Constancy of 𝝓-Holomorphic Sectional Curvature for an Indefinite Generalized 𝑔⋅𝑓⋅𝑓-Space Form

Abstract

Bonome et al., 1997, provided an algebraic characterization for an indefinite Sasakian manifold to reduce to a space of constant 𝜙-holomorphic sectional curvature. In this present paper, we generalize the same characterization for indefinite 𝑔⋅𝑓⋅𝑓-space forms.

1. Introduction

For an almost Hermitian manifold (𝑀^2𝑛,𝑔,𝐽) with dim(𝑀)=2𝑛>4, Tanno [1] has proved the following.

Theorem 1.1. Let dim(𝑀)=2𝑛>4, and assume that almost Hermitian manifold (𝑀^2𝑛,𝑔,𝐽) satisfies 𝑅(𝐽𝑋,𝐽𝑌,𝐽𝑍,𝐽𝑋)=𝑅(𝑋,𝑌,𝑍,𝑋) (1.1) for every tangent vector 𝑋, 𝑌, and 𝑍. Then (𝑀^2𝑛,𝑔,𝐽) has a constant holomorphic sectional curvature at 𝑥 if and only if 𝑅(𝑋,𝐽𝑋)𝑋isproportionalto𝐽𝑋 (1.2) for every tangent vector 𝑋 at 𝑥∈𝑀.

Tanno [1] has also proved an analogous theorem for Sasakian manifolds as follows.

Theorem 1.2. A Sasakian manifold ≥5 has a constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋isproportionalto𝜙𝑋 (1.3) for every tangent vector 𝑋 such that 𝑔(𝑋,𝜉)=0.

Nagaich [2] has proved the generalized version of Theorem 1.1, for indefinite almost Hermitian manifolds as follows.

Theorem 1.3. Let (𝑀^2𝑛,𝑔,𝐽)(𝑛>2) be an indefinite almost Hermitian manifold that satisfies (1.1), then (𝑀^2𝑛,𝑔,𝐽) has a constant holomorphic sectional curvature at 𝑥 if and only if 𝑅(𝑋,𝐽𝑋)𝑋isproportionalto𝐽𝑋 (1.4) for every tangent vector 𝑋 at 𝑥∈𝑀.

Bonome et al. [3] generalized Theorem 1.2 for an indefinite Sasakian manifold as follows.

Theorem 1.4. Let (𝑀^2𝑛+1,𝜙,𝜂,𝜉,𝑔)(𝑛≥2) be an indefinite Sasakian manifold. Then 𝑀^2𝑛+1 has a constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋isproportionalto𝜙𝑋 (1.5) for every vector field 𝑋 such that 𝑔(𝑋,𝜉)=0.

In this paper, we generalize Theorem 1.4 for an indefinite generalized 𝑔⋅𝑓⋅𝑓-space form by proving the following.

Theorem 1.5. Let (𝑀^2𝑛+𝑟,𝐹₁,𝐹₂,ℱ) be an indefinite generalized 𝑔⋅𝑓⋅𝑓-space form. Then 𝑀^2𝑛+𝑟 is of constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋isproportionalto𝜙𝑋 (1.6) for every vector field 𝑋 such that 𝑔(𝑋,𝜉_𝛼)=0, for any 𝛼∈{1,…,𝑟}.

2. Preliminaries

A manifold 𝑀 is called a globally framed f-manifold (or 𝑔⋅𝑓⋅𝑓-manifold) if it is endowed with a nonnull (1,1)-tensor field 𝜙 of constant rank, such that ker𝜙 is parallelizable; that is, there exist global vector fields 𝜉_𝛼, 𝛼∈{1,…,𝑟}, with their dual 1-forms 𝜂^𝛼, satisfying 𝜙²=−𝐼+∑^𝑟_𝛼=1𝜂^𝛼⊗𝜉_𝛼 and 𝜂^𝛼(𝜉_𝛽)=𝛿^𝛼_𝛽.

The 𝑔⋅𝑓⋅𝑓-manifold (𝑀^2𝑛+𝑟,𝜙,𝜉_𝛼,𝜂^𝛼), 𝛼∈{1,…,𝑟}, is said to be an indefinite metric 𝑔⋅𝑓⋅𝑓-manifold if 𝑔 is a semi-Riemannian metric with index 𝜈(0<𝜈<2𝑛+𝑟) satisfying the following compatibility condition: 𝑔𝜙𝑋,𝜙𝑌=𝑔(𝑋,𝑌)−^𝑟_𝛼=1𝜖_𝛼𝜂^𝛼(𝑋)𝜂^𝛼(𝑌), (2.1) for any 𝑋,𝑌∈Γ(𝑇𝑀), being 𝜖_𝛼=±1 according to whether 𝜉_𝛼 is spacelike or timelike. Then, for any 𝛼∈{1,…,𝑟}, one has 𝜂^𝛼(𝑋)=𝜖_𝛼𝑔(𝑋,𝜉_𝛼). Following the notations in [4, 5], we adopt the curvature tensor 𝑅, and thus we have 𝑅(𝑋,𝑌,𝑍)=∇_𝑋∇_𝑌𝑍−∇_𝑌∇_𝑋𝑍−∇_[𝑋,𝑌]𝑍 and 𝑅(𝑋,𝑌,𝑍,𝑊)=𝑔(𝑅(𝑍,𝑊,𝑌),𝑋), for any 𝑋, 𝑌, 𝑍, 𝑊∈Γ(𝑇𝑀).

We recall that, as proved in [6], the Levi-Civita connection ∇ of an indefinite 𝑔⋅𝑓⋅𝑓-manifold satisfies the following formula:2𝑔∇_𝑋𝜙𝑌,𝑍=3𝑑Φ𝑋,𝜙𝑌,()+𝜙𝑍−3𝑑Φ𝑋,𝑌,𝑍𝑔𝑁(𝑌,𝑍),𝜙𝑋+𝜖_𝛼𝑁^𝜙_𝛼(𝑌,𝑍)𝜂^𝛼(𝑋)+2𝜖_𝛼𝑑𝜂^𝛼𝜙𝑌,𝑋𝜂^𝛼(𝑍)−2𝜖_𝛼𝑑𝜂^𝛼𝜙𝑍,𝑋𝜂^𝛼(𝑌), (2.2) where 𝑁^𝜙_𝛼 is given by 𝑁^𝜙_𝛼(𝑋,𝑌)=2𝑑𝜂^𝛼(𝜙𝑋,𝑌)−2𝑑𝜂^𝛼(𝜙𝑌,𝑋).

An indefinite metric 𝑔⋅𝑓⋅𝑓-manifold is called an indefinite 𝒮-manifold if it is normal and 𝑑𝜂^𝛼=Φ, for any 𝛼∈{1,…,𝑟}, where Φ(𝑋,𝑌)=𝑔(𝑋,𝜙𝑌) for any 𝑋,𝑌∈Γ(𝑇𝑀). The normality condition is expressed by the vanishing of the tensor field 𝑁=𝑁_𝜙+∑^𝑟_𝛼=12𝑑𝜂^𝛼⊗𝜉_𝛼, 𝑁_𝜙 being the Nijenhuis torsion of 𝜙.

Furthermore, the Levi-Civita connection of an indefinite 𝒮-manifold satisfies∇_𝑋𝜙𝑌=𝑔𝜙𝑋,𝜙𝑌𝜉+𝜂(𝑌)𝜙²(𝑋), (2.3) where 𝜉=∑^𝑟_𝛼=1𝜉_𝛼 and 𝜂=∑^𝑟_𝛼=1𝜖_𝛼𝜂^𝛼. We recall that ∇_𝑋𝜉_𝛼=−𝜖_𝛼𝜙𝑋 and ker𝜙 is an integrable flat distribution since ∇_𝜉𝛼𝜉_𝛽=0 (see more details in [6]).

A plane section in 𝑇_𝑝𝑀 is a 𝜙-holomorphic section if there exists a vector 𝑋∈𝑇_𝑝𝑀 orthogonal to 𝜉₁,…,𝜉_𝑟 such that {𝑋,𝜙𝑋} span the section. The sectional curvature of a 𝜙-holomorphic section, denoted by 𝑐(𝑋)=𝑅(𝑋,𝜙𝑋,𝜙𝑋,𝑋), is called a 𝜙-holomorphic sectional curvature.

Proposition 2.1 (see [7]). An indefinite Sasakian manifold (𝑀^2𝑛+1,𝜙,𝜉,𝜂,𝑔) has 𝜙-sectional curvature 𝑐 if and only if its curvature tensor verifies 𝑅()()𝑋,𝑌𝑍=𝑐+34𝑔(𝑌,𝑍)𝑋−𝑔()𝑌+()𝑋,𝑍𝑐−14Φ(𝑋,𝑍)𝜙𝑌−Φ(𝑌,𝑍)𝜙𝑋+2Φ(𝑋,𝑌)−𝜙𝑍𝑔(𝑍,𝑌)𝜂(𝑋)𝜉+𝑔(𝑍,𝑋)𝜂(𝑌)𝜉−𝜂(𝑌)𝜂(𝑍)𝑋+𝜂(𝑍)𝜂(𝑋)𝑌 (2.4) for any vector fields 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀).
A Sasakian manifold 𝑀^2𝑛+1 with constant 𝜙-sectional curvature 𝑐∈ℝ is called a Sasakian space form, denoted by 𝑀^2𝑛+1(𝑐).

Definition 2.2. An almost contact metric manifold (𝑀^2𝑛+1,𝜙,𝜉,𝜂,𝑔) is an indefinite generalized Sasakian space form, denoted by 𝑀^2𝑛+1(𝑓₁,𝑓₂,and𝑓₃), if it admits three smooth functions 𝑓₁, 𝑓₂, 𝑓₃ such that its curvature tensor field verifies 𝑅(𝑋,𝑌)𝑍=𝑓₁𝑔(𝑌,𝑍)𝑋−𝑔(𝑋,𝑍)𝑌+𝑓₂Φ(𝑋,𝑍)𝜙𝑌−Φ(𝑌,𝑍)𝜙𝑋+2Φ(𝑋,𝑌)𝜙𝑍+𝑓₃−𝑔(𝑍,𝑌)𝜂(𝑋)𝜉+𝑔(𝑍,𝑋)𝜂(𝑌)𝜉−𝜂(𝑌)𝜂(𝑍)𝑋+𝜂(𝑍)𝜂(𝑋)𝑌 (2.5) for any vector fields 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀).

Remark 2.3. Any indefinite generalized Sasakian space form has 𝜙-sectional curvature 𝑐=𝑓₁+3𝑓₂. Indeed, 𝑓₁=(𝑐+3)/4 and 𝑓₂=𝑓₃=(𝑐−1)/4.

Proposition 2.4 (see [6]). An indefinite 𝒮-manifold 𝑀^2𝑛+𝑟 has 𝜙-sectional curvature 𝑐 if and only if its curvature tensor verifies 𝑅()()𝑋,𝑌𝑍=𝑐+3𝜖4𝑔𝜙𝑋,𝜙𝑍𝜙²𝑌−𝑔𝜙𝑌,𝜙𝑍𝜙²𝑋+(𝑐−𝜖)4Φ(𝑍,𝑌)𝜙𝑋−Φ(𝑍,𝑋)𝜙𝑌+2Φ(𝑋,𝑌)+𝜙𝑍𝜂(𝑍)𝜂(𝑋)𝜙²𝑌−𝜂(𝑌)𝜂(𝑍)𝜙²𝑋+𝑔𝜙𝑍,𝜙𝑌𝜂(𝑋)𝜉−𝑔𝜙𝑍,𝜙𝑋𝜂(𝑌)𝜉 (2.6) for any vector fields 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀) and 𝜖=∑𝜖_𝛼.
An indefinite 𝒮-manifold 𝑀^2𝑛+𝑟 with constant 𝜙-sectional curvature 𝑐∈ℝ is called a 𝒮-space form, denoted by 𝑀^2𝑛+𝑟(𝑐). One remarks that for 𝑟=1 (2.6) reduces to (2.4).

3. An Indefinite Generalized 𝑔⋅𝑓⋅𝑓-Manifold

Let ℱ denote any set of smooth functions 𝐹_𝑖𝑗 on 𝑀^2𝑛+𝑟 such that 𝐹_𝑖𝑗=𝐹_𝑗𝑖 for any 𝑖,𝑗∈{1,…,𝑟}.

Definition 3.1. An indefinite generalized 𝑔⋅𝑓⋅𝑓-space-form, denoted by (𝑀^2𝑛+𝑟,𝐹₁,𝐹₂,ℱ), is an indefinite 𝑔⋅𝑓⋅𝑓-manifold (𝑀^2𝑛+𝑟,𝜙,𝜉_𝛼,𝜂^𝛼,𝑔) which admits smooth function 𝐹₁,𝐹₂,ℱ such that its curvature tensor field verifies 𝑅(𝑋,𝑌)𝑍=𝐹₁𝑔𝜙𝑋,𝜙𝑍𝜙²𝑌−𝑔𝜙𝑌,𝜙𝑍𝜙²𝑋+𝐹₂Φ(𝑍,𝑌)𝜙𝑋−Φ(𝑍,𝑋)𝜙𝑌+2Φ(𝑋,𝑌)+𝜙𝑍_{𝑟𝛼,𝛽=1}𝐹_𝛼𝛽𝜂^𝛼(𝑋)𝜂^𝛽(𝑍)𝜙²𝑌−𝜂^𝛼(𝑌)𝜂^𝛽(𝑍)𝜙²𝑋+𝑔𝜙𝑍,𝜙𝑌𝜂^𝛼(𝑋)𝜉_𝛽−𝑔𝜙𝑍,𝜙𝑋𝜂^𝛼(𝑌)𝜉_𝛽 (3.1) for any vector fields 𝑋,𝑌,𝑍,𝑊∈Γ(𝑇𝑀).
For 𝑟=1, we obtain an indefinite Sasakian space form 𝑀^2𝑛+1(𝑓₁,𝑓₂,𝑓₃) with 𝑓₁=𝐹₁, 𝑓₂=𝐹₂, and 𝑓₃=𝐹₁−𝐹₁₁. In particular, if the given structure is Sasakian, (3.1) holds with 𝐹₁₁=1, 𝐹₁=(𝑐+3)/4, 𝐹₃=(𝑐−1)/4, and 𝑓₃=𝐹₁−𝐹₁₁=(𝑐−1)/4=𝑓₂.

Theorem 3.2. Let (𝑀^2𝑛+𝑟,𝐹₁,𝐹₂,ℱ) be an indefinite generalized 𝑔⋅𝑓⋅𝑓-space form. Then 𝑀^2𝑛+𝑟 is of constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋isproportionalto𝜙𝑋 (3.2) for every vector field 𝑋 such that 𝑔(𝑋,𝜉_𝛼)=0, for any 𝛼∈{1,…,𝑟}.

Proof. Let (𝑀^2𝑛+𝑟,𝐹₁,𝐹₂,ℱ) be an indefinite generalized 𝑔⋅𝑓⋅𝑓-space form. To prove the theorem for 𝑛≥2, we will consider cases when 𝑛=2 and when 𝑛>2, that is, when 𝑛≥3.
Case 1 (𝑔(𝑋,𝑋)=𝑔(𝑌,𝑌)). The proof is similar as given by Lee and Jin [8], so we drop the proof. Case 2 (𝑔(𝑋,𝑋)=−𝑔(𝑌,𝑌)). Here, if 𝑋 is spacelike, then 𝑌 is timelike or vice versa. First of all, assume that 𝑀 is of constant 𝜙-holomorphic sectional curvature. Then (3.1) gives 𝑅𝑋,𝜙𝑋𝑋=𝐹₁+3𝐹₂𝜙𝑋=𝑐𝜙𝑋. (3.3) Conversely, let {𝑋,𝑌} be an orthonormal pair of tangent vectors such that 𝑔(𝜙𝑋,𝑌)=𝑔(𝑋,𝑌)=𝑔(𝑌,𝜉_𝛼)=0, 𝛼∈{1,…,𝑟}, and 𝑛≥3. Then 𝑋̈=(𝑋+𝑖𝑌)/√2 and 𝑌̈=(𝑖𝜙𝑋+𝜙𝑌)/√2 also form an orthonormal pair of tangent vectors such that 𝑔(𝜙𝑋̈,𝑌̈)=0. Then (3.1) and curvature properties give 0=𝑅𝑋̈,𝜙̈̈𝑋̈=𝑋,𝑌,𝑔𝑅𝑋,𝜙𝑋𝑋,𝜙𝑋−𝑔𝑅𝑌,𝜙𝑌𝑌,𝜙𝑌−2𝑔𝑅𝑋,𝜙𝑌𝑌,𝜙𝑌+2𝑔𝑅𝑋,𝜙𝑋𝑌,𝜙𝑋. (3.4) From the assumption, we see that the last two terms of the right-hand side vanish. Therefore, we get 𝑐(𝑋)=𝑐(𝑌).
Now, if span{𝑈,𝑉} is 𝜙-holomorphic, then for 𝜙𝑈=𝑎𝑈+𝑏𝑉, where 𝑎 and 𝑏 are constant, we have span𝑈,𝜙𝑈=span{𝑈,𝑎𝑈+𝑏𝑉}=span{𝑈,𝑉}. (3.5)
Similarly, span𝑉,𝜙𝑉=span{𝑈,𝑉},span𝑈,𝜙𝑈=span𝑉,𝜙𝑉. (3.6) These imply 𝑅𝑈,𝜙𝑈,𝑈,𝜙𝑈=𝑅𝑉,𝜙𝑉,𝑉,𝜙𝑉,or𝑐(𝑈)=𝑐(𝑉). (3.7) If span{𝑈,𝑉} is not 𝜙-holomorphic section, then we can choose unit vectors 𝑋∈span{𝑈,𝜙𝑈}^⟂ and 𝑌∈span{𝑉,𝜙𝑉}^⟂ such that span{𝑋,𝑌} is 𝜙-holomorphic. Thus we get 𝑐(𝑈)=𝑐(𝑋)=𝑐(𝑌)=𝑐(𝑉), (3.8) which shows that any 𝜙-holomorphic section has the same 𝜙-holomorphic sectional curvature.
Now, let 𝑛=2, and let {𝑋,𝑌} be a set of orthonormal vectors such that 𝑔(𝑋,𝑋)=−𝑔(𝑌,𝑌) and 𝑔(𝑋,𝜙𝑋)=0, and we have 𝑐(𝑋)=𝑐(𝑌) as before. Using the property (3.2), we get 𝑅𝑋,𝜙𝑋𝑋=−𝐹₁+3𝐹₂𝜙𝑋=−𝑐(𝑋)𝜙𝑋,𝑅𝑋,𝜙𝑋𝑌=−2𝐹₂𝜙𝑌,𝑅𝑋,𝜙𝑌𝑋=−𝐹₁𝜙𝑌,𝑅𝑋,𝜙𝑌𝑌=𝐹₂𝜙𝑋,𝑅𝑌,𝜙𝑋𝑌=𝐹₁𝜙𝑋,𝑅𝑌,𝜙𝑋𝑋=−𝐹₂𝜙𝑌,𝑅𝑌,𝜙𝑌𝑋=2𝐹₂𝜙𝑋,𝑅𝑌,𝜙𝑌𝑌=𝐹₁+3𝐹₂𝜙=𝑐(𝑌)𝜙𝑌=𝑐(𝑋)𝜙𝑌. (3.9) Now, define 𝑋=𝑎𝑋+𝑏𝑌 such that 𝑎²−𝑏²=1 and 𝑎²≠𝑏². Using the above relations, we get 𝑅𝑋,𝜙𝑋𝑋=𝐶₁𝜙𝑋+𝐶₂𝜙𝑌. (3.10)
Therefore, we have 𝐶₁=−𝑎³𝑐(𝑋)+𝑎𝑏²𝑐(𝑋),𝐶₂=𝑏³𝑐(𝑋)−𝑎²𝑏𝑐(𝑋). (3.11) On the other hand, 𝑅𝑋,𝜙𝑋𝑋=𝑐𝑋𝜙𝑋=𝑐𝑋𝑎𝜙𝑋+𝑏𝜙𝑌. (3.12) Comparing (3.11) and (3.12), we get −𝑎²𝑐(𝑋)+𝑏²𝑐(𝑋)𝑋,𝑏=𝑐₂𝑐(𝑋)−𝑎²𝑐(𝑋)=𝑐𝑋. (3.13) On solving (3.13), we have 𝑐(𝑋)=𝑐𝑋. (3.14) Similary, we can prove 𝑐(𝑌)=𝑐𝑌. (3.15) Therefore, 𝑀 has constant 𝜙-holomorphic sectional curvature. Case 3 (𝑔(𝑈,𝑈)=0). It is enough to show a sufficient condition. Let 𝑌_𝛼 be a unit vector tangent to 𝜉_𝛼, for any 𝛼∈{1,…,𝑟}, such that 𝑔(𝑌_𝛼,𝑌_𝛼)=−𝑔(𝜉_𝛼,𝜉_𝛼)=−𝜖_𝛼, and consider the null vector 𝑈_𝛼=𝜉_𝛼+𝑌. From (3.2), 𝑐𝑈_𝛼𝜙𝑈_𝛼=𝑐𝑈_𝛼𝜙𝜉_𝛼+𝑌_𝛼=𝑅𝜉_𝛼+𝑌_𝛼,𝜙𝜉_𝛼+𝑌_𝛼𝜉_𝛼+𝑌_𝛼. (3.16) Therefore, 𝑐𝑈_𝛼=𝑔𝑐𝑈_𝛼𝜙𝜉_𝛼+𝑌_𝛼,𝜖_𝛼𝜙𝑌_𝛼=𝜖_𝛼𝑔𝑅𝜉_𝛼+𝑌_𝛼,𝜙𝜉_𝛼+𝑌_𝛼𝜉_𝛼+𝑌_𝛼,𝜙𝑌_𝛼=𝜖_𝛼𝑔𝑅𝜉_𝛼,𝜙𝜉_𝛼𝜉_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝜉_𝛼,𝜙𝑌_𝛼𝜉_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝜉_𝛼,𝜙𝜉_𝛼𝑌_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝜉_𝛼,𝜙𝑌_𝛼𝑌_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝜉_𝛼𝜉_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝑌_𝛼𝜉_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝜉_𝛼𝑌_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝑌_𝛼𝑌_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝜉_𝛼,𝜙𝑌_𝛼𝜉_𝛼,𝜙𝑌_𝛼+2𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝑌_𝛼𝜉_𝛼,𝜙𝑌_𝛼+𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝑌_𝛼𝑌_𝛼,𝜙𝑌_𝛼=𝜖_𝛼𝑔𝑅𝑌_𝛼,𝜙𝑌_𝛼𝑌_𝛼,𝜙𝑌_𝛼. (3.17) From Cases 1 and 2, depending on the sign of 𝜖_𝛼, 𝑔(𝑅(𝑌_𝛼,𝜙𝑌_𝛼)𝑌_𝛼,𝜙𝑌_𝛼)=𝜖_𝛼𝑐(𝑌_𝛼) is constant, and hence 𝑐(𝑈_𝛼)=𝑐(𝑌_𝛼) is constant.

Theorem 3.3 (see [9]). Let (𝑀^2𝑛+𝑟,𝜙,𝜂^𝛼,𝜉_𝛼,𝑔)(𝑛≥2) be an indefinite 𝒮-manifold. Then 𝑀^2𝑛+𝑟 is of constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋𝑖𝑠𝑝𝑟𝑜𝑝𝑜𝑟𝑡𝑖𝑜𝑛𝑎𝑙𝑡𝑜𝜙𝑋 (3.18) for every vector field 𝑋 such that 𝑔(𝑋,𝜉_𝛼)=0, for any 𝛼∈{1,…,𝑟}.

Proof. An 𝒮-space form is a special case of 𝑔⋅𝑓⋅𝑓-space form, and hence the proof follows from Theorem 3.2 and (2.6).

Theorem 3.4 (cf. Bonome et al. [3]). Let (𝑀^2𝑛+1,𝜙,𝜂,𝜉,𝑔)(𝑛≥2) be an indefinite Sasakian manifold. Then 𝑀^2𝑛+1 is of constant 𝜙-sectional curvature if and only if 𝑅𝑋,𝜙𝑋𝑋isproportionalto𝜙𝑋 (3.19) for every vector field 𝑋 such that 𝑔(𝑋,𝜉)=0.

Proof. When 𝑟=1, an indefinite 𝒮-space form 𝑀^2𝑛+1(𝑐) reduces to a Sasakian space form. The proof follows from (2.4) and Theorem 3.3.