Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications

Lamia Saeed Alqahtani can be contacted at: Lalqahtani@kau.edu.sa

In this paper some characterizations for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms are obtained. Moreover, a sharp estimate for the squared norm of the second fundamental form is investigated, the equality case is also discussed. By the application of derived inequality, we compute an expression for Dirichlet energy of the involved warping function. Finally, we also proved some classifications for these warped product submanifolds in terms of Ricci solitons and Ricci curvature. A non-trivial example of these warped product submanifolds is provided.

1. Introduction

The study of warped product manifolds has been a favourite topic in the field of geometry due to its applications in Physics and relativistic theories [1]. Many basic solutions to Einstein field equations are given by warped products [1]. The concept of modelling of space–time near black holes uses the idea of warped product manifolds [2]. Schwartzschild space–time is an example of warped product $P \times_{r} S^{2}$ where the base $P = R \times R^{+}$ is a half plane $r > 0$ and fibre $S^{2}$ is the unit sphere. Under certain conditions, the Schwartzschild space–time becomes black hole. A cosmological model to model the universe as a space–time known as Robertson–Walker model is a warped product [3].

One of the important task in Physics and Engineering is to find the Dirichlet energy of smooth functions. Dirichlet energy is analogous to Kinetic energy. On a compact manifold $M$ ⁠, the Dirichlet energy of any smooth function $λ : M \to R$ is given by

E (λ) = \frac{1}{2} \int_{M} {‖ \nabla λ ‖}^{2} d V,

where $\nabla λ$ is the gradient of $λ$ and $d V$ is the volume element. It is obvious that $E (λ) \geq 0$ for any smooth function $λ$ ⁠. We know that the manifolds of non-zero (constant) curvature cannot be represented as a product manifold. So, considering the fact that a Riemannian product of manifolds cannot has negative curvature, the idea of warped product of manifolds came into existence. To construct the class of manifolds of negative or non-positive curvature, R. L. Bishop and B. O’Neill [4] introduced this idea of warped product manifolds. Warped product manifolds (see definition in Section 2) are a generalized setting of product manifolds. Since warping functions of the warped product manifolds are positive valued smooth functions, our interest is to find the Dirichlet energy of these functions.

Some intrinsic properties of warped product manifolds were studied in [4]. Initial extrinsic studies of warped product manifolds in the almost complex setting were performed by B. Y. Chen [5,6] while obtaining some existence results for CR-submanifolds to be CR-warped product submanifolds in Kaehler manifolds. On the other hand, in the almost contact settings contact CR-warped product submanifolds were explored by Hasegawa et al. [7]. Many other geometers have also explored warped product manifolds in contact settings and various existence results have been obtained [8–11].

Warped product pointwise semi-slant submanifold is another generalized class of warped product semi-slant submanifolds and contact CR-warped product submanifolds. In [12], Park studied the warped product pointwise semi-slant warped product submanifolds. After that, Ali and Ozel [13] extended this study in the setting of cosymplectic manifolds and they obtained some optimal inequalities related to the second fundamental form and warping function. Warped product pointwise semi-slant submanifolds for almost contact and almost complex manifolds were explored in (see [14–16]).

On the other hand the Gradient Ricci soliton are extensively investigated in the theory of relativity, physics as well as in the differential geometry. The classification results related to Ricci Soliton and Gradient Ricci solitons with the warped product structure have been established in [17–20]. Moreover, the Ricci curvature has a significant nature in Riemannian geometry, for example Ricci flat is a solution of Einstein field equation on a Riemannian manifold in which cosmological constant vanishes. More clearly, in the theory of general relativity the Ricci tensor is correlated with Einstein’s field equation to study the material contents of universe. So, in comparison with Riemannian curvature, the Ricci curvature is more significant in the theory of relativity and physics.

In the present article, we study warped product pointwise semi-slant submanifolds of cosymplectic space forms and obtain some interesting inequalities for warped product pointwise semi-slant submanifolds. We estimate the squared norm of the second fundamental form in terms of warping function and slant function. The equality case is discussed accordingly. We explored some applications of equality case of the derived inequality. More precisely, we calculate Dirichlet energy of the warping functions by using our derived inequality. Finally, we obtain the classification of the warped product pointwise semi-slant submanifolds admitting the gradient Ricci soliton, in terms of Ricci curvature and second fundamental form, some existence results are also established.

The paper is organized as follows: Section 2 is devoted to basic definitions, formulae and preliminary results which are required for the subsequent study of the paper. In Section 3, we explore the existence of warped product pointwise semi-slant submanifolds in cosymplectic space forms and prove our main results. A non-trivial example is given for the warped product pointwise semi-slant submanifolds in a cosymplectic manifold. Using the derived inequality, formulae for Dirichlet energy of warping function is obtained in Section 4. Conclusions are presented in Section 5. Bibliography is given at the end of the paper.

2. Preliminaries

A $(2 n + 1)$ -dimensional $C^{\infty}$ -manifold $\bar{M}$ is said to have an almost contact structure if on $\bar{M}$ there exist a tensor field $Ψ$ of type $(1,1)$ ⁠, a vector field $η$ and a 1-form $ζ$ satisfying the following properties [21]

Ψ^{2} = - I + ζ η \otimes η, Ψ η = 0, ζ \circ Ψ = 0, ζ (η) = 1.

(2.1)

The manifold $\bar{M}$ with the structure $(Ψ, η, ζ)$ is called almost contact metric manifold. There exists a Riemannian metric $g$ on an almost contact metric manifold $\bar{M}$ ⁠, satisfying the following

ζ (Y) = g (Y, η), g (Ψ Y, Ψ V) = g (Y, V) - ζ (Y) ζ (V),

(2.2)

for all $Y, V \in T \bar{M}$ where $T \bar{M}$ is the tangent bundle of $\bar{M}$ ⁠.

An almost contact metric structure $(Ψ, η, ζ, g)$ is said to be cosymplectic manifold if it satisfies the following tensorial equation [21]

({\bar{\nabla}}_{Y} Ψ) V = 0,

(2.3)

for any $Y, V \in T \bar{M}$ ⁠, where $\bar{\nabla}$ denotes the Riemannian connection of the metric $g$ ⁠. Moreover, for a cosymplectic manifold

{\bar{\nabla}}_{Y} η = 0.

(2.4)

A cosymplectic manifold $\bar{M}$ is said to be a cosymplectic space form [21] if it has constant $Ψ$ -holomorphic sectional curvature $c$ and is denoted by $\bar{M} (c)$ ⁠. The curvature tensor $\bar{R}$ of cosymplectic space form $\bar{M} (c)$ is given by

\begin{matrix} \bar{R} (Y_{1}, Y_{2}) U = \frac{c}{4} {g (Y_{2}, V) Y_{1} - g (Y_{1}, V) Y_{2} + g (Y_{1}, Ψ V) Ψ Y_{2}, \\ - g (Y_{2}, Ψ V) Ψ Y_{1} + 2 g (Y_{1}, Ψ Y_{2}) Ψ V + ζ (Y_{1}) ζ (V) Y_{2} \\ - ζ (Y_{2}) ζ (V) Y_{1} + g (Y_{1}, V) ζ (Y_{2}) η - g (Y_{2}, V) ζ (Y_{1}) η} \end{matrix}

(2.5)

for any vector fields $Y_{1}, Y_{2}, V$ on $\bar{M}$ ⁠.

Let $M$ be a submanifold of an almost contact metric manifold $\bar{M}$ with induced metric $g$ ⁠. The Riemannian connection $\bar{\nabla}$ of $\bar{M}$ induces canonically the connections $\nabla$ and $\nabla^{⊥}$ on the tangent bundle $T M$ and the normal bundle $T^{⊥} M$ of $M$ respectively, then the Gauss and Weingarten formulae are given by

{\bar{\nabla}}_{Y_{1}} Y_{2} = \nabla_{Y_{1}} Y_{2} + σ (Y_{1}, Y_{2}),

(2.6)

{\bar{\nabla}}_{Y_{1}} ξ = - A_{ξ} Y_{1} + \nabla_{Y_{1}}^{⊥} ξ,

(2.7)

for each $Y_{1}, Y_{2} \in T M$ and $ξ \in T^{⊥} M$ ⁠, where $σ$ and $A_{ξ}$ are the second fundamental form and the shape operator respectively for the immersion of $M$ into $\bar{M}$ ⁠, they verify the relation

g (σ (Y_{1}, Y_{2}), ξ) = g (A_{ξ} Y_{1}, Y_{2}),

(2.8)

where $g$ denotes the Riemannian metric on $\bar{M}$ as well as the induced metric on $M$ ⁠.

For a submanifold $M$ of a Riemannian manifold $\bar{M}$ ⁠, the equation of Codazzi is given by

\begin{matrix} {(\bar{R} (Y_{1}, Y_{2}) V)}^{⊥} = \nabla_{Y_{1}}^{⊥} σ (Y_{1}, V) - \nabla_{Y_{2}}^{⊥} σ (Y_{1}, V) + σ (\nabla_{Y_{2}} Y_{1}, V) - σ (\nabla_{Y_{1}} Y_{2}, V) \\ + σ (Y_{1}, \nabla_{Y_{2}} V) - σ (Y_{2}, \nabla_{Y_{1}} V) \end{matrix}

where ${(\bar{R} (Y_{1}, Y_{2}) V)}^{⊥}$ denotes the normal component of the curvature tensor $\bar{R} (Y_{1}, Y_{2}) V$ ⁠.

If $P Y$ and $F Y$ denote the tangential and normal component of $Ψ Y$ respectively for any $Y \in T M$ ⁠, we can write

Ψ Y = P Y + F Y .

(2.9)

Similarly, for any $ξ \in T^{⊥} M$ ⁠, we write

Ψ ξ = t ξ + f ξ,

(2.10)

where $t ξ$ and $f ξ$ are the tangential and normal components of $Ψ ξ$ respectively. Thus $P$ (resp. $f$ ⁠) is 1–1 tensor field on $T M$ (resp. $T^{⊥} M$ ⁠) and $t$ (resp. $F$ ⁠) is a tangential (resp. normal) valued 1-form on $T^{⊥} M$ (resp. $T M$ ⁠). The covariant derivatives of the tensor fields $Ψ$ ⁠, $P$ and $F$ are defined as

({\bar{\nabla}}_{Y_{1}} Ψ) Y_{2} = \nabla_{Y_{1}} Ψ Y_{2} - Ψ \nabla_{Y_{1}} Y_{2}

(2.11)

({\bar{\nabla}}_{Y_{1}} P) Y_{2} = \nabla_{Y_{1}} P Y_{2} - P \nabla_{Y_{1}} Y_{2},

(2.12)

({\bar{\nabla}}_{Y_{1}} F) Y_{2} = \nabla_{Y_{1}}^{⊥} F Y_{2} - F \nabla_{Y_{1}} Y_{2} .

(2.13)

From Eqs. (2.3), (2.6), (2.7), (2.9) and (2.10), we have

({\bar{\nabla}}_{Y_{1}} P) Y_{2} = A_{F Y_{2}} Y_{1} + t σ (Y_{1}, Y_{2}) - g (Y_{1}, Y_{2}) η

(2.14)

({\bar{\nabla}}_{Y_{1}} F) Y_{2} = f σ (Y_{1}, Y_{2}) - σ (Y_{1}, P Y_{2}) .

(2.15)

The mean curvature vector $H$ of $M$ is given by

H = \frac{1}{m} \sum_{s = 1}^{m} σ (e_{s}, e_{s}),

where $m$ is the dimension of $M$ and ${e_{1}, e_{2}, \dots, e_{m}}$ is a local orthonormal frame of vector fields on $M$ ⁠. The squared norm of the second fundamental form $σ$ is defined as

{‖ σ ‖}^{2} = \sum_{r, s = 1}^{m} g (σ (e_{r}, e_{s}), σ (e_{r}, e_{s})) .

(2.16)

A submanifold $M$ of $\bar{M}$ is said to be a totally geodesic submanifold if $σ (Y_{1}, Y_{2}) = 0$ and totally umbilical submanifold if $σ (Y_{1}, Y_{2}) = g (Y_{1}, Y_{2}) H$ ⁠, for each $Y_{1}, Y_{2} \in T M .$

Definition.

([22]). A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said to be slant submanifold if for any $y \in M$ and $Y_{1} \in T_{y} M - 〈 η 〉$ ⁠, the angle between $Y_{1}$ and $Ψ Y_{1}$ is constant. The constant angle $θ \in [0, π / 2]$ is then called slant angle of $M$ in $\bar{M}$ ⁠. If $θ = 0$ ⁠, the submanifold is invariant submanifold and if $θ = π / 2$ then it is anti-invariant submanifold. If $θ \neq 0, π / 2$ ⁠, it is proper slant submanifold.

For slant submanifolds of the contact metric manifolds J. L. Cabrerizo et al. [22] proved the following lemma.

Lemma 2.1.

Let $M$ be a submanifold of an almost contact metric manifold $\bar{M}$ such that $η \in T M$ , then $M$ is a slant submanifold if and only if there exists a constant $λ \in [0,1]$ such that

P^{2} = λ (I - ζ \otimes η),

(2.17)

where $λ = - \cos^{2} θ$ .

The notion of pointwise slant submanifolds was introduced by F. Etayo [23] as a natural generalization of the slant submanifolds in the setting of almost Hermitian manifolds. Later, B. Y. Chen and O. J. Garay [24] investigated pointwise slant submanifolds for almost Hermitian manifolds and obtained some fundamental results. A step forward, K. S. Park [12] extended the concept of pointwise slant submanifolds in almost contact metric manifolds. Recently, Uddin and Al-Khalidi [25] modified the definition of pointwise slant submanifolds for almost contact metric manifolds. More precisely, a submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said to be pointwise slant submanifold if for any $X \in T_{x} M$ such that $η$ is tangential to $M$ ⁠, the angle $θ (X)$ between $Ψ X$ and $T_{x} M - {0}$ is independent of the choice of non zero vector field $X \in T_{p} M - {0}$ ⁠. In this case $θ$ is treated as the function on $M$ ⁠, which is called the slant function of the point wise slant function. Now, we have the following characterizing theorem

Theorem 2.2.

([25]). Let $M$ be a submanifold of an almost contact metric manifold $\bar{M}$ such that $η \in T M$ . Then, $M$ is pointwise slant if and only if

P^{2} = \cos^{2} θ (- I + ζ \otimes η)

(2.18)

for some real valued function $θ$ on $T M$

Thus, one has the following consequences of the above formula.

g (P Y_{1}, P Y_{2}) = \cos^{2} θ [g (Y_{1}, Y_{2}) - ζ (Y_{1}) ζ (Y_{2})],

(2.19)

g (F Y_{1}, F Y_{2}) = \sin^{2} θ [g (Y_{1}, Y_{2}) - ζ (Y_{1}) ζ (Y_{2})]

(2.20)

for all $Y_{1}, Y_{2} \in T M$ ⁠.

Pointwise semi-slant submanifolds were defined and studied by Park [12] as a natural generalization of contact CR-submanifolds in terms of slant function. Now, we have the following definition

Definition.

A submanifold $M$ of an almost contact metric manifold $\bar{M}$ is said to be a pointwise semi-slant submanifold if there exist two orthogonal complementary distributions $D$ and $D^{θ}$ on $M$ such that

The tangent bundle $T M$ can be written as $T M = D \oplus D^{θ} 〈 η 〉,$
The distribution $D$ is invariant,
The distribution $D^{θ}$ is pointwise slant with a slant function $θ$ ⁠.

As a generalization of the product manifolds, one can consider the warped product of manifolds which are defined as follows.

Let $(C_{1}, g_{C_{1}})$ and $(C_{2}, g_{C_{2}})$ be the two Riemannian manifolds with $g_{C_{1}}$ and $g_{C_{2}}$ as their Riemannian metrics resp. and $Φ$ be a positive differentiable function on $C_{1}$ ⁠. Let $π_{1} : C_{1} \times C_{2} \to C_{1}$ ⁠, $π_{2} : C_{1} \times C_{2} \to C_{2}$ are the projection maps given by $π_{C_{1}} (c_{1}, c_{2}) = c_{1}$ and $π_{C_{2}} (c_{1}, c_{2}) = c_{2}$ for every $(c_{1}, c_{2}) \in C_{1} \times C_{2}$ ⁠. The warped product $M = C_{1} \times_{Φ} C_{2}$ [4] is the manifold $C_{1} \times C_{2}$ equipped with the Riemannian structure such that

g (Y_{1}, Y_{2}) = g_{1} (π_{1 *} Y_{1}, π_{1 *} Y_{2}) + {(Φ \circ π_{1})}^{2} g_{2} (π_{2 *} Y_{1}, π_{2 *} Y_{2}),

for all $Y_{1}, Y_{2} \in T M$ ⁠, where $π_{i *}$ denotes the tangent map corresponding to $π_{i}$ for each $i$ ⁠. The function $Φ$ is called the warping function of the warped product manifold. If the warping function is constant then the warped product manifold $M$ is said to be trivial warped product.

Let $Y_{1}$ be a vector field on $C_{1}$ and $Y_{2}$ be a vector field on $C_{2}$ ⁠, then from Lemma 7.3 of [3], we have

\nabla_{Y_{1}} Y_{2} = \nabla_{Y_{2}} (\frac{Y_{1} Φ}{Φ}) Y_{2}

(2.21)

where $\nabla$ is the Levi-Civita connection on $M$ ⁠. For a warped product $M = C_{1} \times_{Φ} C_{2}$ it is easy to observe that

\nabla_{Y_{1}} Y_{2} = \nabla_{Y_{2}} Y_{1} = (Y_{1} l n Φ) Y_{2}

(2.22)

for $Y_{1} \in T C_{1}$ and $Y_{2} \in T C_{2}$ ⁠.

$\nabla Φ$ is the gradient of $Φ$ and is defined as

g (\nabla Φ, Y) = Y Φ,

(2.23)

for all $Y \in T M$ ⁠.

Let $M$ be an $m$ -dimensional Riemannian manifold with the Riemannian metric $g$ and let ${e_{1}, e_{2}, \dots, e_{m}}$ be an orthogonal basis of $T M$ ⁠. Then as a result of (2.23), we get

{‖ \nabla Φ ‖}^{2} = \sum_{i = 1}^{m} {(e_{i} (Φ))}^{2} .

(2.24)

The Laplacian of $Φ$ is defined by

Δ Φ = \sum_{i = 1}^{m} {(\nabla_{e_{i}} e_{i}) Φ - e_{i} e_{i} Φ} .

(2.25)

The Hessian tensor for a differentiable function $Φ$ is symmetric covariant tensor of rank $2$ and is defined as

Δ Φ = - t r a c e H^{Φ}

(2.26)

or we can also write

H e s s^{Φ} = - Δ Φ

(2.27)

Now, we state the Hopf’s Lemma.

Hopf’s Lemma [26]. If $M$ is an $m$ -dimensional connected compact Riemannian manifold. If $Φ$ is a differentiable function on $M$ s. t. $Δ Φ \geq 0$ everywhere on $M$ (or $Δ Φ \leq 0$ everywhere on $M$ ⁠), then $Φ$ is a constant function.

For a compact orientable Riemannian manifold $M$ with or without boundary and as a consequences of the integration theory of manifolds, we have [27]

\int_{M} Δ Φ d V = 0,

(2.28)

where $Φ$ is a function on $M$ and $d V$ is the volume element of $M$ ⁠.

The Ricci soliton idea was given by Hamilton [28]. It is regarded as the natural generality of Einstein metrics and they are the self similar solution of the Ricci flow $\frac{\partial}{\partial t} g (t) = - 2 R i c (t)$ ⁠. If there exists a smooth vector field $Y$ such that the Ricci tensor meets the following condition

R i c + \frac{1}{2} L_{Y} g = α,

(2.29)

for any constant $α$ ⁠, where $L_{Y}$ is the Lie derivative, then the Riemannian metric $g$ on a complete Riemannian manifold $\bar{M}$ is named as Ricci Soliton. If $α < 0, α = 0$ and $α > 0$ then the Ricci soliton is called expanding, steady and Shrinking respectively. If we specify $Y = \nabla Φ$ for a smooth function $Φ$ defined on $\bar{M}$ ⁠, then $g$ admits gradient Ricci soliton with the potential function $Φ$ ⁠. For this case (2.29) takes the form

R i c + \nabla^{2} Φ = α g .

(2.30)

Since the Laplacian $Δ$ and the gradient $\nabla^{2}$ are related as $Δ = \nabla^{2}$ ⁠. Thus, in terms of Hessian (2.29) can be expressed as

R i c = α g + H e s s^{Φ} .

(2.31)

Note 2.1.

If the potential function $Φ$ is constant on a gradient Ricci soliton, then $(M, g, \nabla Φ, λ)$ is an Einstein manifold.

3. Warped product pointwise semi-slant submanifolds

In [12], K. S. Park, proved the existence of the warped product pointwise semi-slant warped product submanifolds of the type $N_{T} \times_{Φ} N_{θ}$ of cosymplectic manifolds and achieved the following lemma

Lemma 3.1.

Let $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold of a cosymplectic manifold $\bar{M}$ such that $η \in T N_{T}$ , where $N_{T}$ and $N_{θ}$ are invariant and pointwise slant submanifolds of $\bar{M}$ , respectively. Then

g (σ (Y, W), F P Z) = - \cos^{2} θ Y l n Φ g (W, Z) - Φ Y l n Φ g (W, P Z),

(3.1)

and

g (σ (Ψ Y, W), F Z) = Y l n Φ g (W, Z) - Ψ Y l n Φ g (W, T Z),

(3.2)

for any $Y \in T N_{T}$ and $Z, W \in T N_{θ}$ .

Now let $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold of a cosymplectic manifold $\bar{M}$ and we consider the vector field $η$ tangent to $N_{T}$ ⁠. If $D$ is invariant distribution and $D^{θ}$ is proper point wise slant distribution with the slant function $θ$ ⁠, then the tangent bundle $T M$ and $T^{⊥} M$ are decomposed (resp.) as follows

T M = D \oplus D^{θ} \oplus 〈 η 〉,

T^{⊥} M = F D^{θ} \oplus μ,

where $μ$ is the orthogonal complementary distribution of $F D^{θ}$ in $T^{⊥} M$ ⁠. It is easy to see that $μ$ is an invariant subbundle of $T^{⊥} M$ with respect to $Ψ$ ⁠.

In view of the above direct decomposition, the second fundamental form $σ$ can be written as

σ (U_{1}, U_{2}) = σ_{F D^{θ}} (U_{1}, U_{2}) + σ_{μ} (U_{1}, U_{2}),

(3.3)

for $U_{1}, U_{2} \in T M$ ⁠, where $σ_{F D^{θ}} (U_{1}, U_{2})$ and $σ_{μ} (U_{1}, U_{2})$ are the components of $σ (U_{1}, U_{2})$ in the normal sub-bundles $F D^{θ}$ and $μ$ respectively. Moreover if ${V_{1}, V_{2}, \dots, V_{q}}$ be a local orthonormal frame of vector fields of $D^{θ}$ ⁠, then

σ_{F D^{θ}} (U_{1}, U_{2}) = \sum_{r = 1}^{q} σ^{r} (U_{1}, U_{2}) F V_{r}

(3.4)

where

σ^{r} (U_{1}, U_{2}) = \csc^{2} θ g (σ (U_{1}, U_{2}), F V_{r})

(3.5)

To ensure the existence, we construct an example of a warped product pointwise semi-slant submanifold of the type $M = N_{T} \times_{Φ} N_{θ}$ in cosymplectic manifold with $η$ tangent to $N_{T}$ ⁠.

Example.

Let $\bar{M} = C^{5} \times R$ be a Riemannian product of Euclidean space $C^{5}$ with line $R$ such that the structure vector field $η = \frac{\partial}{\partial t}$ ⁠, 1-form $ζ = d t$ and metric $g = g_{1} + d t^{2}$ ⁠, where $g_{1}$ is the metric on Euclidean space. Then $(\bar{M}, Ψ, η, ζ, g)$ is a cosymplectic manifold. Let $φ : M^{5} \to {\bar{M}}^{11}$ be a point wise semi-slant submanifold such that $0 < u, v < 1$ is defined as

e_{1} = u \tanh x, e_{2} = v \tanh x, e_{3} = u \tanh y, e_{4} = v \tanh y, e_{5} = u sech x,

e_{6} = v sech x, e_{7} = u sech y, e_{8} = v sech y, e_{9} = x, e_{10} = y, e_{11} = 0.

The tangent space $T M$ is spanned by the vector fields $X_{1}$ ⁠, $X_{2}$ ⁠, $X_{3}$ ⁠, $X_{4}$ ⁠, $X_{5}$ such that

X_{1} = sech y \frac{\partial}{\partial e_{8}} + sech x \frac{\partial}{\partial e_{6}} + \tanh y \frac{\partial}{\partial e_{4}} + \tanh x \frac{\partial}{\partial e_{2}},

X_{2} = sech y \frac{\partial}{\partial e_{7}} + sech x \frac{\partial}{\partial e_{5}} + \tanh y \frac{\partial}{\partial e_{3}} + \tan x \frac{\partial}{\partial e_{1}},

X_{3} = η = \frac{\partial}{\partial t},

X_{4} = \frac{\partial}{\partial e_{9}} + v \tanh x \frac{\partial}{\partial e_{6}} + u \tan x \frac{\partial}{\partial e_{5}} - v sech x \frac{\partial}{\partial e_{2}} - u sech x \frac{\partial}{\partial e_{1}},

X_{5} = \frac{\partial}{\partial e_{10}} + v \tanh y \frac{\partial}{\partial e_{7}} + u \tan y \frac{\partial}{\partial e_{8}} - v sech y \frac{\partial}{\partial e_{4}} - u sech y \frac{\partial}{\partial e_{3}} .

Then $D^{θ} = span {X_{4}, X_{5}}$ is a pointwise slant distribution with slant function $\cos^{- 1} (\frac{1}{u^{2} + v^{2} + 1})$ and invariant distribution $D = span {X_{1}, X_{2}, X_{3}}$ ⁠. Thus $M^{5} = N_{T} \times_{Φ} N_{θ}$ is a non-trivial warped product pointwise semi-slant submanifold of $M^{11}$ with the warping function $Φ = \sqrt{(u^{2} + v^{2} + 1)}$ ⁠.

In this section, for convention, we denote by $Y, X \in T N_{T}$ and $V, U \in T N_{θ}$ as the vector fields of respective tangent bundles of $N_{T}$ and $N_{θ}$ ⁠. At first, some initial results to be proved.

Lemma 3.2.

Let $N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold of a cosymplectic manifold $\bar{M}$ . Then

$η l n Φ = 0$ ,
$g (σ (Ψ Y, V), F V) = Y l n Φ {‖ V ‖}^{2}$ ,
$g (σ (Ψ Y, V), Ψ σ (Y, V)) = {‖ σ_{μ} (Y, V) ‖}^{2} + \cos^{2} θ {(Y l n Φ)}^{2} {‖ V ‖}^{2}$ ,

for all $Y \in T N_{T}$ and $V \in T N_{θ}$ ⁠, where $σ_{μ}$ is the $μ$ component of the second fundamental form $σ$ ⁠.

Proof.

From (2.4), (2.6), and (2.22), it is easy to see that $η l n Φ = 0$ ⁠. Moreover, part (ii) is a particular case of (3.2). To prove part (iii), on making use of (2.6) and (2.3), we get

σ (Ψ Y, V) = Ψ σ (Y, V) + Ψ \nabla_{V} Y - \nabla_{V} Ψ Y .

Now using (2.22), the above equation can be written as

σ (Ψ Y, V) = Ψ σ (Y, V) + Y l n Φ Ψ V - Ψ Y l n Φ V .

Comparing the normal parts

σ (Ψ Y, V) = Ψ σ_{μ} (Y, V) + Y l n Φ F V,

taking inner product with $Ψ σ (Y, V)$ ⁠, we get

g (σ (Ψ Y, V), Ψ σ (Y, V)) = {‖ σ_{μ} (Y, V) ‖}^{2} + Y l n Φ g (Ψ σ (Y, V), F V) .

(3.6)

Calculating the last term of above equation by using (2.6), (2.3), and (2.20) as follows

g (Ψ σ (Y, V), F V) = g (σ (Ψ Y, V), F V) - \sin^{2} θ Y l n Φ {‖ V ‖}^{2} .

Utilizing part (ii), we get

g (Ψ σ (Y, V), F V) = \cos^{2} θ Y l n Φ {‖ V ‖}^{2},

using in (3.6), we get the required result.

Lemma 3.3.

Let $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold of a cosymplectic manifold $\bar{M}$ . Then

g (σ (Y, P V), F V) = - g (σ (Y, V), F P V) = - \cos^{2} θ Y \ln Φ {‖ V ‖}^{2}

for all $Y \in T N_{T}$ ⁠, $V \in T N_{θ}$ ⁠.

Proof.

From the part (ii) of Lemma 3.2, we may obtain

g (σ (P Y, V), F U) + g (σ (P Y, U), F V) = 2 Y l n Φ g (V, U)

for any $Y \in T N_{T}$ and $V, U \in T N_{θ}$ ⁠. Replacing $U$ by $P V \in D^{θ}$ and using the fact that $V$ and $P V$ are orthogonal, we have

g (σ (Y, P V), F V) = - g (σ (Y, V), F P V) .

(3.7)

By using (2.12), (2.14), and (2.22), we have

P Y l n Φ V - Y l n Φ P V = t σ (Y, V) .

Now taking inner product with $U \in T N_{θ}$ in the above equation, we have

P Y l n Φ g (V, U) - Y l n Φ g (P V, U) = - g (σ (Y, V), F U) .

Interchanging $V$ and $U$ and subtracting the resultant from the above equation, we get

- g (σ (Y, V), F U) + g (σ (Y, U), F V) = 2 Y l n Φ g (V, P U) .

In particular, replacing $U$ by $P V \in D^{θ}$ ⁠, the above equation yields

g (σ (Y, V), F P V) - g (σ (Y, P V), F V) = - 2 \cos^{2} θ Y l n Φ {‖ V ‖}^{2} .

(3.8)

Using (3.7), we obtain

g (σ (Y, P V), F V) = - g (σ (Y, V), F P V) = - \cos^{2} θ Y l n Φ {‖ V ‖}^{2} . □

(3.9)

Lemma 3.4.

On a warped product pointwise semi-slant submanifold $M = N_{T} \times_{Φ} N_{θ}$ of a cosymplectic manifold $\bar{M}$ , we have

\begin{array}{l} \sum_{i = 1}^{p} [\sum_{j, k = 1}^{2 q} g (σ (Ψ e_{i}, e^{k}), F e^{j}) g (σ (e_{i}, P e^{k}), F e^{j}) \\ - g (σ (e_{i}, e^{k}), F e^{j}) g (σ (Φ e_{i}, P e^{k}), F e^{j})] = - 4 q \cos^{2} θ {‖ \nabla l n Φ ‖}^{2}, \end{array}

where ${e_{0} = η, e_{1}, e_{2}, \dots, e_{p}, Ψ e_{1}, Ψ e_{2}, \dots, Ψ e_{p}}$ and ${e^{1}, e^{2}, \dots, e^{q}, e^{q + 1} = \sec θ P e^{1}, e^{q + 2} = \sec θ P e^{2}, \dots, e^{2 q} = \sec θ P e^{q}}$ are the frames of the orthonormal vector fields on $T N_{T}$ and $T N_{θ}$ respectively.

Proof.

First, we expand the left hand term in the following way

\sum_{i = 1}^{p} [\sum_{j, k = 1}^{2 q} g (σ (Ψ e_{i}, e^{k}), F e^{j}) g (σ (e_{i}, P e^{k}), F e^{j})]

= \sum_{i = 1}^{p} [\sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), F e^{j}) g (σ (e_{i}, P e^{j}), F e^{j})

+ \sum_{j \neq k = 1}^{2 q} g (σ (Ψ e_{i}, e^{k}), F e^{j}) g (σ (e_{i}, P e^{k}), F e^{j})]

= \sum_{i = 1}^{p} [\sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), F e^{j}) g (σ (e_{i}, P e^{j}), F e^{j})

+ \sum_{j = 1}^{q} g (σ (Ψ e_{i}, e^{j}), F e^{j + q}) g (σ (e_{i}, P e^{j}), F e^{j + q})

+ \sum_{j = 1}^{q} g (σ (Ψ e_{i}, e^{j + q}), F e^{j}) g (σ (e_{i}, P e^{j + q}), F e^{j})]

= \sum_{i = 1}^{p} [\sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), F e^{j}) g (σ (e_{i}, P e^{j}), F e^{j})

+ \sec^{2} θ \sum_{j = 1}^{q} g (σ (Ψ e_{i}, e^{j}), F P e^{j}) g (σ (e_{i}, P e^{j}), F P e^{j})

- \sum_{j = 1}^{q} g (σ (Ψ e_{i}, P e^{j}), F e^{j}) g (σ (e_{i}, e^{j}), F e^{j})] .

Using part (ii) of Lemmas 3.2, 3.3 and utilizing (2.24), we get

\sum_{i = 1}^{p} [\sum_{j, k = 1}^{2 q} g (σ (Ψ e_{i}, e^{k}), F e^{j}) g (σ (e_{i}, P e^{k}), F e^{j})]

= \sum_{i = 1}^{p} [- 2 q \cos^{2} θ {(e_{i} l n Φ)}^{2} - 2 q \cos^{2} θ {(Ψ e_{i} l n Φ)}^{2}]

= - 2 q \cos^{2} θ {‖ \nabla l n Φ ‖}^{2} .

Replacing $e_{i}$ by $Ψ e_{i}$ in above equation, we get

\sum_{i = 1}^{p} [\sum_{j, k = 1}^{2 q} g (σ (e_{i}, e^{k}), F e^{j}) g (σ (Ψ e_{i}, P e^{k}), F e^{j})] = 2 q \cos^{2} θ {‖ \nabla l n Φ ‖}^{2} .

Subtracting the above two findings, we get the required result. $□$

Now, we prove the following characterization.

Theorem 3.5.

Let $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold of a cosymplectic space form $\bar{M} (c)$ such that $N_{T}$ is a compact submanifold. Then $M$ is a Riemannian product submanifold if the following inequalities hold

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} \leq c p q \sin^{2} θ - 2 q (\cos^{2} θ + 2 \cos^{2} θ) {‖ \nabla l n Φ ‖}^{2}

and

\sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ_{μ} (Ψ e_{i}, e^{j}), σ_{μ} (e_{i}, P e^{j})) \geq 0,

where $σ_{μ}$ denotes the component of $σ$ in $μ$ , and $(2 p + 1)$ and $2 q$ are the dimensions of $N_{T}$ , and $N_{θ}$ respectively.

Proof.

For any unit vector fields $Y \in T N_{T}$ and $V \in T N_{θ}$ ⁠, using (2.1), (2.5), and (2.20), we have

\bar{R} (Y, Ψ Y, V, F V) = - \frac{c}{2} \sin^{2} θ {‖ Y ‖}^{2} {‖ V ‖}^{2}

(3.10)

On the other hand by Codazzi equation

\begin{array}{l} \bar{R} (Y, Ψ Y, V, F V) = g ((\nabla_{Y}^{⊥} σ (Ψ Y, V), F V) - g (\nabla_{Ψ Y}^{⊥} σ (Y, V), F V) \\ + g (σ (Y, \nabla_{Ψ Y} V), F V) - g (σ (Ψ Y, \nabla_{Y} V), F V) \\ - g (σ (\nabla_{Y} Ψ Y, V), F V) + g (σ (\nabla_{Ψ Y} Y, V), F V) . \end{array}

(3.11)

Now, we compute the values of the terms involved in (3.11). First, we have

g (\nabla_{Y}^{⊥} σ (Ψ Y, V), F V) = Y g (σ (Ψ Y, V), F V) - g (σ (Ψ Y, V), \nabla_{Y}^{⊥} F V) .

Using the part (ii) of Lemma 3.2 in the above equation, we get

\begin{array}{l} g (\nabla_{Y}^{⊥} σ (Ψ Y, V), F V) = Y^{2} l n Φ {‖ V ‖}^{2} + 2 {(Y l n Φ)}^{2} {‖ V ‖}^{2} \\ - g (σ (Ψ Y, V), \nabla_{Y}^{⊥} F V) . \end{array}

(3.12)

We calculate the last term of (3.12) using (2.9) as follows

g (σ (Ψ Y, V), \nabla_{Y}^{⊥} F V) = g (σ (Ψ Y, V), {\bar{\nabla}}_{Y} (Ψ V - P V)) .

By the use of (2.6) and (2.11) the above equation takes the form

\begin{matrix} g (σ (Ψ Y, V), \nabla_{Y}^{⊥} F V) = g (σ (Ψ Y, V), ({\bar{\nabla}}_{Y} Ψ) V + Ψ {\bar{\nabla}}_{Y} V) \\ - g (σ (Ψ Y, V), σ (Y, P V)) . \end{matrix}

Using (2.3), (2.6), (2.22), part (ii), and (iii) of Lemma 3.2, we obtain

\begin{array}{l} g (σ (Ψ Y, V), \nabla_{Y}^{⊥} F V) = {(Y l n Φ)}^{2} (1 + \cos^{2} θ) {‖ V ‖}^{2} + {‖ σ_{μ} (Y, V) ‖}^{2} \\ - g (σ (Ψ Y, V), σ (Y, P V)) . \end{array}

(3.13)

Utilizing (3.13) in (3.12), we get

\begin{array}{l} g (\nabla_{Y}^{⊥} σ (Ψ Y, V), F V) = Y^{2} l n Φ {‖ V ‖}^{2} + {(Y l n Φ)}^{2} \sin^{2} θ {‖ V ‖}^{2} \\ - {‖ σ_{μ} (Y, V) ‖}^{2} + g (σ (Ψ Y, V), σ (Y, P V)) . \end{array}

(3.14)

Similarly, we can write

\begin{array}{l} g (\nabla_{Ψ Y}^{⊥} σ (Y, V), F V) = - {(Ψ Y)}^{2} l n Φ {‖ V ‖}^{2} - {(Ψ Y l n Φ)}^{2} \sin^{2} θ {‖ V ‖}^{2} \\ + {‖ σ_{μ} (Ψ Y, V) ‖}^{2} + g (σ (Y, V), σ (Ψ Y, P V)) . \end{array}

(3.15)

From the part (ii) of Lemma 3.2, we have

g (A_{F V} V, Ψ Y) = Y l n Φ {‖ V ‖}^{2} .

Replacing $Y$ by $\nabla_{Y} Y$ (using the totally geodesicness of $N_{T}$ ⁠, $\nabla_{Y} Y \in T N_{T}$ ⁠) in the above equation, we have

g (A_{F V} V, Ψ \nabla_{Y} Y) = \nabla_{Y} Y l n Φ {‖ V ‖}^{2} .

By using (2.6), the above equation takes the form

g (A_{F V} V, Ψ {\bar{\nabla}}_{Y} Y - Ψ σ (Y, Y)) = \nabla_{Y} Y l n Φ {‖ V ‖}^{2} .

By using the fact that the first factor $N_{T}$ is totally geodesic in $M$ ⁠, it can be easily verified that $σ {(Y_{1}, Y)}_{2} \in μ$ ⁠, for all $Y_{1}, Y_{2} \in T N_{T}$ ⁠. Using this and (2.11) in the above equation, we get

g (σ (\nabla_{Y} Ψ Y, V), F V) = \nabla_{Y} Y l n Φ {‖ V ‖}^{2} .

(3.16)

Similarly, we can write

g (σ (\nabla_{Ψ Y} Y, V), F V) = - \nabla_{Ψ Y} Ψ Y l n Φ {‖ V ‖}^{2} .

(3.17)

By use of (2.22) and the part (ii) of Lemma 3.2, we have

g (σ (Ψ Y, \nabla_{Y} V), F V) = {(Y l n Φ)}^{2} {‖ V ‖}^{2}

(3.18)

and

g (σ (Y, \nabla_{Ψ Y} V), F V) = - {(Ψ Y l n Φ)}^{2} {‖ V ‖}^{2} .

(3.19)

Substituting values from (3.10), (3.14), (3.15), (3.16), (3.17), (3.18), and (3.19) in (3.11), we obtain

\begin{array}{l} - \frac{c}{2} \sin^{2} θ {‖ Y ‖}^{2} {‖ V ‖}^{2} = Y^{2} l n Φ {‖ V ‖}^{2} + {(Ψ Y)}^{2} l n Φ {‖ V ‖}^{2} \\ - {(Y l n Φ)}^{2} \cos^{2} θ {‖ Y ‖}^{2} - {(Ψ Y l n Φ)}^{2} \cos^{2} θ {‖ V ‖}^{2} \\ - {‖ σ_{μ} (Y, V) ‖}^{2} - {‖ σ_{μ} (Ψ Y, V) ‖}^{2} \\ - \nabla_{Y} Y l n Φ {‖ V ‖}^{2} - \nabla_{Ψ Y} Ψ Y l n Φ {‖ V ‖}^{2} \\ + g (σ (Ψ Y, V), σ (Y, P V)) - g (σ (Y, V), σ (Ψ Y, P V)) . \end{array}

Let ${e_{0} = η, e_{1}, e_{2}, \dots, e_{p}, e_{p + 1} = Ψ e_{1}, e_{p + 2} = Ψ e_{2}, \dots, e_{2 p} = Ψ e_{p}}$ be the orthonormal frame on $T N_{T}$ and ${e^{1}, e^{2}, \dots, e^{q}, e^{q + 1} = \sec θ P e^{1}, e^{q + 2} = \sec θ P e^{2}, \dots, e^{2 q} = \sec θ P e^{q}}$ be an orthonormal frame on $T N_{θ}$ ⁠. Now, using the decomposition (3.3), formula (3.5), (2.6) and (2.3), the above equation takes the form

\begin{array}{l} - \frac{c}{2} \sin^{2} θ {‖ Y ‖}^{2} {‖ V ‖}^{2} = Y^{2} l n Φ {‖ V ‖}^{2} + {(Ψ Y)}^{2} l n Φ {‖ V ‖}^{2} \\ - {(Y l n Φ)}^{2} \cos^{2} θ {‖ V ‖}^{2} - {(Ψ Y l n Φ)}^{2} \cos^{2} θ {‖ V ‖}^{2} \\ - {‖ σ_{μ} (Y, V) ‖}^{2} - {‖ σ_{μ} (Ψ Y, V) ‖}^{2} \\ - \nabla_{Y} Y l n Φ {‖ V ‖}^{2} - \nabla_{Ψ Y} Ψ Y l n Φ {‖ V ‖}^{2} \\ + \csc^{2} θ \sum_{j = 1}^{2 q} [g (σ (Ψ Y, V), F V_{j}) g (σ (X, P V), F V_{j}) \\ - g (σ (Ψ Y, V), F V_{j}) g (σ (Ψ Y, P V), F V_{j})] {‖ V_{j} ‖}^{2} \\ + 2 g (σ_{μ} (Ψ Y, V), σ_{μ} (Y, P V)) . \end{array}

Now summing the above equation over $i = 1,2, \dots, p$ and $j = 1,2, \dots 2 q$ ⁠, using (2.24), (2.25) and part (iii) of Lemma 3.2, one can get

\begin{array}{l} 2 q Δ (l n Φ) = c p q \sin^{2} θ - 2 q \cos^{2} θ {‖ \nabla l n Φ ‖}^{2} - \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} \\ - 4 q \cot^{2} θ {‖ \nabla l n Φ ‖}^{2} + 2 \sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ_{μ} (Ψ e_{i}, e^{j}), σ_{μ} (e_{i}, P e^{j})) . \end{array}

(3.20)

From (3.20) if

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} \leq c p q \sin^{2} θ - 2 q (\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2}

and

\sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ_{μ} (Ψ e_{i}, e^{j}), σ_{μ} (e_{i}, P e^{j})) \geq 0,

then $Δ l n Φ \geq 0$ ⁠, so by the Hopf’s Lemma, $l n Φ$ is constant that mean $Φ$ is constant, which proves the theorem. □

In the next theorem, we obtain the squared norm of the second fundamental form in terms of the warping function and slant function.

Theorem 3.6.

Let $\bar{M} (c)$ be a $(2 n + 1)$ -dimensional cosymplectic space form and $M = N_{T} \times_{Φ} N_{θ}$ be an $m$ -dimensional warped product pointwise semi-slant submanifold such that $N_{T}$ is a $(2 p + 1)$ -dimensional invariant submanifold and $N_{θ}$ be a $2 q$ -dimensional proper pointwise slant submanifold of $\bar{M} (c)$ . If

\sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), σ (e_{i}, P e^{j})) \geq 0,

then

The squared norm of the second fundamental form $σ$ satisfies
${‖ σ ‖}^{2} \geq c p q \sin^{2} θ + 2 q \sin^{2} θ {‖ \nabla l n Φ ‖}^{2} - 2 q Δ (l n Φ) .$
(3.21)
The equality sign of (3.21) holds identically if and only if
- $N_{T}$ is totally geodesic invariant submanifold of $\bar{M} (c)$ . Hence $N_{T}$ is a cosymplectic space form.
- $N_{θ}$ is totally umbilical submanifolds of $\bar{M} (c)$ .
- $\sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), σ (e_{i}, P e^{j})) = 0$ .

Proof.

From (3.20), we have

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} \geq c p q \sin^{2} θ - 2 q (\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} - 2 q Δ (l n Φ) .

(3.22)

For the orthonormal frames ${e_{0} = η, e_{1}, e_{2}, \dots, e_{p}, e_{p + 1} = Ψ e_{1}, e_{p + 2} = Ψ e_{2}, \dots, e_{2 p} = Ψ e_{p}}$ and ${e^{1}, e^{2}, \dots, e^{q}, \sec θ P e^{1}, \sec θ P e^{2}, \dots, \sec θ P e^{q}}$ ⁠, in view of the formulae (3.4), (3.5), part (ii) of Lemma 3.2, we get

\begin{array}{l} \sum_{i = 0}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{F D_{θ}} (e_{i}, e^{j}) ‖}^{2} = \sum_{i = 0}^{2 p} \sum_{j, k = 1}^{2 q} \csc^{2} θ g {(σ (e_{i}, e^{j}), F e^{k})}^{2} \\ = \csc^{2} θ \sum_{i = 0}^{2 p} [\sum_{j = 1}^{2 q} g {(σ (e_{i}, e^{j}), F e^{j})}^{2} \\ + \sum_{j \neq k = 1}^{2 q} g {(σ (e_{i}, e^{j}), F e^{k})}^{2}] \\ = \csc^{2} θ \sum_{i = 0}^{2 p} [2 q {(e_{i} l n Φ)}^{2} + \sec^{2} θ \sum_{j = 1}^{q} {g {(σ (e_{i}, e^{j}), F P e^{j})}^{2} + g {(σ (e_{i}, P e^{j}), F e^{j})}^{2}}] . \end{array}

Further using Lemma 3.3 and (2.24), the above equation reduced to

\sum_{i = 0}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{F D_{θ}} (e_{i}, e^{j}) ‖}^{2} = 2 q \csc^{2} θ {‖ \nabla l n Φ ‖}^{2} + 2 q \cot^{2} θ {‖ \nabla l n Φ ‖}^{2} .

(3.23)

From (3.22), (3.23) we get the required inequality.

To prove the part (ii), let $σ^{'}$ be the second fundamental form for the immersion of $N_{θ}$ in $M$ ⁠. Then for any $U, V \in T N_{θ}$ and $Y \in T N_{T}$ ⁠, using Gauss formula, we have

g (σ^{'} (U, V), V) = g (\nabla_{U} V, Y) = - Y l n Φ g (U, V) .

Using (2.23), we have

g (σ^{'} (U, V), Y) = - g (U, V) g (\nabla l n Φ, Y),

σ^{'} (U, V) = - g (U, V) \nabla l n Φ .

(3.24)

If the equality sign of (3.21) holds identically, then we obtain

σ (D, D) = 0, σ (D^{θ}, D^{θ}) = 0,

(3.25)

g (σ_{μ} (Ψ D, D^{θ}), σ_{μ} (D, P D^{θ})) = 0.

(3.26)

The first condition of (3.25) implies that $N_{T}$ is totally geodesic submanifold in $M$ ⁠. On the other hand it is easy to see that $g (σ (Y_{1}, Ψ Y_{2}), F V) = 0$ ⁠, for all $Y_{1}, Y_{2} \in T N_{T}$ ⁠, $V \in T N_{θ}$ ⁠. It follows that $N_{T}$ is totally geodesic in $\bar{M} (c)$ and hence is a cosymplectic space form. Moreover, the second condition of (3.25) together with (3.24) implies that $N_{θ}$ is a totally umbilical submanifold.

This proves the theorem. $□$

4. Some applications

Theorem 3.6 motivates us to obtain formulae to calculate the Dirichlet energy involving warping function $Φ$ ⁠. We denote by $E (Φ)$ the Dirichlet energy of a function $Φ$ ⁠. For a compact orientable warped product pointwise semi-slant submanifold $M = N_{T} \times_{Φ} N_{θ}$ in a cosymplectic space form $\bar{M} (c)$ ⁠, we compute the Dirichlet energy of the warping function $Φ$ in the following theorem.

Theorem 4.1.

Let $M = N_{T} \times_{Φ} N_{θ}$ be a compact orientable warped product pointwise semi-slant submanifold of a cosymplectic space form $\bar{M} (c)$ , such that $N_{T}$ be a $(2 p + 1)$ -dimensional invariant submanifold tangent to the structure vector field $η$ and $N_{θ}$ be a $2 q$ -dimensional pointwise slant submanifold of $\bar{M} (c)$ . Then for each $x \in N_{θ}$ , the Dirichlet energies of the warping function satisfy the following

E (l n Φ) = \frac{1}{4 q \sin^{2} θ} \int_{N_{T} \times {x}} {‖ σ ‖}^{2} d V + \frac{c p}{4} V o l (N_{T})

if and only if

$N_{T}$ is totally geodesic invariant submanifold of $\bar{M} (c)$ and is a cosymplectic space form,
$N_{θ}$ is totally umbilical submanifolds of $\bar{M} (c)$ ,
$\sum_{i = 1}^{p} \sum_{j = 1}^{2 q} g (σ (Ψ e_{i}, e^{j}), σ (e_{i}, P e^{j})) = 0$ ,

Proof.

On integrating the equality case of the inequality (3.21) and using the definition of Dirichlet energy and (2.28), we get the required result. $□$

If $θ = π / 2$ ⁠, then the compact orientable warped product pointwise semi-slant submanifolds become contact CR-warped product submanifolds. The following can be deduced from Theorem 4.1.

Corollary 4.2.

Let $M = N_{T} \times_{Φ} N_{⊥}$ be a compact orientable contact CR-warped product submanifold of a cosymplectic space form $\bar{M} (c)$ , such that $N_{T}$ be a $(2 p + 1)$ -dimensional invariant submanifold tangent to the structure vector field $η$ and $N_{⊥}$ be an $2 q$ -dimensional anti-invariant submanifold of $\bar{M} (c)$ . Then for each $x \in N_{⊥}$ , we have

E (l n Φ) = \frac{1}{4 q} \int_{N_{T} \times {x}} {‖ σ ‖}^{2} d V + \frac{c p}{4} V o l (N_{T})

if and only if

$N_{T}$ is totally geodesic invariant submanifold of $\bar{M} (c)$ and is a cosymplectic space form,
$N_{⊥}$ is a totally umbilical anti-invariant submanifold of $\bar{M} (c)$ .

If the equality sign of (3.21) holds, then

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = c p q \sin^{2} θ - 2 q (\csc^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} - 2 q Δ l n Φ .

(4.1)

Since, the Laplacian of a smooth function $Φ$ is the trace of the Hessian of the function. In terms of Hessian, (4.1) can be written as follows

\begin{array}{l} \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = c p q \sin^{2} θ - 2 q (\csc^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} \\ + 2 q \sum_{i = 1}^{p} [H e s s^{l n Φ} (e_{i}, e_{i}) + H e s s^{l n Φ} (Ψ e_{i}, Ψ e_{i})] \end{array}

(4.2)

Now, we have the following classification theorem for the warped product pointwise semi-slant submanifolds admitting the gradient Ricci soliton satisfying the equality case of (3.21).

Theorem 4.3.

Let $\bar{M} (c)$ be a 2n $+$ 1-dimensional cosymplectic space form and $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold admitting a shrinking gradient Ricci soliton. If

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} + 4 p q = c p q \sin^{2} θ + 2 q \sum_{i = 1}^{2 p} R i c (e_{i}, e_{i}),

(4.3)

then one of the following is true

(i)The slant function $θ = π / 2$ i.e., M is a contact CR-warped product submanifold,
(ii)The warping function $Φ$ is constant i.e., M is trivial Riemannian product pointwise semi-slant submanifold.

Proof.

Suppose that warped product pointwise semi-slant submanifold $M = N_{T} \times_{Φ} N_{θ}$ satisfies the basic equation of the Ricci soliton, such that the potential function $τ = l n Φ$ ⁠, then

R i c (X, Y) = λ g (X, Y) + H e s s^{τ} (X, Y),

(4.4)

for all $X, Y \in T N_{T}$ ⁠. Considering that ${e_{1}, e_{2}, \dots, e_{p}, e_{p + 1} = Ψ e_{1}, \dots, e_{2 p} = Ψ e_{p}}$ be an orthonormal frame of the vector fields on $T N_{T}$ ⁠. Now, taking summation over $i = 1,2, \dots, p$ for $X = Y$ in (4.4), we have

\sum_{i = 1}^{p} R i c (e_{i}, e_{i}) = λ p + + \sum_{i = 1}^{p} H e s s^{τ} (e_{i}, e_{i}) .

(4.5)

Replacing $e_{i}$ by $Ψ e_{i}$ in above equation, we get

\sum_{i = 1}^{p} R i c (Ψ e_{i}, Ψ e_{i}) = λ p + + \sum_{i = 1}^{p} H e s s^{τ} (Ψ e_{i}, Ψ e_{i}) .

(4.6)

From (4.5) and (4.6), we have

\sum_{i = 1}^{2 p} R i c (e_{i}, e_{i}) = 2 λ p + + \sum_{i = 1}^{p} (H e s s^{τ} (e_{i}, e_{i}) + H e s s^{τ} (Ψ e_{i}, Ψ e_{i})) .

(4.7)

By the assumption that the equality case of (3.21) holds, then by (4.2)

\begin{array}{l} \frac{1}{2 q} \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = \frac{c p}{2} \sin^{2} θ - (\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} \\ + \sum_{i = 1}^{2 p} R i c (e_{i}, e_{i}) - 2 λ p, \end{array}

(4.8)

\begin{array}{l} \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = c p q \sin^{2} θ - 2 q (\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} \\ + 2 q \sum_{i = 1}^{2 p} R i c (e_{i}, e_{i}) - 4 p q λ \end{array}

By the assumption (4.3), we get

(\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} = 0.

From the last equation it is evident that $θ = π / 2$ or the warping function is constant, which proves the theorem.

If the submanifold $M = N_{T} \times_{Φ} N_{θ}$ admits the steady gradient Ricci soliton, then from last theorem, it is easy to conclude the following

Theorem 4.4.

Let $\bar{M} (c)$ be a 2n $+$ 1-dimensional cosymplectic space form and $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold admitting a steady gradient Ricci soliton. If

\sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = c p q \sin^{2} θ - 2 q \sum_{i = 1}^{2 p} R i c (e_{i}, e_{i}),

(4.9)

then one of the following is true

The slant function $θ = π / 2$ i.e., M is a contact CR-warped product submanifold,
The warping function $Φ$ is constant i.e., M is trivial Riemannian product pointwise semi-slant submanifold.

In terms of Ricci curvature, we have the following classification

Theorem 4.5.

Let $\bar{M} (c)$ be a $2 n + 1$ -dimensional cosymplectic space form and $M = N_{T} \times_{Φ} N_{θ}$ be a warped product pointwise semi-slant submanifold with the equality case of (3.21) holds. If the following holds

2 q \int_{M} R i c (\nabla l n Φ, -) d V = c p q \sin^{2} θ - \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2},

(4.10)

then one of the following statement is true

(i)The slant function $θ = π / 2$ i.e., M is a contact CR-warped product submanifold,
(ii)The warping function $Φ$ is constant i.e., M is trivial Riemannian product pointwise semi-slant submanifold.

Proof.

For a connection $\nabla$ on a smooth manifold $M$ ⁠, there exists a tensor $R$ of type (1, 3) called the curvature tensor of the connection $\nabla$ defined by

- \nabla_{U} \nabla_{V} W + \nabla_{V} \nabla_{U} W - \nabla_{[U, V]} W = R (U, V) W

(4.11)

for all $U, V, W \in T M$ ⁠.

For a warping function $τ = l n Φ$ ⁠, from (4.11), we have

\nabla^{2} d (τ) (V, U, W) - \nabla^{2} d (τ) (U, V, W) = d (τ) R (U, V) W .

(4.12)

By the smoothness property of $Φ$ on $N_{T}$ and $\nabla_{U V}^{2} = \nabla_{U} \nabla_{V} - \nabla_{\nabla_{U} V}$ ⁠, then the curvature tensor $R (U, V) W$ behaves like a derivative. Since $d τ$ is closed, then it is easy to see that $\nabla^{2} d (τ) (U, V, W) = \nabla^{2} d (τ) (V, U, W)$ ⁠, for any vector fields $U, V, W \in T N_{T}$ ⁠. Now, for a local orthonormal frame ${e_{1}, e_{2}, \dots, e_{2 p}}$ on $N_{T}$ and for a fixed point $t \in N_{T}$ such that $\nabla_{e i} (e_{j}) (t) = 0$ ⁠, for $1 \leq i, j \leq 2 p + 1$ ⁠. If we specify $\nabla_{e i} (U) (t) = 0$ ⁠, for any $U \in T N_{T}$ and taking trace with respect to $V$ and $W$ in the following equation

\nabla^{2} d (τ) (V, U, W) = \nabla^{2} d (τ) (U, V, W),

and utilizing (4.12), we have

\sum_{i = 1}^{p} (\nabla^{2} d (τ)) (e_{i}, e_{j}, U) = - d (Δ (τ)) (U) + R i c (\nabla, U) .

(4.13)

Further solving left hand side, the above equation takes the form

d i v (H e s s^{τ}) (U) + d (Δ (τ)) (U) = R i c (\nabla τ, U),

(4.14)

d i v (H e s s^{τ}) + d (Δ (τ)) = R i c (\nabla τ, -) .

(4.15)

As $M = N_{T} \times_{Φ} N_{θ}$ is a compact orientable warped product submanifold, then on integrating

Δ (τ) + \int_{M} d i v (H e s s^{τ}) d V = \int_{M} R i c (\nabla τ, -) d V,

where $d V$ is the volume element.

Since $Δ Φ = - d i v (\nabla Φ)$ [28] and $\int_{M} d i v (U) d V = 0$ for any $U \in T N_{T}$ ⁠. So, it is easy to conclude that $\int_{M} d i v (H e s s^{τ}) d V = 0$ ⁠. Then

Δ (τ) = \int_{M} R i c (\nabla τ, -) d V .

(4.16)

Utilizing above equation in (4.1), we have

\begin{array}{l} \sum_{i = 1}^{2 p} \sum_{j = 1}^{2 q} {‖ σ_{μ} (e_{i}, e^{j}) ‖}^{2} = c p q \sin^{2} θ - 2 q (\csc^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} \\ - 2 q \int_{M} R i c (\nabla l n Φ, -) d V . \end{array}

(4.17)

By the assumption (4.10), we get

(\cos^{2} θ + 2 \cot^{2} θ) {‖ \nabla l n Φ ‖}^{2} = 0.

From the above equation it is evident that $θ = π / 2$ or the warping function $Φ$ is constant, which proves the theorem.

5. Conclusion

In this paper, by using Hopf’s Lemma, we obtained the characterizing inequalities for the existence of warped product pointwise semi-slant submanifolds of cosymplectic space forms. Moreover, we also worked out an estimation for the squared norm of the second fundamental form in terms of the warping function and slant function. To strengthen our results, we provided a non-trivial example of a warped product pointwise semi-slant submanifold in a cosymplectic manifold. Moreover, some applications in the form of the Dirichlet energy of the warping function are derived. The results obtained may be helpful in further studies on the Dirichlet energy of smooth functions.

The author is highly thankful to anonymous referee for his/her valuable suggestions and comments which have improved the paper. Declaration of Competing Interest: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. The publisher wishes to inform readers that the article “Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications” was originally published by the previous publisher of the Arab Journal of Mathematical Sciences and the pagination of this article has been subsequently changed. There has been no change to the content of the article. This change was necessary for the journal to transition from the previous publisher to the new one. The publisher sincerely apologises for any inconvenience caused. To access and cite this article, please use Alqahtani, L. S. (2019), “Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications”, Arab Journal of Mathematical Sciences, Vol. 27 No. 1, pp. 53-72. The original publication date for this paper was 19/12/2019.

References

[1]

J.K.

Beem

Ehrlich

T.G.

Powell

Warped Product Manifolds in Relativity

, in:

Selected Studies, North-Holland, Amsterdam-New York

1982

Warped product pointwise semi-slant submanifolds of cosymplectic space forms and their applications

1. Introduction

2. Preliminaries

3. Warped product pointwise semi-slant submanifolds

4. Some applications

5. Conclusion

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