Certain investigations of sequential warped product submanifolds on cosymplectic manifolds

Abstract

In a special class of almost contact metric manifolds known as cosymplectic manifolds, the current study aims to establish the existence result and a few inequalities for sequential warped product submanifolds. These results and inequalities represent fruitful connections between the primary intrinsic and extrinsic invariants. Furthermore, findings related to Dirichlet energy have been addressed. Finally, some exceptional cases resulting in several inequities are examined.

1 Introduction

The Nash theorem was created to ensure that Riemannian manifolds are always perceived as Riemannian submanifolds [23]. Gromov in [16], however, noted that this hope had not come true. The fundamental cause of this is that the extrinsic aspects of the submanifolds are not within the control of the known intrinsic facts. The establishment of meaningful connections between the primary intrinsic and extrinsic invariants of submanifolds is a fundamental concern in submanifold theory. There is growing interest in stabilizing this concern by developing several kinds of geometric equalities and inequalities that admit invariants. The research along this path gained momentum after Chen [7–9], employed the warped product technique as a method to develop inequalities involving invariants and deduced some significant information. After that, several differential geometers established geometric inequalities to analyze the relationships between extrinsic and intrinsic parameters not only for warped products but also for their generalizations, i.e., double and multiple warped products in different ambient spaces (see [10, 12, 13, 24, 32, 33]). The sequential warped product (briefly, SWP), whose base remains a warped product manifold, is a new generalization of warped products (in short, WP) [5]. Shenawy revealed SWP in [31]. Thereafter, De, Shenawy, and Unal did a full investigation of SWP in [14]. With certain examples, Pahan and Pal [25], Karaca and Ozgür [18] extended the study of sequential warped space with (quasi-) Einstein’s structure. Sahin [28] discovered the presence of SWP submanifolds in Kähler manifolds and presented geometric inequalities and equalities such that the associations between the intrinsic and extrinsic invariants are illustrated. In [19], the author recently generalized the Chen inequality involving invariants in nearly Kähler manifolds. Recently, Perktas–Blaga [27] continued the study on Sasakian manifolds and provided some nonexistence results. Motivated by these developments, in this article, the authors first present a numerical example in support of the existence of SWP of the form $( \Omega _{T}\times _{f}\Omega _{\perp } ) \times _{h} \Omega _{\theta }$ on cosymplectic manifolds: an odd-dimensional counterpart of a Kaehler manifold, and then generalized Chen’s type inequality and equalities involved Dirichlet energy and curvature to analyze the geometric invariants using the SWP technique for such type of SWP on a cosymplectic manifold. Other types of SWP on a cosymplectic manifold may be thought of as open problems. A succinct summary of the article is provided below. We first recall some key concepts and definitions of an almost contact, cosymplectic, and their submanifolds in Sect. 2 and Sect. 3. Then, in Sect. 4 we derive and review a few results related to SWP for future use. Finally, in Sect. 5 we establish some important results that extend inequalities for various warped products, inequalities involving Dirichlet energy and curvature in cosymplectic manifolds. We also look at the specific instances and corresponding equality.

2 Preliminaries

Let Ω̄ be a $2m+1$-dimensional smooth manifold and $\mathfrak{X}(\bar{\Omega})$ be the Lie algebra of smooth vector fields on $\bar{\Omega}^{2m+1}$. Then, $(\varphi ,\xi ,\eta )$ is said to have an almost contact structure on $\bar{\Omega}^{2m+1}$ [22], if there exist an endomorphism φ of type $(1,1)$, a smooth global vector field ξ and a 1-form η satisfying

$$ \varphi ^{2}=-\mathcal{I}+\eta \otimes \xi \quad \text{and} \quad \eta \varphi =0, $$

(1)

where $\mathcal{I}$ is the identity map. If an almost contact structure admits Riemann metric g such that

$$\begin{aligned}& g(\varphi \cdot ,\varphi \cdot )=g(\cdot ,\cdot )-\eta (\cdot )\eta ( \cdot ), \end{aligned}$$

(2)

$$\begin{aligned}& \eta (\xi )=1,\qquad \varphi \xi =0,\quad \text{and}\quad g(\cdot ,\varphi \cdot )+g( \varphi \cdot ,\cdot )=0, \end{aligned}$$

(3)

then it is called an almost contact Riemann structure $(\varphi ,\xi ,\eta ,g)$ to $\bar{\Omega}^{2m+1}$. Furthermore, $\bar{\Omega}^{2m+1}$ associated with the structure $(\varphi ,\xi ,\eta ,g)$ is called an almost contact metric manifold $\bar{\Omega}(\varphi ,\xi ,\eta ,g)$.

Also, $g(\cdot ,\varphi \cdot )=\Phi (\cdot ,\cdot )$ where Φ is termed the fundamental 2-form of $\bar{\Omega}^{2m+1}$. If both η and Φ are closed then $\bar{\Omega}(\varphi ,\xi ,\eta ,g)$ is an almost cosymplectic manifold.

Definition 1

An almost contact Riemannian manifold $\bar{\Omega}^{2m+1}$ is cosymplectic [22] if Φ is parallel, that is, $\bar{\nabla}\Phi =0$. In this context, ∇̄ stands for the connection Levi–Civita on $\bar{\Omega}^{2m+1}$ w.r.t. g.

From the above definition and (1), it is easy to obtain that

$$ \bar{\nabla}_{U}\xi =(\bar{\nabla}_{U}\eta )V=0,\quad \text{for all } U,V \in \mathfrak{X}(\bar{\Omega}). $$

(4)

Let $\bar{\Omega}(k)$ be a cosymplectic space form, then the Riemannian curvature on $\bar{\Omega}(k)$ for arbitrary $U,V,W,X\in \mathfrak{X}(\bar{\Omega})$ is given by

$$\begin{aligned} &\bar{R}(U,V,W,X) \\ &\quad =\frac{k}{4} \bigl(g(V,W)g(U,X)-g(U,W)g(V,X)-\eta (V) \eta (W)g(U,X) \\ &\quad \quad{} + \eta (U)\eta (W)g(V,X)+\eta (V)\eta (X)g(U,W)-\eta (U)\eta (X)g(V,W) \\ &\quad \quad{} + g(\varphi V,W)g(\varphi U,X)-g(\varphi U,W)g(\varphi V,X)+2g( \varphi U,V)g( \varphi W,X) \bigr). \end{aligned}$$

(5)

3 Geometry of submanifolds

Now, we review some important formulas and definitions for future usage concerning submanifolds Ω of dimension d, immersed isometrically in $\bar{\Omega}^{2m+1}$. (For details refer to [6].) The formulas of Gauss–Weingarten are defined by the expressions:

$$\begin{aligned}& \bar{\nabla}_{U}V =\nabla _{U}V+\sigma (U,V), \end{aligned}$$

(6)

$$\begin{aligned}& \bar{\nabla}_{U}N =\nabla _{U}^{\bot }N -A_{N }U, \end{aligned}$$

(7)

$\forall U,V \in \mathfrak{X}(\Omega )$: a space tangent to Ω and $N \in \mathfrak{X}(\Omega ^{\bot })$: a space normal to Ω. The induced tangent and normal connections on $\mathfrak{X}(\Omega )$ and $\mathfrak{X}(\Omega ^{\bot })$ are denoted by the symbols ∇ and $\nabla ^{\bot }$, respectively. Then, the shape operator and the second fundamental form (abbreviated SFF) are represented by the $A_{N }$ at N and σ, respectively, in such a way that

$$\begin{aligned} g(A_{N }U,V)=g \bigl(\sigma (U,V),N \bigr). \end{aligned}$$

(8)

If p is any point in Ω and $\{x_{1},\ldots ,x_{d}, x_{d+1},\ldots ,x_{2m+1}\}$ is an orthonormal frame of the tangent space $\mathfrak{X}_{p}(\bar{\Omega})=\mathfrak{X}_{p}(\Omega )\oplus \mathfrak{X}_{p}(\Omega ^{\bot })$ such that $\{x_{1},\ldots ,x_{d}\}\in \mathfrak{X}_{p}(\Omega )$ and $\{x_{d+1},\ldots ,x_{2m+1}\}\in \mathfrak{X}_{p}(\Omega ^{\bot })$, then $\mathcal{H}(p)=\frac{1}{d}\mathrm{trace} \sigma$, gives the mean curvature vector of Ω and $\|\sigma \|^{2}$ is computed by

$$ \|\sigma \|^{2}=\sum_{a,b=1}^{d}g \bigl(\sigma (x_{a},x_{b}),\sigma (x_{a},x_{b}) \bigr). $$

(9)

By setting $\sigma _{ab}^{c}=g(\sigma (x_{a},x_{b}),x_{c})$, $a,b\in \{x_{1}, \ldots ,x_{d}\}$, $c\in \{x_{d+1},\ldots ,x_{2m+1}\}$, (9) can be represented as

$$ \|\sigma \|^{2}=\sum_{c=d+1}^{2m+1} \sum_{a,b=1}^{d}g \bigl(\sigma (x_{a},x_{b}),x_{c} \bigr). $$

(10)

If $\sigma (U,V)$ equals $g(U,V)\mathcal{H}$ (0), then Ω is umbilical (geodesic). For $\mathcal{H}=0$, Ω is minimal. Moreover, if

$$\begin{aligned}& \varphi {U} =t{U}+n{U}, \end{aligned}$$

(11)

$$\begin{aligned}& \varphi {N} =t^{\prime }{N}+n^{\prime }{N}, \end{aligned}$$

(12)

for all $U\in \mathfrak{X}(\Omega )$, where tU ($t^{\prime }{N}$) and nU ($n^{\prime }{N}$) are tangential (normal) parts of φU (φN), then employing (1), (2), and (11), we obtain

$$ g(U,tV)=-g(tU,V)\quad \text{for all }U,V\in \mathfrak{X}(\Omega ). $$

(13)

In view of Definition 1 and (6) and (12), we have

$$\begin{aligned}& (\nabla _{U}t)V =A_{nV}U+t^{{\prime }}\sigma (U,V), \end{aligned}$$

(14)

$$\begin{aligned}& (\nabla _{U}n)V =-\sigma (U,tV)+n^{{\prime }}\sigma (U,V), \end{aligned}$$

(15)

where $(\nabla _{U}\varphi )V$, $(\nabla _{U}t)V$ and $(\nabla _{U}n)V$ are defined by

$$\begin{aligned}& (\nabla _{U}\varphi )V =\nabla _{U}\varphi V-\varphi \nabla _{U}V, \end{aligned}$$

(16)

$$\begin{aligned}& (\nabla _{U}t)V =\nabla _{U}tV-t\nabla _{U}V, \end{aligned}$$

(17)

$$\begin{aligned}& (\nabla _{U}n)V =\nabla _{U}^{\perp }nV-n \nabla _{U}V, \end{aligned}$$

(18)

for all $U,V\in \mathfrak{X}(\Omega )$. Next, the Gauss–Codazzi equations are defined, respectively, by,

$$\begin{aligned}& \begin{aligned}[b] \bar{R}(U,V,W,X)& =g \bigl(\sigma (U,W),\sigma (V,X) \bigr)-g \bigl(\sigma (U,X), \sigma (V,W) \bigr) \\ &\quad{} + R(U,V,W,X), \end{aligned} \end{aligned}$$

(19)

$$\begin{aligned}& \bigl(\bar{R}(U,V)W \bigr)^{\perp }) =(\nabla _{U}\sigma ) (V,W)-(\nabla _{V} \sigma ) (U,W), \end{aligned}$$

(20)

for every U, V, W and $X\in \mathfrak{X}(\Omega )$, where R and R̄ denote the Riemann curvature on Ω and $\bar{\Omega}^{2m+1}$. Next, the gradient of the smooth function f on Ω is given by

$$ g(\nabla f,U)=Uf \quad \text{and}\quad \Vert \nabla f \Vert ^{2}=\sum_{a=1}^{d} \bigl(U_{a}(f) \bigr)^{2}, $$

(21)

for any $U_{a}\in \mathfrak{X}(\Omega )$.

Here, we refresh certain key definitions and findings from [26] for further usage.

Definition 2

Let Ω be a submanifold of $\bar{\Omega}^{2m+1}$. If at a point p in Ω, any tangent vector field $W/\{0\}$ not proportional to the characteristic vector field $\xi \in \mathfrak{X}(\Omega )$, the angle symbolized by θ, between $\varphi (W)$ and $\mathfrak{X}_{p}(\Omega )$ does not depend on p nor W, then Ω is pointwise slant. Consequently, θ is identified as a function on Ω, and thus known as the slant function.

Remark 1

For θ constant on Ω, Ω is simply called a slant submanifold, specifically, invariant when $\theta =0$ and antinvariant when $\theta =\pi /2$.

Next, from the definition and [20, 30], we write the iff condition for a submanifold Ω of $\bar{\Omega}^{2m+1}$ to be pointwise slant, given as follows for any vector field:

$$ t^{2}W=\cos ^{2}(\theta )\varphi ^{2}W \mbox{:} \quad W \in \mathfrak{X}( \Omega ). $$

(22)

Additionally, we have for $W_{1}, W_{2}\in \mathfrak{X}(\Omega )$ that

$$\begin{aligned}& g(tW_{1},tW_{2}) =\cos ^{2}g(\varphi W_{1},\varphi W_{2}), \end{aligned}$$

(23)

$$\begin{aligned}& g(nW_{1},nW_{2}) =\sin ^{2}g(\varphi W_{1},\varphi W_{2}). \end{aligned}$$

(24)

Definition 3

A submanifold Ω is pointwise semislant (resp., pseudoslant), if there are distributions $\mathfrak{D}_{T}$: an invariant and $\mathfrak{D}_{\theta }$: a pointwise slant on Ω (resp., $\mathfrak{D}_{\bot }$: an antiinvariant and $\mathfrak{D}_{\theta }$) with slant function θ and characteristic vector field ξ such that

• i. The tangent space $\mathfrak{X}(\Omega )$ admits the orthogonal direct decomposition $\mathfrak{X}(\Omega ) = \mathfrak{D}_{T}\oplus \mathfrak{D}_{\theta} \oplus <\xi >$ (resp., $\mathfrak{X}(\Omega ) = \mathfrak{D}_{\bot}\oplus \mathfrak{D}_{ \theta} \oplus <\xi >$);

• ii. The distribution $\mathfrak{D}_{T}$ is invariant, i.e., $\varphi (\mathfrak{D}_{T})\subseteq \mathfrak{D}_{T}$ (resp., $\mathfrak{D}_{\bot}$ is antiinvariant, i.e., $\varphi (\mathfrak{D}_{\bot})\subseteq \mathfrak{X}(\Omega ){^{\bot}}$).

4 Sequential warped product submanifolds

Shenawy in [31] and De et al. in [14] defined SWP as: consider three Riemannian manifolds B, $F_{1}$, and $F_{2}$. If $f:B\rightarrow \mathbb{R}^{+}$ and $h:B\times F_{1}\rightarrow \mathbb{R}^{+}$, then the SWP $(B\times _{f}F_{1})\times _{h}F_{2}$ of B, $F_{1}$ and $F_{2}$ is the product manifold $\bar{\Omega}=B\times F_{1}\times F_{2}$ endowed with Riemannian metric $\bar{g}=(g_{B}\oplus _{f^{2}}g_{1})\oplus _{h^{2}}g_{2}$. Ω̄ is a Riemannian triple product for f, h constants; Ω̄ is a WP with a product base manifold for exactly one of f, h constant. Ω̄ is SWP, if f, h are nonconstants.

Sahin recently analyzed the possibility of warped products of type $( \Omega _{T}\times _{f}\Omega _{\perp } ) \times _{h} \Omega _{\theta }$ for Kaehler manifolds and discovered some significant findings [28]. Motivated by the work of Sahin, in this section we continue the study for cosymplectic manifolds: an odd-dimensional counterpart of a Kaehler manifold, and derive several geometric characterizations for such types of submanifolds in cosymplectic manifold.

For a vector field on a factor manifold and its lift to the sequential warped product manifold, we use the same notation. We start by keeping in mind the following statements about SWP manifolds for future use.

Proposition 1

[14] For $U_{a},V_{a},W_{a}\in \mathfrak{X}(\Omega _{a})$, $a\in \{1,2,3\}$, we have on $\Omega = ( \Omega _{1}\times _{f}\Omega _{2} ) \times _{h} \Omega _{3}$ that

• 1. $\nabla _{U_{a}}U_{3}=\nabla _{U_{3}}U_{a}=U_{3}(\ln h)U_{a}= \frac{U_{3}(h)}{h}U_{a}$, $a\in \{1,2\}$;

• 2. $\nabla _{U_{1}}U_{2}=\nabla _{U_{2}}U_{1}=U_{1}(\ln f)U_{2}= \frac{U_{1}(f)}{f}U_{2}$;

• 3. $R(U_{a},V_{3})W_{b}=\frac{-1}{h}H^{h}(U_{a},W_{b})V_{3}$, $a,b\in \{1,2 \}$.

Proposition 2

[14] In a SWP $\Omega = ( \Omega _{1}\times _{f}\Omega _{2} ) \times _{h} \Omega _{3}$, $\Omega _{1}$ and $\Omega _{1}\times _{f}\Omega _{2}$ are totally geodesic submanifolds in $\Omega _{1}\times _{f}\Omega _{2}$ and Ω. Also, $\Omega _{2}$ and $\Omega _{3}$ are totally umbilical in $\Omega _{1}\times _{f}\Omega _{2}$ and Ω.

Now, we present a nonexistence result or SWP submanifolds by including ξ in the second or the third factor manifolds.

Theorem 1

If $\Omega = ( \Omega _{T}\times _{f}\Omega _{\perp} ) \times _{h}\Omega _{\theta }$ is a SWP immersion in cosymplectic manifold $\bar{\Omega}^{2m+1}$. This means

• 1. if $\xi \in \mathfrak{X}(\Omega _{\perp })$, then Ω is a CR-slant WP submanifold;

• 2. if $\xi \in \mathfrak{X}(\Omega _{\theta })$, then Ω is a single WP submanifold.

Proof

Consider $\xi \in \mathfrak{X}(\Omega _{\perp })$, then by the use of Proposition 1, we attain

$$ \nabla _{U}\xi =U(\ln f)\xi ,\quad \forall U\in \mathfrak{X}(\Omega _{T}). $$

In view of (4), (6), and (7), we have $U(\ln f)=0$. This shows that f is constant on $\Omega _{T}$. From the above discussion, we conclude that Ω is a CR-slant warped product submanifold. On the other hand, $\xi \in \mathfrak{X}(\Omega _{\theta })$, then by the use of Proposition 1, we achieve

$$ \nabla _{U}\xi =U(\ln h)\xi \quad \text{and}\quad \nabla _{W}\xi =W(\ln h)\xi , $$

(25)

for every $U\in \mathfrak{X}(\Omega _{T})$ and $V\in \mathfrak{X}(\Omega _{\perp })$. In light of (4), (6), (7), and (25), we observe that $U(\ln h)=V(\ln h)=0$. This shows that h is constant on both the factors $\Omega _{T}$ and $\Omega _{\perp }$. This completes the proof. □

Next, we first present a numerical example that shows the existence of SWP of the form $( \Omega _{T}\times _{f}\Omega _{\perp } ) \times _{h} \Omega _{\theta }$ and then continue the study by presenting several important results for such submanifolds.

Example 1

Consider a 13-dimensional Euclidean space $\mathbb{E}^{13}$ with coordinates $(x_{1}, \ldots , x_{6}, y_{1}, \ldots , y_{6}, t)$ and Euclidean metric g. The almost contact structure $(\varphi , \xi , \eta )$ on $\mathbb{E}^{13}$ is described as

$$\begin{aligned} &\varphi \biggl(\frac{\partial}{\partial x_{a}} \biggr)=- \frac{\partial}{\partial y_{a}},\qquad \varphi \biggl( \frac{\partial}{\partial y_{a}} \biggr)= \frac{\partial}{\partial x_{a}}, \qquad \varphi \biggl( \frac{\partial}{\partial t} \biggr)=0, \end{aligned}$$

(26)

$$\begin{aligned} &\xi =\frac{\partial}{\partial t}, \qquad \eta =dt,\qquad g=dt^{2}+\sum _{a=1}^{12}\,dx ^{2}_{a}. \end{aligned}$$

(27)

One can easily verify that the Euclidean space $\mathbb{E}^{13}$ with almost contact structure is a cosymplectic manifold. Consider a subset $\Omega \subset \mathbb{E}^{13}$ immersed as a submanifold by the following immersion

$$\begin{aligned}& x_{1}=u \cos \alpha ,\qquad x_{2}=v \cos \beta ,\qquad x_{3}=v \sin \alpha ,\qquad x_{4}=u\sin \beta ,\qquad x_{5}=\alpha \sin \beta , \\& x_{6}=w, \qquad y_{1}=v \cos \alpha ,\qquad y_{2}=u \cos \beta ,\qquad y_{3}=u \sin \alpha ,\qquad y_{4}=v\sin \beta , \\& y_{5}=\alpha \cos \beta ,\qquad y_{6}=\beta ,\qquad t=t. \end{aligned}$$

The tangent subspace of Ω at each point is spanned by the basis

$$\begin{aligned}& Z_{\alpha } = -u\sin \alpha \frac{\partial}{\partial x_{1}}+v\cos \beta \frac{\partial}{\partial x_{3}}+\sin \beta \frac{\partial}{\partial x_{5}}-v\sin \alpha \frac{\partial}{\partial y_{1}}+u\cos \alpha \frac{\partial}{\partial y_{3}}+\cos \beta \frac{\partial}{\partial y_{5}}, \\& \begin{aligned} Z_{\beta }& = -v \sin \beta \frac{\partial}{\partial x_{2}}+u\cos \beta \frac{\partial}{\partial x_{4}}+\alpha \cos \beta \frac{\partial}{\partial x_{5}}-u \sin \beta \frac{\partial}{\partial y_{2}} \\ &\quad{} + v \cos \beta \frac{\partial}{\partial y_{4}}-\alpha \sin \beta \frac{\partial}{\partial y_{5}}+ \frac{\partial}{\partial y_{6}}, \end{aligned} \\& Z_{u} = \cos \alpha \frac{\partial}{\partial x_{1}}+\sin \beta \frac{\partial}{\partial x_{4}}+\cos \beta \frac{\partial}{\partial y_{2}}+\sin \alpha \frac{\partial}{\partial y_{3}},\qquad Z_{w}= \frac{\partial}{\partial x_{6}}, \\& Z_{v} = \cos \beta \frac{\partial}{\partial x_{2}}+\sin \alpha \frac{\partial}{\partial x_{3}}+\cos \alpha \frac{\partial}{\partial y_{1}}+\sin \beta \frac{\partial}{\partial y_{4}},\qquad Z_{t}=\xi . \end{aligned}$$

By straightforward computation, we observed that the distribution $\mathfrak{D}_{T}=\operatorname{span}\lbrace Z_{u}, Z_{v}, Z_{t}\rbrace$ is invariant, the distribution $\mathfrak{D}_{\perp}=\operatorname{span}\lbrace Z_{\alpha }\rbrace$ is antiinvariant and distribution $\mathfrak{D}_{\theta}=\operatorname{span}\lbrace Z_{\beta }, Z_{w }\rbrace$ is pointwise slant with slant angle

$$\begin{aligned} \theta =\cos ^{-1} \biggl( \frac{1}{\sqrt{1+u^{2}+v^{2}+\alpha ^{2}}} \biggr). \end{aligned}$$

The induced metric on Ω is given by

$$\begin{aligned} g_{\Omega}=du^{2}+dv^{2}+dt^{2}+f^{2}\,d \alpha ^{2}+h^{2} \bigl(d\beta ^{2}+dw ^{2} \bigr). \end{aligned}$$

This shows that Ω is a $SWP$ manifold of $\mathbb{E}^{13}$ with warping functions $f=\sqrt{u^{2}+v^{2}+1}$ and $h=\sqrt{u^{2}+v^{2}+\alpha ^{2}+1}$.

Remark 2

From here onwards, we consider and study the SWP submanifolds of the type $( \Omega _{T}\times _{f}\Omega _{\perp } ) \times _{h} \Omega _{\theta }$ when the characteristic vector field ξ is tangent to $\Omega _{T}$.

Proposition 3

Let $( \Omega _{T}\times _{f}\Omega _{\perp } ) \times _{h} \Omega _{\theta }$ be a SWP submanifold of cosymplectic manifold $\bar{\Omega}^{2m+1}$. Then, $\xi (\ln f)=\xi (\ln h)=0$.

Proof

Consider $U\in \mathfrak{X}(\Omega _{T})$ and $V\in \mathfrak{X}(\Omega _{\perp})$, then by consequence of the first part of Proposition 1, we have $\nabla _{U}V=\nabla _{V}U=U(\ln f)V=U(f)/f.V$. Since $\xi \in \mathfrak{U}(\Omega _{T})$, then the previous equation becomes $\nabla _{V}\xi =\xi (\ln f)V$. Hence, applying (4) and (6) in the above expression, we obtain the first part. By the use of the second part of Proposition 1, we derive $\nabla _{U}W=U(f)/f*W$, $W \in \mathfrak{X}(\Omega _{\theta})$. With the help of (4) and (6) in the last relation, we obtain the second part. □

Here, we present some crucial findings for later use.

Lemma 1

Assume $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a SWP submanifold of cosymplectic manifold $\bar{\Omega}^{2m+1}$. Then, for all $U_{1}, U_{2}\in \mathfrak{X}(\Omega _{T})$ and $V_{1},V_{2}\in \mathfrak{X}(\Omega _{\perp })$ we have

• i. $g(\sigma (U_{1}, U_{2}), \varphi V_{1})=0$;

• ii. $g(\sigma (U_{1},V_{1}),\varphi V_{2})=-\varphi U_{1}(f)/f.g(V_{1},V_{2})$.

Proof

By the utilization of (3), (6), (16), and Definition 1, we concede that

$$ g \bigl(\sigma (U_{1}, U_{2}),\varphi V_{1} \bigr)=-g(\bar{\nabla}_{U_{1}} \varphi V,V_{1}). $$

(28)

Inserting the covariant derivative’s characteristic into (28), we arrive at

$$ g \bigl(\sigma (U_{1}, U_{2}),\varphi V_{1} \bigr)=g(\varphi V,\bar{\nabla}_{U_{1}}V_{1})-Ug( \varphi V,V_{1}). $$

(29)

By the use of the first part of Proposition 1 in (29), we obtain the first part. For the second part, we utilize (3), (6), and (11) to obtain $g(\sigma (U_{1},V_{1}),\varphi V_{2})=-g(\varphi \bar{\nabla}_{V_{1}}U_{1},V_{2})$. With the help of the first part of Proposition 1, we obtain the second part. □

Lemma 2

If $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a SWP of cosymplectic manifold $\bar{\Omega}^{2m+1}$, then we attain for all $U_{1}, U_{2}\in \mathfrak{X}(\Omega _{T})$ and $W_{1},W_{2}\in \mathfrak{X}(\Omega _{\theta })$ that

• i. $g(\sigma (U_{1}, U_{2}), nW_{1})=0$;

• ii. $g(\sigma (U_{1},W_{1}), nW_{2})=-\varphi U_{1}(h)/h.g(W_{1},W_{2})+U_{1}(h)/h.g(tW_{1},W_{2})$.

Proof

By the result of (6) and (11), we obtain $g(\sigma (U_{1}, U_{2}),nW_{1})=g(\bar{\nabla}_{U_{1}}U_{2},\varphi W_{1}-g( \bar{\nabla}_{U_{1}}U_{2},tW_{1})$. The previous expression simplifies to the following form when (3) is used:

$$ g \bigl(\sigma (U_{1}, U_{2}),nW_{1} \bigr)=-g(\bar{\nabla}_{U_{1}}\varphi U_{2},W_{1})-g( \bar{\nabla}_{U_{1}}U_{2},tW_{1}). $$

Applying Definition 1 and the second part of Proposition 1 into the above expression, we obtain the first part. By reusing (6) and (11), we have $g(\sigma (U_{1},W_{1}),nW_{2})=g(\bar{\nabla}_{W_{1}}U_{1},\varphi W_{2}-g( \bar{\nabla }_{W_{1}}U_{2}, tW_{2})$. Next, employing (3) and Proposition 1(ii), we obtain the second part. □

Moreover, employing (1) in Lemma 2, we obtain that

$$\begin{aligned}& g \bigl(\sigma (\varphi U_{1}, U_{2}),nW_{1} \bigr) =0, \end{aligned}$$

(30)

$$\begin{aligned}& g \bigl(\sigma (\varphi U_{1},W_{1}),nW_{2} \bigr) =U_{1}(h)/h.g(W_{1},W_{2})+ \varphi U_{1}(h)/h.g(tW_{1},W_{2}). \end{aligned}$$

(31)

Replacing $W_{1}$ by $tW_{1}$ in Lemma 2, then utilizing (22), we obtain

$$ g \bigl(h(U_{1},tW_{1}),nW_{2} \bigr)=-U_{1}(h)/h.g(tW_{1},W_{2})-\cos ^{2} \theta \varphi U_{1}(h)/h.g(W_{1},W_{2}). $$

(32)

Replacing $W_{1}$ by $tW_{1}$ in (28) and (29), and then employing (22) in the obtained expression, we arrive at

$$ g \bigl(h(\varphi U_{1},tW_{1}),nW_{2} \bigr)=U_{1}(h)/h.g(tW_{1},W_{2})-\cos ^{2} \theta \varphi U_{1}(h)/h.g(W_{1},W_{2}). $$

(33)

Replacing $W_{2}$ by $tW_{2}$ in (29), (32), and (33) and using (23), yields

$$ g \bigl(h(\varphi U_{1},W_{1}),ntW_{2} \bigr)=U_{1}(h)/h.g(W_{1},tW_{2})+\cos ^{2} \theta \varphi U_{1}(h)/h.g(W_{1},W_{2}). $$

(34)

Replacing $U_{1}$ by $\varphi U_{1}$ in (34), we achieve that

$$ g \bigl(h(U_{1},W_{1}),ntW_{2} \bigr)=- \varphi U_{1}(h)/h.g(W_{1},tW_{2})-\cos ^{2} \theta U_{1}(h)/h.g(W_{1},W_{2}). $$

(35)

Lemma 3

If $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a SWP of cosymplectic manifold $\bar{\Omega}^{2m+1}$, then we derive for all $U \in \mathfrak{X}(\Omega _{T})$, $V\in \mathfrak{X}(\Omega _{\perp })$ and $W \in \mathfrak{X}(\Omega _{\theta })$ that

• i. $g(\sigma (U, V), nU)$ vanishes;

• ii. $g(\sigma (U, W), \varphi V)$ vanishes;

• iii. $\sigma (V, W)=\sigma (W, W)$.

Proof

As a result of (6) and (11), we obtain $g(\sigma (U,V),nW)=g(\bar{\nabla}_{V}U,\varphi W)-g(\nabla _{V}U,tW)$. Now, employing the first part of Lemma 2 into the last expression, we achieve the first part. Likewise, we can prove the second part. By the consequence of (3), (6), and (11), we obtain

$$ g \bigl(\sigma (V,tW),nW \bigr)=-g(\varphi \bar{\nabla}_{V}tW,W)-g( \nabla _{V}tW,tW). $$

With the help of Definition 1 and the third part of Proposition 2, we have

$$ g \bigl(\sigma (V,tW),nW \bigr)=-g(\bar{\nabla}_{V}\varphi tW,W)-V(h)/h.g(tW,tW). $$

Applying (11), (22), and (23), we achieve the third part. □

Lemma 4

If $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a SWP of cosymplectic manifold $\bar{\Omega}^{2m+1}$, then we deduce for all $V\in \mathfrak{X}(\Omega _{\perp })$ and $W\in \mathfrak{X}(\Omega _{\theta })$ that

$$ g \bigl(\sigma (W,V),ntW \bigr)=\cos ^{2}\theta V(h)/h.g(W,W)+g \bigl( \sigma (W, tW), \varphi V \bigr). $$

(36)

Proof

By use of (3) and (6), we obtain $g(\sigma (W,tW),\varphi V)=-g(\varphi \bar{\nabla}_{W}tW,V)$. Now, utilizing (11) and Definition 1 in the last expression, we observe that

$$ g \bigl(\sigma (W,tW),\varphi V \bigr)=-g \bigl(\bar{\nabla}_{W}t^{2}W,V \bigr)-g( \bar{\nabla}_{W}ntW,V). $$

By virtue of (8) and (22), we concede that

$$ g \bigl(\sigma (W,tW),\varphi V \bigr)=-\cos ^{2}\theta g(\bar{ \nabla}_{W}W,V)+g \bigl( \sigma (W,V),ntW \bigr). $$

Using the second part of Proposition 1 and the property of the covariant derivative, we achieve (36). □

Lemma 5

If $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a SWP of cosymplectic manifold $\bar{\Omega}^{2m+1}$, then we have for all $U\in \mathfrak{X}(\Omega _{T})$, $V\in \mathfrak{X}(\Omega _{\perp })$ and $W \in \mathfrak{X}(\Omega _{\theta })$ that

$$\begin{aligned} & \bigl\Vert \sigma _{\nu }(U,V) \bigr\Vert ^{2}=g \bigl(\sigma ( \varphi U,V),\varphi \sigma (U,V) \bigr), \end{aligned}$$

(37)

$$\begin{aligned} & \bigl\Vert \sigma _{\nu }(U,W) \bigr\Vert ^{2}=2\cos ^{2}\theta U(h)^{2}/h.g(W,W)+g \bigl( \sigma (\varphi U,W),\varphi \sigma (U,W) \bigr). \end{aligned}$$

(38)

Proof

By the use of (16) and (6), we have $\varphi \nabla _{V}U-\nabla _{V}\varphi U=\sigma (\varphi U, V)- \varphi \sigma (U, V)$. Employing the fact that $\Omega _{T}$ is an invariant submanifold, then by the application of Proposition 1(i) the above expression reduces to the following form:

$$\begin{aligned} U(h)/h.\varphi V-\varphi U(h)/h.V=\sigma (\varphi U, V)-\varphi \sigma (U, V). \end{aligned}$$

(39)

Now, by taking an inner product with $\varphi \sigma (U, V)$ into (39), we achieve the first part. Reusing (16) and (6), we have $\varphi \nabla _{W}U-\nabla _{W}\varphi U=\sigma (\varphi U, W)- \varphi \sigma (W, V)$. By the application of Proposition 1(i) in the above expression, we obtain

$$\begin{aligned} U(h)/h.\varphi W-\varphi U(h)/h.W=\sigma (\varphi U, W)-\varphi \sigma (U,W). \end{aligned}$$

(40)

Now, by taking the inner product with $\varphi \sigma (U, W)$ into (39), we have

$$\begin{aligned} U(h)/h.g \bigl(\varphi \sigma (U, W), nW \bigr)=g \bigl(\sigma (\varphi U, W), \varphi \sigma (U, W) \bigr)- \bigl\Vert \sigma _{\nu}(U, W) \bigr\Vert ^{2}. \end{aligned}$$

(41)

In view of (35) and (41), we obtain the second part. □

5 Main results

In this section, we show several significant findings and their geometric applications. Let Ω be a SWP submanifold of cosymplectic manifold $\bar{\Omega}^{2m+1}$.

Then, we can express $\mathfrak{X}(\bar{\Omega})=\mathfrak{X}(\Omega )\oplus \mathfrak{X}( \Omega )^{\bot }$ with $\mathfrak{X}(\Omega )=\mathfrak{D}_{T}\oplus \mathfrak{D}_{\perp }\oplus \mathfrak{D}_{\theta }$ and $\mathfrak{X}(\Omega )^{\bot }=n\mathfrak{D}_{\perp }\oplus n \mathfrak{D}_{\theta }\oplus \nu$.

Take orthonormal frames as follows. For tangent distributions $\mathfrak{D}_{T}$, $\mathfrak{D}_{\perp }$, and $\mathfrak{D}_{\theta }$:

$$\begin{aligned}& \{x_{1},\ldots ,x_{p },x_{p +1}=\varphi x_{1},\ldots ,x_{2p }= \varphi x_{p },\xi \},\\& \bigl\{ x_{2p +1}=x_{1}^{\ast },x_{2p +2}=x_{2}^{\ast }, \ldots ,x_{2p +q }=x_{q }^{\ast } \bigr\} \end{aligned}$$

and

$$ \{x_{2p +q +1}=\hat{x}_{1},\ldots ,x_{2p +q +r }= \hat{x}_{r },x_{2p +q +r +1}=\sec \theta t\hat{x}_{1}, \ldots ,x_{2p +q +2r }=\sec \theta t \hat{x}_{r }\} $$

and for normal distributions $n\mathfrak{D}_{\perp}$, $n\mathfrak{D}_{\theta }$, and ν:

$$\begin{aligned}& \bigl\{ \tilde{x}_{1}=\varphi x_{1}^{\ast }, \tilde{x}_{2}=\varphi x_{2}^{ \ast },\ldots , \tilde{x}_{q }=\varphi x_{q }^{\ast } \bigr\} ,\\& \{\tilde{x}_{q +1}=\csc \theta n\hat{x}_{1},\ldots , \tilde{x}_{q +r }= \csc \theta \hat{x}_{r }, \tilde{x}_{q +r +1}=\csc \theta \sec \theta nt \hat{x}_{1}, \ldots ,\tilde{x}_{q +2r }=\csc \theta \sec \theta nt \hat{x}_{r } \} \end{aligned}$$

and

$$ \{\tilde{x}_{q +2r }=\overline{x}_{1},\ldots , \tilde{x}_{q +2r +s}= \overline{x}_{s}, \tilde{x}_{q +2r +s+1}=\overline{x}_{s+1},\ldots , \tilde{x}_{q +2r +2s}=\overline{x}_{2s}\}. $$

Then, we deduce $\dim(\mathfrak{D}_{T})=2p +1$, $\dim(\mathfrak{D}_{\perp })=q$, $\dim(\mathfrak{D}_{\theta })=2r$, $\dim(n\mathfrak{D}_{\perp })=q$, $\dim(n\mathfrak{D}_{\theta })=2r$ and $\dim(\nu )=2s$. Hence, Ω is $d=2p +q +2r +1$-dimensional and $\Omega ^{\bot }$ is $2m-d+1=2s+q +2r$-dimensional.

Here, we present our first main result that represents a relationship between the warping function and the second fundamental form and acts as a generalization for earlier existing literature in this regard.

Theorem 2

Assume $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is a d-dimensional SWP submanifold of cosymplectic manifold $\bar{\Omega}^{2m+1}$, then

$$ \Vert \sigma \Vert ^{2}\geq 2q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}+4r \bigl(1+2\cot ^{2}\theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}+2r \cot ^{2}\theta \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2} $$

(42)

holds, where σ symbolizes the SFF, ${\nabla ^{T}(f)}$: gradient(f) on $\Omega _{T}$, ${\nabla ^{T}(h)}$ and ${\nabla ^{\perp}(h)}$: gradients(h) on $\Omega _{T}$ and $\Omega _{\perp }$.

Proof

By the use of (10), we have

$$ \Vert \sigma \Vert ^{2}=\sum_{a,b=1}^{d} \sum_{c=1}^{2m+1-d}g \bigl( \sigma (x_{a},x_{b}),\tilde{x}_{c} \bigr)^{2}. $$

Since $\mathfrak{X}(\Omega ^{\perp })=\mathfrak{X}(n\mathfrak{D}_{\perp }) \oplus \mathfrak{X}(n\mathfrak{D}_{\theta })\oplus \nu$, then the preceding expression is simplified into its subsequent form as

$$\begin{aligned} \Vert \sigma \Vert ^{2}& =\sum_{a,b=1}^{d} \sum_{c=1}^{q }g \bigl(\sigma (x_{a},x_{b}), \tilde{x}_{c} \bigr)^{2}+\sum_{a,b=1}^{d}\sum _{c=1}^{2r }g \bigl(\sigma (x_{a},x_{b}), \tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + \sum_{c=1}^{2s }\sum _{a,b=1}^{d}g \bigl(\sigma (x_{a},x_{b}), \bar{x}_{c} \bigr)^{2}. \end{aligned}$$

(43)

The $n\mathfrak{D}_{\perp }$ and $n\mathfrak{D}_{\theta }$ components are found in the first and second terms, while ν-components are found in the final term in the above expression. Further, we only use the frame fields described above to calculate the components in the first and second terms as follows:

$$\begin{aligned} \Vert \sigma \Vert ^{2}&\geq \sum_{a,b=1}^{d} \sum_{c=1}^{q }g \bigl( \sigma (x_{a},x_{b}),\tilde{x}_{c} \bigr)^{2}+\sum_{a,b=1}^{d}\sum _{c=1}^{2r }g \bigl(\sigma (x_{a},x_{b}),\tilde{x}_{q+c} \bigr)^{2}. \end{aligned}$$

(44)

By using the above-mentioned frame field, we concede that

$$\begin{aligned} \Vert \sigma \Vert ^{2}& \geq \sum_{a,b=1}^{2p } \sum_{c=1}^{q }g \bigl( \sigma (x_{a},x_{b}),\tilde{x}_{c} \bigr)^{2}+\sum_{c=1}^{q }g \bigl(\sigma ( \xi ,\xi ),\tilde{x}_{c} \bigr)^{2}+\sum_{a,b=1}^{2p } \sum_{c=1}^{2r }g \bigl(\sigma (x_{a},x_{b}), \tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + \sum_{a,b,c=1}^{q }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{b}^{\ast } \bigr), \tilde{x}_{c} \bigr)^{2}+\sum _{c=1}^{2r }g \bigl(\sigma (\xi ,\xi ), \tilde{x}_{q+c} \bigr)^{2}+\sum _{a,b=1}^{q }\sum_{c=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{b}^{ \ast } \bigr),\tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + \sum_{a,b,c=1}^{2r }g \bigl(\sigma ( \hat{x}_{a},\hat{x}_{b}), \tilde{x}_{q+c} \bigr)^{2}+\sum_{a,b=1}^{2r } \sum_{c=1}^{q }g \bigl(\sigma ( \hat{x}_{a}, \hat{x}_{b}), \tilde{x}_{c} \bigr)^{2} \\ &\quad{} + 2\sum_{a,c=1}^{q }\sum _{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast }, \hat{x}_{b} \bigr),\tilde{x}_{c} \bigr)^{2}+2\sum_{a=1}^{q } \sum_{b,c=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{ \ast },\hat{x}_{b} \bigr),\tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + 2\sum_{a=1}^{2p }\sum _{b=1}^{2r }\sum_{c=1}^{q }g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{c} \bigr)^{2}+2\sum _{a=1}^{2p }\sum_{b=1}^{q } \sum_{c=1}^{2r }g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + 2\sum_{a=1}^{2p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{q+c} \bigr)^{2}+2\sum_{a=1}^{2p } \sum_{b,c=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{ \ast } \bigr),\tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(45)

Now, employing the first and second parts of Lemma 1 and the first and second parts of Lemma 3 in the above expression

$$\begin{aligned} \Vert \sigma \Vert ^{2}& \geq \sum_{c=1}^{q }g \bigl(\sigma (\xi ,\xi ), \tilde{x}_{c} \bigr)^{2}+\sum _{a,b,c=1}^{q }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{b}^{ \ast } \bigr),\tilde{x}_{c} \bigr)^{2}+\sum _{c=1}^{2r }g \bigl(\sigma (\xi ,\xi ), \tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + \sum_{a,b=1}^{q }\sum _{c=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{b}^{ \ast } \bigr),\tilde{x}_{q+c} \bigr)^{2} +\sum _{a,b,c=1}^{2r }g \bigl(\sigma (\hat{x}_{a}, \hat{x}_{b}),\tilde{x}_{q+c} \bigr)^{2} \\ &\quad {}+2\sum_{a=1}^{q } \sum_{b,c=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{ \ast },\hat{x}_{b} \bigr),\tilde{x}_{q+c} \bigr)^{2} \\ &\quad{} + \sum_{a,b=1}^{2r }\sum _{c=1}^{q }g \bigl(\sigma (\hat{x}_{a}, \hat{x}_{b}),\tilde{x}_{c} \bigr)^{2}+2\sum_{a,c=1}^{q } \sum_{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{ \ast },\hat{x}_{b} \bigr),\tilde{x}_{c} \bigr)^{2} \\ &\quad{} + 2\sum_{a=1}^{2p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{q+c} \bigr)^{2}+2\sum_{a=1}^{2p } \sum_{b,c=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{ \ast } \bigr),\tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(46)

With the help of (4) and (6), we have $\sigma (\xi ,\xi )=0$. In order to reduce the complexity of the computation, we extract the final three terms from the equation above, and by using the orthonormal frame previously defined, we reach

$$\begin{aligned} \Vert \sigma \Vert ^{2}& \geq 2\sum _{a=1}^{p } \sum_{b,c=1}^{q }g \bigl( \sigma \bigl(x_{a},x_{b}^{\ast } \bigr),ne_{c}^{\ast } \bigr)^{2}+2\sum _{a=1}^{p } \sum_{b,c=1}^{q }g \bigl(\sigma \bigl( \varphi x_{a},x_{b}^{\ast } \bigr),ne_{c}^{ \ast } \bigr)^{2} \\ &\quad{} + 2\csc ^{2}\theta \Biggl( \sum_{a=1}^{p } \sum_{b,c=1}^{2r }g \bigl( \sigma (x_{a},\hat{x}_{b}),n\hat{x}_{c} \bigr)^{2}+g \bigl(\sigma (\varphi x_{a}, \hat{x}_{b}),n \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad{} + 2\sec ^{2}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a}, \hat{x}_{b}), nt\hat{x}_{c} \bigr)^{2}+g \bigl( \sigma (\varphi x_{a},\hat{x}_{b}),nt \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad{} + 2\sec ^{2}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a},t \hat{x}_{b}),n\hat{x}_{c} \bigr)^{2}+g \bigl( \sigma (\varphi x_{a},t\hat{x}_{b}),n \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad{} + 2\sec ^{4}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a},t \hat{x}_{b}),nt\hat{x}_{c} \bigr)^{2}+g \bigl( \sigma (\varphi x_{a},t\hat{x}_{b}),nt \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad{} + 2\sec ^{2}\theta \csc ^{2}\theta \Biggl( \sum _{b,c=1}^{2r }\sum _{a=1}^{q }g \bigl(\sigma \bigl(x_{a}^{\ast }, \hat{x}_{b} \bigr), nt\hat{x}_{c} \bigr)^{2} \\ &\quad {}+ \sum_{b,b=1}^{2r }\sum _{a=1}^{q }g \bigl(\sigma \bigl(x_{a}^{\ast },t \hat{x}_{b} \bigr),n\hat{x}_{c} \bigr)^{2} \Biggr) . \end{aligned}$$

(47)

Next, utilizing (31)–(35) and the second parts of Lemmas 1 and 2, we have

$$\begin{aligned} \Vert \sigma \Vert ^{2}& \geq 2\sum _{a=1}^{p } \sum_{b,c=1}^{q } \biggl( \biggl( \varphi \frac{x_{a}(f)}{f} \biggr)^{2}+ \biggl( \frac{x_{a}(f)}{f} \biggr)^{2} \biggr)g \bigl(x_{b}^{ \ast },x_{c}^{\ast } \bigr)^{2} \\ &\quad{} + 4\csc ^{2}\theta \sum_{a=1}^{p } \sum_{b,c=1}^{r } \biggl( \biggl(\varphi \frac{x_{a}(h)}{h} \biggr)^{2}+ \biggl(\frac{x_{a}(h)}{h} \biggr)^{2} \biggr)g(\hat{x}_{b},\hat{x}_{c})^{2} \\ &\quad{} + 4\cot ^{2}\theta \sum_{a=1}^{p } \sum_{b,c=1}^{r } \biggl( \biggl(\varphi \frac{x_{a}(h)}{h} \biggr)^{2}+ \biggl(\frac{x_{a}(h)}{h} \biggr)^{2} \biggr)g(\hat{x}_{b},\hat{x}_{c})^{2} \\ &\quad{} + 2\cot ^{2}\theta \sum_{b,c=1}^{r } \sum_{a=1}^{q } \frac{x_{a}^{\ast}(h)}{h}g( \hat{x}_{b},\hat{x}_{b})^{2} \\ &\quad {}+2\sec ^{2}\theta \csc ^{2}\theta \sum _{b, c=1}^{r }\sum_{a=1}^{q }g \bigl(\sigma \bigl(x_{a}^{\ast },t\hat{x}_{b} \bigr),n \hat{x}_{c} \bigr)^{2}. \end{aligned}$$

(48)

Leaving the last term and by adding and subtracting the same quantity in (48), and by the application of Proposition 3, we have

$$\begin{aligned} \Vert \sigma \Vert ^{2}& \geq 2\sum _{a=1}^{2p +1} \sum_{b, c=1}^{q } \biggl( \varphi \frac{x_{a}(f)}{f} \biggr)^{2}g \bigl(x_{b}^{\ast },x_{c}^{\ast } \bigr)^{2} \\ &\quad{} + 4\csc ^{2}\theta \sum_{a=1}^{2p +1} \sum_{b, c=1}^{r } \biggl( \frac{x_{a}(h)}{h} \biggr)^{2}g(\hat{x}_{b}, \hat{x}_{c})^{2} \\ &\quad{} + 4\cot ^{2}\theta \sum_{a=1}^{2p +1} \sum_{b, c=1}^{r } \biggl( \frac{x_{a}(h)}{h} \biggr)^{2}g(\hat{x}_{b}, \hat{x}_{c})^{2} \\ &\quad{} + 2\cot ^{2}\theta \sum_{b, c=1}^{r } \sum_{a=1}^{q } \biggl( \frac{x_{a}^{\ast}(f)}{f} \biggr)^{2}g(\hat{x}_{b}, \hat{x}_{b})^{2}. \end{aligned}$$

(49)

Using (21) in the above expression, we derive (42). Hence, the theorem is proved. □

5.1 Particular cases

If in ineq. (42) of Theorem 2:

• i. $r =0$ and the norm of $\frac{\nabla ^{T}(h)}{h}$ and $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP becomes a CR-warped product [34], and

  $$ \Vert \sigma \Vert ^{2}\geq 2q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}. $$

• ii. $q=0$ and the norm of $\frac{\nabla ^{T}(f)}{f}$ vanishes then SWP becomes WP $\Omega _{T}\times _{h}\Omega _{\theta }$ with

  $$ \Vert \sigma \Vert ^{2}\geq 4r \bigl(1+2\cot ^{2} \theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}. $$

  It is to be noted that the same result was obtained at [26] for Sasakian manifolds, but now for cosymplectic ones.

• iii. $p=0$ and the norm of $\frac{\nabla ^{T}(f)}{f}$ and $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP becomes pointwise pseudoslant WP $\Omega _{\bot }\times _{h}\Omega _{\theta }$ [2, Th. 4.1] and

  $$ \Vert \sigma \Vert ^{2}\geq 2r \cot ^{2}\theta \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2}. $$

• iv. $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP becomes a biwarped product $\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{\theta }$ [11] such that

  $$ \Vert \sigma \Vert ^{2}\geq 2q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}+4r \bigl(1+2\cot ^{2}\theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}. $$

• v. $\frac{\nabla ^{T}(f)}{f}$ vanishes, then SWP becomes CR-slant WP $(\Omega _{T}\times \Omega _{\perp })\times _{h}\Omega _{\theta }$ [1] satisfying

  $$ \Vert \sigma \Vert ^{2}\geq 4r \biggl( \bigl(\csc ^{2} \theta +\cot ^{2} \theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2} +\cot^{2} \theta \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2} \biggr). $$

Theorem 3

Consider a d-dimensional SWP submanifold $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ of cosymplectic manifold $\bar{\Omega}^{2m+1}$. If the SFF σ satisfies

$$ \Vert \sigma \Vert ^{2}=2q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}+4r \bigl(1+2\cot ^{2}\theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}+2r \cot ^{2}\theta \biggl\Vert \frac{\nabla ^{\perp }(h)}{h} \biggr\Vert ^{2}, $$

(50)

then, the claims that follow hold:

• (i) Ω is totally geodesic in $\bar{\Omega}^{2m+1}$;

• (ii) Ω is $\Omega _{T}$–$\Omega _{\perp}$ ($\Omega _{T}$–$\Omega _{\theta}$) mixed totally geodesic;

• (iii) $\Omega _{\perp }\times _{h}\Omega _{\theta }$ (resp. $\Omega _{\perp }$) is totally mixed geodesic (resp., umbilical) in $\bar{\Omega}^{2m+1}$;

• (iv) The $\mathcal{H}=-\nabla f$ for $\Omega _{\perp}$;

• (v) $\Omega _{\theta }$ is totally umbilical in cosymplectic manifold $\bar{\Omega}^{2m+1}$;

• (vi) The $\mathcal{H}=-\nabla f$ for $\Omega _{\theta}$.

The present theorem can be proved using the same steps as Theorem 4.2 of [19].

Proof

Suppose that σ satisfies (50), then by (44), we have

$$\begin{aligned} g \bigl(\sigma (T\Omega , T\Omega ), \nu \bigr)=0. \end{aligned}$$

(51)

Hence, straightforwardly from (51), Lemma 1, Lemma 2, and Proposition 2, we obtain $(1)$. Also from (46), we obtain that

$$\begin{aligned} g \bigl(\sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp}), n \mathfrak{D}_{ \theta} \bigr)=0, \qquad g \bigl(\sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp}), n \mathfrak{D}_{\perp} \bigr)=0. \end{aligned}$$

(52)

By the use of (46), we have

$$\begin{aligned} g \bigl(\sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\theta}), n \mathfrak{D}_{ \theta} \bigr)=0, \qquad g \bigl(\sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\theta}), n \mathfrak{D}_{\perp} \bigr)=0. \end{aligned}$$

(53)

From (51) and (53), we have

$$\begin{aligned} \sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\theta})=0. \end{aligned}$$

(54)

By the consequence (54), we accomplished the proof of the second part. By virtue of (51) and (52), we have

$$\begin{aligned} \bigl\Vert \sigma (\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp}) \bigr\Vert =0. \end{aligned}$$

(55)

Since $\Omega _{\perp}$ is a totally umbilical submanifold in $\Omega _{T}\times _{f}\Omega _{\perp}$ using this fact in (55), we conclude that $\Omega _{\perp}$ is totally umbilical in M̅. Similarly, we prove the fourth part. Let $\sigma ^{\perp}$ be the second fundamental form on $\Omega _{\perp}$, then by the utilization of Proposition 1, we have $g(\sigma ^{\perp}(Z_{1}, Z_{2}), X)=-g(\nabla _{X}Z_{1}, Z_{2})=-X( \ln f)g(Z_{1}, Z_{2})$. Now, assume σ and $\sigma ^{\theta}$ are the second fundamental tensors of M̅ and $\Omega _{\theta}$, respectively, then we have $\sigma (W_{1}, W_{2})=\sigma (W_{1}, W_{2})+\sigma ^{\theta}(W_{1}, W_{2})$. By the consequence of (49) and Proposition 2, we obtain $\sigma (W_{1}, W_{2})=\sigma ^{\theta}(W_{1}, W_{2})=g(W_{1}, W_{2})H^{ \theta}$, where $H^{\theta}$ is the mean curvature of $\Omega _{\theta}$. Using (1), we obtain $g(\sigma ^{\theta}(W_{1}, W_{2}), X)=g(\nabla _{W_{1}}W_{2}, X)=-X( \ln h)g(W_{1}, W_{2})$. Now, by the definition of gradient, we achieve the last part. This completes the proof. □

Our next result presents one of the ways to analyze Dirichlet problems. Several authors in different senses have analyzed the existence of solutions for Dirichlet problems [3, 11]. Here, we investigate the relation involving Dirichlet energy of functions and the second fundamental form for SWP. Jackson et al. in [17] stated this as: The Dirichlet energy of smooth function ψ and compact submanifold Ω over its volume element dV is defined by

$$ E(\psi )=\frac{1}{2} \int _{\Omega } \Vert \nabla \psi \Vert ^{2}\,dV . $$

(56)

As a result, inspired by the publications mentioned above, we derive an important finding:

Theorem 4

Let $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ be a d-dimensional connected, compact SWP submanifold of cosymplectic manifold $\bar{\Omega}^{2m+1}$. Then, the Dirichlet energy of f and h satisfies

$$\begin{aligned} &4q E(\ln f)+4r \cot ^{2}\theta E(\ln h)+ 4r \bigl( 2+3\cot ^{2} \theta \bigr) E^{\Omega _{T}}(\ln h) \\ &\quad \leq \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV -16r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E^{\Omega _{T}}(\ln h) \bigr)\,dV \\ &\quad \quad{} - 4r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E^{\Omega _{\perp }}(\ln h) \bigr)\,dV , \end{aligned}$$

(57)

where $E^{\Omega _{T}}(\ln h)$, $E^{\Omega _{\perp }}(\ln h)$ and $E(\ln f)$, $E(\ln h)$ are the Dirichlet energy of lnh on $\Omega _{T}$, $\Omega _{\perp }$, and lnf and lnh on Ω, respectively.

Proof

Integrating the relation (42) over Ω, we have

$$\begin{aligned} \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV &\geq 2q \int _{\Omega } \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}\,dV +4r \int _{\Omega } \bigl(1+2 \cot ^{2}\theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2} \,dV \\ &\quad{} + 2r \int _{\Omega }\cot ^{2}\theta \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2}\,dV . \end{aligned}$$

(58)

By using the property of integration in (58), we obtain

$$\begin{aligned} \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV & \geq 2q \int _{\Omega } \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}\,dV +4r \bigl(1+2\cot ^{2} \theta \bigr) \int _{\Omega } \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}\,dV \\ &\quad{} + 2r \cot ^{2}\theta \int _{\Omega } \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2}\,dV \\ &\quad {}-16r \int _{\Omega } \cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \biggl( \int _{\Omega } \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2} \biggr)\,dV \\ &\quad{} - 4r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \biggl( \int _{\Omega } \biggl\Vert \frac{\nabla ^{\perp}(h)}{h} \biggr\Vert ^{2} \biggr)\,dV . \end{aligned}$$

(59)

By utilization of $\nabla =\nabla ^{T}+\nabla ^{\perp }+\nabla ^{\theta }$ and (56) in the above equation, we obtain (57). □

5.2 Particular cases

In inequality (57) of Theorem 4 if:

• i. $r =0$ and the norm of $\frac{\nabla ^{T}(h)}{h}$, $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP is a CR-warped product with

  $$\begin{aligned} 4q E(\ln f)\leq \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV . \end{aligned}$$

• ii. $q =0$ and the norm of $\frac{\nabla ^{T}(f)}{f}$ vanishes, then SWP is WP $\Omega _{T}\times _{h}\Omega _{\theta }$ [4] with

  $$\begin{aligned} E(\ln h) \leq \frac{1}{4r ((1+2\cot ^{2}\theta )} \biggl( \int _{ \Omega } \Vert \sigma \Vert ^{2}\,dV -16r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E(\ln h) \bigr)\,dV \biggr). \end{aligned}$$

• iii. $p =0$ and the norm of $\frac{\nabla ^{T}(f)}{f}$, $\frac{\nabla ^{T}(h)}{h}$ vanishes, then SWP is pointwise pseudoslant WP $\Omega _{\bot }\times _{h}\Omega _{\theta }$ and

  $$\begin{aligned} 4r \cot ^{2}\theta E(\ln h)\leq \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV -4r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E(\ln h) \bigr)\,dV . \end{aligned}$$

• iv. the norm of $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP is a biwarped product $\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{\theta }$ satisfying

  $$\begin{aligned} &4q E(\ln f)+4r \bigl(1+2\cot ^{2}\theta \bigr)E(\ln h) \\ &\quad \leq \int _{\Omega} \Vert \sigma \Vert ^{2}\,dV -16r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E(\ln h) \bigr)\,dV . \end{aligned}$$

• v. the norm of $\frac{\nabla ^{T}(f)}{f}$ vanishes, then SWP is CR-slant WP $(\Omega _{T}\times \Omega _{\perp })\times _{h}\Omega _{\theta }$ such that

  $$\begin{aligned} &4r \cot ^{2}\theta E(\ln h)+ 4r \bigl( 2+3\cot ^{2} \theta \bigr) E^{ \Omega _{T}}(\ln h) \\ &\quad \leq \int _{\Omega } \Vert \sigma \Vert ^{2}\,dV -16r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E^{\Omega _{T}}(\ln h) \bigr)\,dV \\ &\quad \quad{} - 4r \int _{\Omega }\cos \theta \csc ^{3}\theta \frac{d\theta }{dV} \bigl(E^{\Omega _{\perp }}(\ln h) \bigr)\,dV . \end{aligned}$$

Remark 3

The Dirichlet energy in the special cases $(1)$, $(2)$, and $(4)$ can be easily reduced to Corollary 6.1, Corollary 6.2, and Theorem 6.1 of [11] after certain computations.

Motivated by Sahin [28, 29], we give the Lawson–Simons-type inequality [21] (possesses important applications in the theory of integral currents [15] for SWP in a cosymplectic manifold with constant holomorphic sectional curvature k (briefly: Cosymplectic space form denoted by $(\bar{\Omega}^{2m+1},k)$). We also extract exceptional cases for the same.

Theorem 5

Let $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ be a d-dimensional SWP of cosymplectic space form $(\bar{\Omega}^{2m+1}, k)$, then

$$\begin{aligned} &\sum_{b=1}^{2p }\sum _{a=1}^{2r } \bigl\Vert \sigma (x_{a}, \hat{x}_{b}) \bigr\Vert ^{2}+\sum _{a=1}^{2p }\sum _{b=1}^{q } \bigl\Vert \sigma \bigl(x_{a},x_{b}^{ \ast } \bigr) \bigr\Vert ^{2}+\sum_{a=1}^{q } \sum _{b=1}^{2r } \bigl\Vert \sigma \bigl(x_{a}^{ \ast },\hat{x}_{b} \bigr) \bigr\Vert ^{2} \\ &\quad{} - \sum_{a=1}^{q }\sum _{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{a}^{ \ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr)\geq 2r \biggl( \frac{\Delta ^{\perp }(h)}{h}-\frac{kq }{4} \biggr) , \end{aligned}$$

(60)

where $\Delta ^{\perp }(h)$ denotes the Laplacian of h on $\Omega _{\perp }$. Moreover, if the equality holds then $\Omega =\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{ \theta }$ is single WP of ($\bar{\Omega}^{2m+1}, k$) satisfying $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\perp })\perp \nu$ and $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\theta })\perp \nu$.

Proof

Substituting $U=V=x_{a}^{\ast }$ and $V=W=\hat{x}_{b}$ into (5), we have

$$ \bar{R} \bigl(x_{a}^{\ast },\hat{x}_{b},x_{a}^{\ast }, \hat{x}_{b} \bigr)=- \frac{kq r }{2}. $$

(61)

By virtue of (19) and (61), we have

$$\begin{aligned} &\sum_{a=1}^{q }\sum _{b=1}^{2r }g \bigl(R \bigl(x_{a}^{\ast }, \hat{x}_{b} \bigr) \hat{x}_{b},x_{a}^{\ast } \bigr)- \sum_{a=1}^{q }\sum _{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{ \ast },x_{a}^{\ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr) \\ &\quad{} + \sum_{a=1}^{q }\sum _{b=1}^{2r }\Vert \sigma (x_{a}^{\ast }, \hat{x}_{b}\Vert ^{2}=-\frac{kq r }{2}. \end{aligned}$$

(62)

Further, utilizing Proposition 1 in the above expression, we compute

$$ \sum_{a=1}^{q }\sum _{b=1}^{2r }\Vert \sigma (x_{a}^{\ast }, \hat{x}_{b} \Vert ^{2}-\sum _{a=1}^{q } \sum_{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{a}^{ \ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr)=2r \sum _{a=1}^{q } \frac{H^{\lambda }(x_{a}^{\ast },x_{a}^{\ast })}{h}- \frac{kq r }{2}. $$

(63)

By the definition of a Laplacian, the above expression reduces to the given form

$$ \sum_{a=1}^{q }\sum _{b=1}^{2r }\Vert \sigma (x_{a}^{\ast }, \hat{x}_{b} \Vert ^{2}-\sum _{a=1}^{q } \sum_{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{a}^{ \ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr)=2r \biggl( \frac{\Delta ^{\perp }(h)}{h}-\frac{kq r }{2} \biggr) . $$

(64)

Now, we compute the norm of $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\perp })$ and $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\theta })$. First, consider

$$\begin{aligned} &\sum_{a=1}^{2p }\sum _{b=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \sigma \bigl(x_{a},x_{b}^{\ast } \bigr) \bigr) \\ &\quad = \sum_{a=1}^{2p }\sum _{b=1}^{q } \sum _{c=1}^{2s}g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \tilde{x}_{c} \bigr)^{2} + \sum_{a=1}^{2p }\sum _{b,c=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \tilde{x}_{c} \bigr)^{2} \\ &\quad \quad{}+\sum _{a=1}^{2p }\sum_{b=1}^{q } \sum_{c=1}^{2s}g \bigl(\sigma \bigl(x_{a},x_{b}^{ \ast } \bigr), \tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(65)

In light of Lemmas 1 and 3, we have

$$\begin{aligned} &\sum_{a=1}^{2p }\sum _{b=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \sigma \bigl(x_{a},x_{b}^{\ast } \bigr) \bigr) \\ &\quad = \sum_{a=1}^{p }\sum _{b,c=1}^{q } \biggl( \biggl( \varphi \frac{x_{a}(f)}{f} \biggr)^{2} + \biggl(\frac{x_{a}(f)}{f} \biggr)^{2} \biggr)g \bigl(x_{b}^{\ast },x_{c}^{\ast } \bigr)^{2}+\sum_{a=1}^{2p } \sum_{b=1}^{q }\sum _{c=1}^{2s}g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

This implies the following relation

$$ \sum_{a=1}^{2p }\sum _{b=1}^{q }g \bigl(\sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \sigma \bigl(x_{a},x_{b}^{\ast } \bigr) \bigr)=q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}+\sum_{a=1}^{2p }\sum _{b=1}^{q }\sum _{c=1}^{2s}g \bigl( \sigma \bigl(x_{a},x_{b}^{\ast } \bigr), \tilde{x}_{c} \bigr)^{2}. $$

(66)

Next, consider $g(\sigma (x_{a},\hat{x}_{b}),\sigma (x_{a},\hat{x}_{b}))$ and using the adopt frame, we obtain

$$\begin{aligned} & \sum_{a=1}^{2p }\sum _{b=1}^{2s}g \bigl(\sigma (x_{a}, \hat{x}_{b}), \sigma (x_{a},\hat{x}_{b}) \bigr) \\ &\quad =\csc ^{2}\theta \Biggl( \sum_{a=1}^{p } \sum_{b,c=1}^{2r }g \bigl( \sigma (x_{a},\hat{x}_{b}),n\hat{x}_{c} \bigr)^{2}+g \bigl(\sigma (\varphi x_{a}, \hat{x}_{b}),n \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad \quad{} + \sec ^{2}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum_{b,c=1}^{2r }g \bigl(\sigma (x_{a},\hat{x}_{b}),nt \hat{x}_{c} \bigr)^{2}+g \bigl(\sigma (\varphi x_{a},\hat{x}_{b}),nt\hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad \quad{} + \sec ^{2}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum_{b,c=1}^{2r }g \bigl(\sigma (x_{a},t\hat{x}_{b}),n \hat{x}_{c} \bigr)^{2}+g \bigl(\sigma (\varphi x_{a},t\hat{x}_{b}),n\hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad \quad{} + \sec ^{4}\theta \csc ^{2}\theta \Biggl( \sum _{a=1}^{p }\sum _{b,c=1}^{2r }g \bigl(\sigma (x_{a},t \hat{x}_{b}),nt\hat{x}_{c} \bigr)^{2}+g \bigl( \sigma (\varphi x_{a},t\hat{x}_{b}),nt \hat{x}_{c} \bigr)^{2} \Biggr) \\ &\quad \quad{} + \sum_{a=1}^{2p }\sum _{b=1}^{2r }\sum_{c=1}^{2s}g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{c} \bigr)^{2}+\sum _{a=1}^{2p } \sum_{b=1}^{2r } \sum_{c=1}^{q }g \bigl(\sigma (x_{a}, \hat{x}_{b}),n\tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(67)

Utilizing Lemmas 1 and 2 in (67) and also utilizing (30)–(35) in (67), we receive that

$$\begin{aligned} &\sum_{a=1}^{2p }\sum _{b=1}^{2r }g \bigl(\sigma (x_{a}, \hat{x}_{b}), \sigma (x_{a},\hat{x}_{b}) \bigr) \\ &\quad = 2\csc ^{2}\theta \sum _{a=1}^{p }\sum_{b,c=1}^{r } \biggl( \biggl( \varphi \frac{x_{a}(h)}{h} \biggr)^{2}+ \biggl( \frac{x_{a}(h)}{h} \biggr)^{2} \biggr)g(\hat{x}_{b}, \hat{x}_{c})^{2} \\ &\quad \quad{} + 2\cot ^{2}\theta \sum_{a=1}^{p } \sum_{b,c=1}^{r } \biggl( \biggl(\varphi \frac{x_{a}(h)}{h} \biggr)^{2}+ \biggl(\frac{x_{a}(h)}{h} \biggr)^{2} \biggr)g(\hat{x}_{b},\hat{x}_{c})^{2} \\ &\quad \quad{} + \sum_{a=1}^{2p }\sum _{b=1}^{2r }\sum_{c=1}^{2s}g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(68)

By the definition of gradient, we concede that

$$\begin{aligned} &\sum_{b=1}^{2p }\sum _{a=1}^{2r }g \bigl(\sigma (x_{a}, \hat{x}_{b}), \sigma (x_{a},\hat{x}_{b}) \bigr) \\ &\quad = 4r \bigl(1+2\cot ^{2} \bigr)\theta \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2} + \sum_{a=1}^{2p }\sum _{b=1}^{2r }\sum_{c=1}^{2s}g \bigl(\sigma (x_{a}, \hat{x}_{b}), \tilde{x}_{c} \bigr)^{2}. \end{aligned}$$

(69)

From (64), (66), and (69), we have (60). If equality holds then we have

$$ 4r \bigl(1+2\cot ^{2}\theta \bigr) \biggl\Vert \frac{\nabla ^{T}(h)}{h} \biggr\Vert ^{2}+q \biggl\Vert \frac{\nabla ^{T}(f)}{f} \biggr\Vert ^{2}=0. $$

The above equation shows that the functions f and h are constant on $\Omega _{T}$. Therefore, Ω reduces to $\Omega _{T}\times \Omega _{\perp }\times _{h}$, $\Omega _{\theta }$ is a single warped product. By the direct consequence of (66) and (69), we conclude that $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\perp })\perp \nu$ and $\sigma (\mathfrak{D}_{T},\mathfrak{D}_{\theta })\perp \nu$. This completes the proof. □

5.3 Particular cases

In ineq. (42) of Theorem 5 if:

• i. $r =0$ and the norm of $\frac{\nabla ^{T}(h)}{h}$, $\frac{\nabla ^{\perp}(h)}{h}$ vanishes then SWP is a CR-warped product with

  $$ \sum_{a=1}^{2p }\sum _{b=1}^{q } \bigl\Vert \sigma \bigl(x_{a},x_{b}^{\ast } \bigr) \bigr\Vert ^{2}\geq 0. $$

• ii. $q =0$ and $\frac{\nabla ^{T}(f)}{f}$ vanishes, then SWP is WP $\Omega _{T}\times _{h}\Omega _{\theta }$ satisfying

  $$ \sum_{b=1}^{2p }\sum _{a=1}^{2r } \bigl\Vert \sigma (x_{a}, \hat{x}_{b}) \bigr\Vert ^{2}\geq 0. $$

• iii. $p =0$ and the norm of $\frac{\nabla ^{T}(f)}{f}$, $\frac{\nabla ^{T}(h)}{h}$ vanishes, then SWP is pointwise pseudoslant WP $\Omega _{\bot }\times _{h}\Omega _{\theta }$ with

  $$ \sum_{a=1}^{q }\sum _{b=1}^{2r }\bigl\Vert \sigma\bigl(x_{a}^{\ast }, \hat{x}_{b}\bigr) \bigr\Vert ^{2}-\sum _{a=1}^{q } \sum_{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{a}^{ \ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr)\geq 2s \frac{\Delta ^{\perp }(h)}{h}. $$

• iv. the norm of $\frac{\nabla ^{\perp}(h)}{h}$ vanishes, then SWP is a biwarped product $\Omega _{T}\times _{f}\Omega _{\perp }\times _{h}\Omega _{\theta }$ with

  $$\begin{aligned} &\sum_{b=1}^{2p }\sum _{a=1}^{2r } \bigl\Vert \sigma (x_{a}, \hat{x}_{b}) \bigr\Vert ^{2}+\sum _{a=1}^{2p }\sum _{b=1}^{q } \bigl\Vert \sigma \bigl(x_{a},x_{b}^{ \ast } \bigr) \bigr\Vert ^{2}+\sum_{a=1}^{q } \sum _{b=1}^{2r }\Vert \sigma (x_{a}^{ \ast },\hat{x}_{b}\Vert ^{2} \\ &\quad{} - \sum_{a=1}^{q }\sum _{b=1}^{2r }g \bigl(\sigma \bigl(x_{a}^{\ast },x_{a}^{ \ast } \bigr),\sigma (\hat{x}_{b},\hat{x}_{b}) \bigr)\geq - \frac{cq r }{2}. \end{aligned}$$

  (70)