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 (M²ⁿ, 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 (M²ⁿ, g, J) satisfies

()

for every tangent vector X, Y, and Z. Then (M²ⁿ, g, J) has a constant holomorphic sectional curvature at x if and only if

()

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

()

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 (M²ⁿ, g, J)(n > 2) be an indefinite almost Hermitian manifold that satisfies (1.1), then (M²ⁿ, g, J) has a constant holomorphic sectional curvature at x if and only if

()

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 (M²ⁿ⁺¹, ϕ, η, ξ, g) (n ≥ 2) be an indefinite Sasakian manifold. Then M²ⁿ⁺¹ has a constant ϕ-sectional curvature if and only if

()

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 be an indefinite generalized g · f · f-space form. Then is of constant ϕ-sectional curvature if and only if

()

for every vector field X such that

, for any α ∈ {1, …, r}.

2. Preliminaries

A manifold is called a globally framed f-manifold (or g · f · f-manifold) if it is endowed with a nonnull (1,1)-tensor field of constant rank, such that is parallelizable; that is, there exist global vector fields , α ∈ {1, …, r}, with their dual 1-forms , satisfying and .

The g · f · f-manifold

, α ∈ {1, …, r}, is said to be an indefinite metric g · f · f-manifold if

is a semi-Riemannian metric with index ν (0 < ν < 2n + r) satisfying the following compatibility condition:

()

for any

, being ϵ_α = ±1 according to whether

is spacelike or timelike. Then, for any α ∈ {1, …, r}, one has

. 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

, 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:

()

where

is given by

An indefinite metric g · f · f-manifold is called an indefinite 𝒮-manifold if it is normal and , for any α ∈ {1, …, r}, where for any . The normality condition is expressed by the vanishing of the tensor field , being the Nijenhuis torsion of .

Furthermore, the Levi-Civita connection of an indefinite 𝒮-manifold satisfies

()

where

and

. We recall that

and

is an integrable flat distribution since

(see more details in [6]).

A plane section in is a -holomorphic section if there exists a vector orthogonal to such that span the section. The sectional curvature of a -holomorphic section, denoted by , is called a -holomorphic sectional curvature.

Proposition 2.1 (see [7].)An indefinite Sasakian manifold has -sectional curvature c if and only if its curvature tensor verifies

()

for any vector fields

A Sasakian manifold with constant -sectional curvature c ∈ ℝ is called a Sasakian space form, denoted by .

Definition 2.2. An almost contact metric manifold is an indefinite generalized Sasakian space form, denoted by , if it admits three smooth functions f₁, f₂, f₃ such that its curvature tensor field verifies

()

for any vector fields

Remark 2.3. Any indefinite generalized Sasakian space form has -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 has -sectional curvature c if and only if its curvature tensor verifies

()

for any vector fields

and ϵ = ∑ϵ_α.

An indefinite 𝒮-manifold with constant -sectional curvature c ∈ ℝ is called a 𝒮-space form, denoted by . 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 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 , is an indefinite g · f · f-manifold which admits smooth function F₁, F₂, ℱ such that its curvature tensor field verifies

()

for any vector fields

For r = 1, we obtain an indefinite Sasakian space form 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 be an indefinite generalized g · f · f-space form. Then is of constant ϕ-sectional curvature if and only if

()

for every vector field X such that

, for any α ∈ {1, …, r}.

Proof. Let 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 is of constant -holomorphic sectional curvature. Then (3.1) gives

()

Conversely, let {X, Y} be an orthonormal pair of tangent vectors such that

, α ∈ {1, …, r}, and n ≥ 3. Then

and

also form an orthonormal pair of tangent vectors such that

. Then (3.1) and curvature properties give

()

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 -holomorphic, then for , where a and b are constant, we have

()

Similarly,

()

These imply

()

If span {U, V} is not

-holomorphic section, then we can choose unit vectors

and

such that span {X, Y} is

-holomorphic. Thus we get

()

which shows that any

-holomorphic section has the same

-holomorphic sectional curvature.

Now, let n = 2, and let {X, Y} be a set of orthonormal vectors such that and , and we have c(X) = c(Y) as before. Using the property (3.2), we get

()

Now, define

such that a² − b² = 1 and a² ≠ b². Using the above relations, we get

()

Therefore, we have

()

On the other hand,

()

Comparing (3.11) and (3.12), we get

()

On solving (3.13), we have

()

Similary, we can prove

()

Therefore,

has constant

-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 , for any α ∈ {1, …, r}, such that , and consider the null vector U_α = ξ_α + Y. From (3.2),

()

Therefore,

()

From Cases 1 and 2, depending on the sign of ϵ_α,

is constant, and hence c(U_α) = c(Y_α) is constant.

Theorem 3.3 (see [9].)Let (n ≥ 2) be an indefinite 𝒮-manifold. Then M^2n+r is of constant ϕ-sectional curvature if and only if

()

for every vector field X such that

, 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 (M²ⁿ⁺¹, ϕ, η, ξ, g)(n ≥ 2) be an indefinite Sasakian manifold. Then M²ⁿ⁺¹ is of constant ϕ-sectional curvature if and only if

()

for every vector field X such that g(X, ξ) = 0.

Proof. When r = 1, an indefinite 𝒮-space form M²ⁿ⁺¹(c) reduces to a Sasakian space form. The proof follows from (2.4) and Theorem 3.3.

References

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

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1. Introduction

2. Preliminaries

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

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

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1. Introduction

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

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