Warped product manifolds are arguably the most natural and significant generalization of Riemannian product manifolds. Bishop and O’Neill in [3] introduced the concept of warped product manifolds to construct manifolds of nonpositive curvature. Later O’Neill in [20] extended the theory of Riemannian warped product manifolds for pseudo-Riemannian warped product manifolds. More precisely, let (B, g_B) and (F, g_F) be two pseudo-Riemannian manifolds and let f : B → (0,∞) be a positive differentiable function. Consider the product manifold B × F with projections

The warped product B ×_f F is the manifold B × F endowed with the pseudo-Riemannian structure such that

(or equivalently, g = g_B + (f)²g_F,) for any X, Y ∈ Γ(TM), where ‘*’ stands for derivation map. The function f is called the warping function of the warped product.

The theory of warped products attained attention when Chen, in [7], introduced a new class of CR-submanifolds [5] called CR-warped product Kaehlerian manifolds and investigated some existence and non-existence results. Hasegawa-Mihai [15] and Munteanu [19] continued this line of research for Sasakian manifold, which can be viewed as an odd-dimensional counterpart of a Kähler manifold. Since then, several differential geometer’s have studied the existence (or non-existence) of warped product submanifolds in Riemannian metric manifolds [9, 10, 25].

In addition to Riemannian metric manifolds, manifolds with neutral metric structures (i.e., para-Hermitian manifolds) have been explored in various geometric and physical problems, and the theory successfully applied to supersymmetric field theory, string theory and black holes [13, 16, 17, 21, 22]. The study of neutral metric manifolds was less prominant until Davidov et al. in [12] presented some analogies and differences between the structures admitting the neutral metric and the Riemannian metric. Thereafter, Chen and others initiated the study of the geometry of pseudo-Riemannian warped product submanifolds in para complex (contact) manifolds [1, 11, 24]. Motivated by the above-mentioned works, in this paper we investigate the existence (or non-existence) of ℘ℛ-semi-slant warped product submanifolds in para-Kähler manifolds by assuming proper and improper slant factor aspects.

The organization of the article is as follows. In Section 2 we recall some fundamental concepts about para-Kähler manifolds and their submanifolds, warped product manifolds, along with a few relevant results required for the present study. Section 3 deals with the definitions of slant, ℘ℛ-semi-slant submanifolds, and derivations of the necessary and sufficient conditions for integrability and totally geodesic foliation of the distributions associated with the definition of ℘ℛ-semi-slant submanifolds. In Section 4, we derive non-existence results for non-trivial ℘ℛ-semi-slant warped product submanifolds of the forms M_T ×_f M_λ and M_λ ×_f M_T, where M_T is an invariant submanifold and M_λ is a proper slant submanifold in M̄. Finally in Section 5, we present an example that illustrates the existence of a ℘ℛ-semi-slant warped product submanifold with an improper slant coefficient (called ℘ℛ-warped product) in M̄.

Let M̄ be an even-dimensional smooth manifold. A smooth manifold M̄ is said to have an almost product structure if

where ℘ is a tensor field of type (1, 1) and I is the identity transformation on M̄. For this, the pair (M̄,℘) is called an almost product manifold. An almost para-Hermitian manifold (M̄,℘, ḡ) [18] is a smooth manifold associated with an almost product structure ℘ and a pseudo-Riemannian metric ḡ satisfying

for any X, Y tangent to M̄. Clearly, from equations (2.1) and (2.2), we conclude that the signature of ḡ is necessarily (m,m), and

for any X, Y ∈ Γ(TM̄); where Γ(TM̄) is the Lie algebra of vector fields in M̄. The fundamental 2-form ω of M̄ is defined by

Definition 2.1

An almost para-Hermitian manifold M̄ is called a para-Kähler manifold [8] if ℘ is parallel with respect to ∇̄ i.e.,

where ∇̄ is the Levi-Civita connection on M̄ with respect to ḡ.

Let us consider that M be an isometrically immersed submanifold of a para-Kähler manifold M̄ in the sense of O’Neill [20]. Let g be the induced metric on M such that g = ḡ|_M [14]. We denote the set of vector fields normal to M by Γ(TM^⊥) and the sections of tangent bundle TM of M by Γ(TM). Then the Gauss-Weingarten formulas are given by, respectively,

for any X, Y ∈ Γ(TM) and ζ ∈ Γ(TM^⊥), where ∇ is the induced connection, ∇^⊥ is the normal connection on the normal bundle Γ(TM^⊥), h is the second fundamental form, and the shape operator A_ζ associated with the normal section ζ is given by [4]

If we write, for all τ ∈ Γ(TM) and ζ ∈ Γ(TM^⊥) that

where tτ (resp., nτ) is tangential (resp., normal) part of ℘τ and t′ζ (resp., n′ζ) is tangential (resp., normal) part of ℘ζ, then the submanifold M is said to be invariant if n is identically zero and anti-invariant if t is identically zero. Now, from equations (2.3) and (2.9), we obtain for any X, Y ∈ Γ(TM) that

Here, we state some important results on warped product manifolds;

Proposition 2.2.([3])

For X, Y ∈ Γ(TB) and Z,W ∈ Γ(TF), we obtain for the warped product manifold M = B ×_f F that

• ∇_XY ∈ Γ(TB),

• ∇XZ=∇ZX=(Xff) Z,

• ∇ZW=-g(Z,W)f∇f,

where ∇ denotes the Levi-Civita connection on M and ∇f is the gradient of f defined by g(∇f,X) = Xf.

Remark 2.3.([3])

It is also important to note that for a warped product M = B ×_f F,

• M is said to be trivial if f is constant,

• B is totally geodesic and F is totally umbilical in M,

• To reduce the complexity, we represent a vector field X on B with its lift X̄ and a vector field Z on F with its lift Z̄.

Now, we present a lemma for later use:

Lemma 2.4

Let M = B ×_f F be a non-trivial warped product submanifold of apara-Kähler manifold M̄, then

for any X, Y ∈ Γ(TB) and Z,W ∈ Γ(TF).

In this section, motivated by [1, 24], we first state the definition of a slant submanifold in a para-Kähler manifold M̄, and then continue the study by introducing a generalized class of a ℘ℛ-submanifold [11] called ℘ℛ-semi-slant submanifold in M̄.

Let M be a non-degenerate submanifold of a para-Kähler manifold M̄ such that t²X = λX = λ℘²X, g(tX, Y) = −g(X, tY) for any X, Y ∈ Γ(TM), where λ is a real coefficient. Then with the help of equation (2.11), we have

On the other hand,

In particular, from equations (3.1) and (3.2), we obtain for X = Y that g(℘X,tX)∣℘X∣ ∣tX∣=λ. Here we call λ a slant coefficient and consequently M a slant submanifold. Conversely, assume that M is a slant submanifold then λ∣℘X∣∣℘X∣=∣tX∣∣℘X∣, where X is a non-lightlike vector field. We obtain by the consequence of previous equation for any X, Y ∈ Γ(TM) that -λg(X,℘2Y)∣℘X∣ ∣tY∣=g(℘X,tY)∣℘X∣ ∣tY∣, which yields g(X, t²Y) = λg(X, ℘²Y), g(tX, Y) = −g(X, tY). Hence, t² = λI, g(tX, Y) = −g(X, tY) by virtue of the fact that structure is para-Kähler and X is any non-lightlike vector field.

Remark 3.1

The slant coefficient λ is sometimes chosen as cos²θ or cosh²θ or −sinh²θ for vector fields tangent to M, where θ is a real constant, called slant angle. The nature of the λ equals to cos²θ or cosh²θ or −sinh²θ depending on the consideration of vector fields (that is, angle between spacelike–spacelike or timelike–timelike or timelike–spacelike vector fields). The authors in [1, 2] distinguished different cases for λ, depending on the behaviour of vector fields.

As a consequence of the above characterization, we define slant submanifold in an almost para-Hermitian manifold;

Definition 3.2

Let M be an isometrically immersed submanifold of an almost para-Hermitian manifold M̄ and let be the distribution on M. Then is said to be slant distribution on M, accordingly M slant submanifold, if there exist a real valued constant λ such that

for any non lightlike tangent vector fields on M. Here, λ will be called slant coefficient that is globally constant and independent of the choice of point on M in M̄.

Remark 3.3

Since the manifold M is non-degenerate (i.e., M includes either spacelike vector fields or timelike vector fields), thus our definition of slant submanifold can be considered as the generalization of definitions given in [6, 23] that covers only the spacelike vector fields which implies λ = cos²θ, where θ is a slant angle.

Now, we have the following definition.

Definition 3.4

Let M be an isometrically immersed submanifold of an almost para-Hermitian metric manifold M̄(℘, ḡ). Then the submanifold M is said to be a ℘ℛ-semi-slant if it is furnished with a pair of non-degenerate orthogonal distribution ( ) satisfying the following conditions:

• ,

• the distribution is invariant under ℘, i.e., and

• the distribution is slant distribution with slant coefficient λ.

A ℘ℛ-semi-slant submanifold is

• ℘ℛ-submanifold if and with λ = 0 [11],

• proper if , and λ ≠ 0, 1.

Here, we give an example for demonstrating the existence of proper ℘ℛ-semi-slant submanifold M in a para-Kähler manifold M̄.

Example 3.5

Let M̄ = ℝ⁸ be a 8-dimensional manifold with the standard Cartesian coordinates

Define the para-Kähler pseudo-Riemannian metric structure (℘, ḡ) on M̄ by

Here, {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈} is a local orthonormal frame for Γ(TM̄), given by ei=∂∂x¯i,ej=∂∂x¯j.

Let M be an isometrically immersed pseudo-Riemannian submanifold in a para-Kähler manifold M̄ defined by

where α ∈ (0, π/2) and k ≠ 0. Then the tangent bundle Γ(TM) of M is spanned by the vector fields

where X₁,X₃ are spacelike and X₂,X₄ are timelike vector fields tangent to M. Now, the space ℘(TM) with respect to the paracosymplectic pseudo-Riemannian metric structure (℘, ḡ) of M̄ becomes

Therefore from equations (5.2), (5.3) and (5.1), we obtain that and , are the distributions defined by span{X₃,X₄}, and span{X₁,X₂} respectively, where is an invariant distribution and is a slant distribution with slant coefficient λ = | cos(α)|. Thus M becomes a proper ℘ℛ-semi-slant submanifold M of a para-Kähler manifold M̄.

Next, we prove the following result as a proposition for later use.

Proposition 3.6

Let M be a slant submanifold of an almost para-Hermitian manifold M̄(℘, ḡ). Then

for any X, Y ∈ Γ(TM).

Theorem 3.7

Let M be a proper ℘ℛ-semi-slant submanifold M of a para-Kähler manifold M̄. Then the distribution ( )

• is involutive if and only if h(X, tY) = h(tX, Y);

• defines totally geodesic foliation if and only if A_nZtY = A_ntZY;

for anyand.

Theorem 3.8

Let M be a proper ℘ℛ-semi-slant submanifold M of a para-Kähler manifold M̄. Then the distribution ( )

• is involutive if and only if g(A_nWZ − A_nZW, tX) = g(A_ntZW − A_ntWZ,X);

• defines totally geodesic if and only if A_ntWX = A_nWtX;

for anyand.

In this section, we investigate the existence or non-existence of non-trivial ℘ℛ-semi-slant warped product submanifolds of the forms M_T ×_f M_λ, M_λ ×_f M_T in para-Kähler manifolds M̄, where M_T and M_λ are invariant and slant submanifolds of M̄, respectively.

Now, we derive the following major results.

Theorem 4.1

Let M → M̄ be an isometric immersion of a proper ℘ℛ-semi-slant submanifold M into a para-Kähler manifold M̄. Then there does not exist any non-trivial ℘ℛ-semi-slant warped product submanifold M = M_T ×_f M_λ in M̄.

Theorem 4.2

Let M be an isometrically immersed ℘ℛ-semi-slant submanifold of a para-Kähler manifold M̄. Then there doesn’t exist a non-trivial ℘ℛ-semi-slant warped product submanifold M = M_λ ×_f M_T in M̄.

In contrast to the results in Section 4, we hereby give a numerical example to illustrates the existence of a ℘ℛ-warped product submanifold M of the form M_T ×_f M_λ with vanishing slant coefficient (i.e. M_T ×_f M_⊥) in a para-Kähler manifold.

Let M̄ = ℝ⁸ be a 8-dimensional manifold with the standard Cartesian coordinates

Define the para-Kähler pseudo-Riemannian metric structure (℘, ḡ) on M̄ by

here {e₁, e₂, e₃, e₄, e₅, e₆, e₇, e₈} is a local orthonormal frame for Γ(TM̄), given by ei=∂∂xi,ej=∂∂yj.

Let M be an isometrically immersed pseudo-Riemannian submanifold in a para-Kähler manifold M̄ defined by

where γ, β ∈ (0, π/2) and either u < v ∈ ℝ₊ or v < u ∈ ℝ₋. Then the tangent bundle Γ(TM) of M is spanned by the vectors

where X_u,X_γ,X_β are spacelike vector fields and X_v is a timelike vector field tangent to M. Now the space ℘(TM) with respect to the para-Kähler pseudo-Riemannian metric structure (℘, ḡ) of M̄ becomes

From equations (5.2) and (5.3), we obtain that ℘(X_γ) and ℘(X_β) are orthogonal to Γ(TM), ℘(X_u) and ℘(X_v) are tangent to M. This means and can be taken as a subspace span{X_u,X_v} and a subspace span{X_γ,X_β}, respectively. Consequently, M becomes a ℘ℛ-semi-slant submanifold. Furthermore, we can say from Theorem 3.7 and Theorem 3.8 that and are integrable. We denote the integral manifolds of and by M_T and M_λ, respectively, then the induced pseudo-Riemannian metric tensor g of M is given by

that is,

Hence, M is a 4-dimensional ℘ℛ-semi-slant warped product submanifold in M̄ with warping function f=(v2-u2) and slant coefficient λ for is 0. Thus, M is a non-trivial ℘ℛ-semi-slant warped product submanifold with improper slant coefficient in M (in particular, a ℘ℛ-warped product submanifold in a para-Kähler manifold [11]).

The author is grateful to the anonymous reviewers for their several valuable comments and suggestions. The author would also like to thank colleagues Vishvesh Mishra, Deepak Grover and Manoj Gaur for their helpful suggestions that improved the presentation of this manuscript.