇) and leaves the connection ∇ invariant, we find that Φ_𝑠𝑞(𝑋)+𝜋𝑠_𝑞∗𝑋=𝜋(𝑋)∀𝑋∈𝜒(𝑀) (25) and hence at 𝑞 we have 𝜋_∣𝑞𝐼−𝑠_𝑞∗𝑞𝑋=Φ_𝑠𝑞(𝑋). (26) So we define a 1-form 𝜋 through the following formula: 𝜋_∣𝑥(𝑋)=Φ_{𝑠𝑥∣𝑥}∘𝐼−𝑠_𝑥∗𝑥⁻¹𝑋∀𝑋∈𝑇_𝑥𝑀. (27)

Theorem 19. Let (𝑀,∇) be an rps-manifold, then (𝑀,∇) is inessential.

Proof. We define a torsion free affine connection ∇ projectively related to ∇ through the fundamental 1-form of 𝑠_𝑥, 𝜋 as follows: 𝜋_∣𝑥(𝑋)=Φ_{𝑠𝑥∣𝑥}∘𝐼−𝑠_𝑥∗𝑥⁻¹𝑋∀𝑋∈𝑇_𝑥𝑀 (28) and prove that the connection ∇ is invariant under all the symmetries of 𝑀.
Let 𝑠_𝑞 be a symmetry at 𝑞 of 𝑀, the condition that ∇ is invariant under 𝑠_𝑞 is equivalent to 𝜋(𝑋)−𝜋𝑠_𝑞∗𝑋=Φ_𝑠𝑞(𝑋). (29) We verify (29) at a point 𝑝 of 𝑀, we have to prove that by (28) Φ_{𝑠𝑝∣𝑝}∘𝐼−𝑠_𝑝∗𝑝⁻¹𝑋−𝜋_{𝑠𝑞(𝑝)}𝑠_{𝑞∗∣𝑝}𝑋=Φ_{𝑠𝑝𝑞∣}(𝑋) (30) so if we put 𝑧=𝑠_𝑞(𝑝), (30) reduces to Φ_{𝑠𝑝∣𝑝}∘𝐼−(𝑠_𝑝)_∗𝑝⁻¹𝑋−Φ_{𝑠𝑧∣𝑧}∘𝐼−𝑠_𝑧∗𝑧⁻¹𝑠_{𝑞∗∣𝑝}𝑋=Φ_{𝑠𝑞∣𝑝}(𝑋). (31) But since 𝑠_𝑞∘𝑠_𝑝=𝑠_𝑧∘𝑠_𝑞, we have Φ_𝑠𝑝(𝑌)+Φ_𝑠𝑞𝑠_𝑝∗𝑌=Φ_𝑠𝑞(𝑌)+Φ_𝑠𝑧𝑠_𝑞∗𝑌. (32) Now evaluate (32) at 𝑝, and let 𝑌=(𝐼−(𝑠_𝑝)_∗𝑝)⁻¹𝑋; then as 𝑠_𝑞∘𝑠_𝑝=𝑠_𝑧∘𝑠_𝑞, we have (31), and we are done.

Remark 20. The authors have studied Finsler homogeneous and symmetric spaces [9]; recently Habibi and the second author generalized them to Finsler s-manifolds and weakly Finsler symmetric spaces [10, 11]. Therefore these concepts can be mixed and find more generalizations which will be the content of other papers.

Acknowledgment

The author would like to thank the anonymous referees for their suggestions and comments, which helped in improving the paper.