$\phi$-pluriharmonicity in quasi bi-slant conformal $\xi^\perp$-submersions: a comprehensive study

Abbreviations: RS: Riemannian submersion; RM: Riemannian Manifold; ACM manifold: Almost contact metric manifold; $\mathcal{QBSC}~\xi^{\perp}$-submersion: Quasi bi-slant conformal $\xi^{\perp}$-submersion; G: gradient

1.   Introduction

Immersions and submersions play crucial roles in differential geometry, with slant submersions being a particularly intriguing subject in the fields of differential, complex and contact geometry. The study of Riemannian submersions between Riemannian manifolds was first explored by O'Neill ^[23] and Gray ^[14], independently, and subsequently led to investigations of Riemannian submersions between almost Hermitian manifolds, known as almost Hermitian submersions, by Watson in 1976 ^[34]. Riemannian submersions have many applications in mathematics and physics, especially in Yang-Mills theory ^[8,35] and in Kaluza-Klein theory ^[18,21].

Semi-invariant submersions, a generalization of holomorphic submersions and anti-invariant submersions, were introduced by Sahin in 2013 ^[30]. In 2016, Tatsan, Sahin, and Yanan studied hemi-slant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds, and presented several decomposition theorems for them ^[33]. R. Prasad et al. further examined quasi bi-slant submersions from almost contact metric manifolds onto Riemannian manifolds ^[25], as well as from Kenmotsu manifolds ^[26], which represents a step forward in the study of Riemannian submersions.

Since then, many authors have explored different types of Riemannian submersions, including anti-invariant submersions ^[4,29], slant submersions ^[11,31], semi-slant submersions ^[17,24] and hemi-slant submersions ^[1,20], from both almost Hermitian manifolds and almost contact metric manifolds. These studies have greatly expanded our understanding of the geometrical structures of Riemannian manifolds.

The concept of almost contact Riemannian submersions from almost contact manifold was introduced by Chinea in ^[9]. Chinea also examined the fibre space, base space and total space using a differential geometric perspective. To generalize Riemannian submersions, Gundmundsson and Wood ^[15,16] presented horizontally conformal submersion, defined as: Let $(M_1, g_1)$ and $(M_2, g_2)$ be two Riemannian manifolds of dimension $m_1$ and $m_2$, respectively. A smooth map $\Psi : (M_1, g_1) \rightarrow (M_2, g_2)$ is called a horizontally conformal submersion, if there is a positive function $\lambda$ such that

for all $X_1, X_2 \in \Gamma(ker \Psi_*)^\perp$. Thus, Riemannian submersion is a particular horizontally conformal submersion with $\lambda = 1$. Later on, Fuglede ^[13] and Ishihara ^[19] separately studied horizontally conformal submersions. Additionally, various other kind of submersions, such as conformal slant submersions ^[3], conformal anti-invariant submersions ^[6], conformal semi-slant submersions ^[2], conformal semi-invariant submersions ^[5] and conformal anti-invariant submersions ^[27] have been studied by Akyol and Sahin and R. Prasad et al. ^[27]. Furthermore, Shuaib and Fatima recently explored conformal hemi-slant Riemannian submersions from almost product manifolds onto Riemannian manifolds ^[32].

In this paper, we study quasi bi-slant conformal $\xi^{\perp}$-submersions from Sasakian manifold onto a Riemannian manifold considering the Reeb vector field $\xi$ horizontal. This paper is divided into six sections. Section 2 contains definitions of almost contact metric manifolds and, in particular, Sasakian manifolds. In section 3, fundamental results for quasi bi-slant conformal submersion are investigated, which are necessary for our main results. The conditions of integrability and total geodesicness of distributions are explored in Section 4. Section 5 provides some condition under which a Riemannian submersion becomes totally geodesic as well as some decomposition theorems for quasi bi-slant conformal submersion are obtained. The last section discusses $\phi$-pluriharmonicity of quasi bi-slant conformal $\xi^{\perp}$-submersions.

2.   Preliminaries

Let $M$ be a $(2n+1)$-dimensional almost contact manifold with almost contact structures $(\phi, \xi, \eta)$, where a $(1, 1)$ tensor field $\phi$, a vector field $\xi$ and a $1$-form $\eta$ satisfying

where $I$ is the identity tensor. The almost contact structure is said to be normal if $N + d\eta \otimes \xi = 0$, where $N$ is the Nijenhuis tensor of $\phi$. Suppose that a Riemannian metric tensor ${g}$ is given in $M$ and satisfies the condition

Then $(\phi, \xi, \eta, {g})$-structure is called an almost contact metric structure. Define a tensor field $\Phi$ of type $(0, 2)$ by $\Phi({\widehat{X}}, {\widehat{Y}}) = {g}(\phi {\widehat{X}}, {\widehat{Y}})$. If $d\eta = \Phi$, then an almost contact metric structure is said to be normal contact metric structure. Let $\Phi$ be the fundamental 2-form on $M$, i.e, $\Phi({\widehat{U}}, {\widehat{V}}) = {g}({\widehat{U}}, \phi {\widehat{V}})$. If $\Phi = d\eta$, $M$ is said to be a contact manifold. A normal contact metric structure is called a Sasakian structure, which satisfies

where $\nabla$ is the Levi-Civita connection of $g$. From above formula, we have for Sasakian manifold

The covariant derivative of $\phi$ is defined by

for any vector fields ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma(TM)$. Now, we provide a definition for conformal submersion and discuss some useful results that help us to achieve our main results.

Definition 2.1. Let ${\Psi}$ be a Riemannian submersion (RS) from an ACM manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a Riemannian manifold (RM) $({\bar Q_2}, g_2)$. Then ${\Psi}$ is called a horizontally conformal submersion, if there is a positive function $\lambda$ such that

for any ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma(ker {\Psi_*})^\perp$. It is obvious that every RS is a particularly horizontally conformal submersion with $\lambda = 1$.

Let ${\Psi} : ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a RS. A vector field $\widehat{X}$ on ${\bar Q_1}$ is called a basic vector field if $\widehat{X} \in \Gamma (\ker {\Psi}_*)^\bot$ and ${\Psi}$-related with a vector field $\widehat{X}$ on ${\bar Q_2}$ i.e ${\Psi}_*({\widehat{X}}(q)) = {\widehat{X}}{\Psi}(q)$ for $q \in {\bar Q_1}$.

The formulas provide the two $(1, 2)$ tensor fields $\mathcal{T}$ and $\mathcal{A}$ by O'Neill are

for any $E_1, F_1 \in \Gamma(T{\bar Q_1})$ and $\nabla$ is Levi-Civita connection of $g_1$. Note that a RS ${\Psi} : ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ has totally geodesic fibers if and only if $\mathcal{T}$ vanishes identically. From Eqs (2.7) and (2.8), we can deduce

for any vector fields ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}}_1, {\widehat{Y}}_1 \in \Gamma(ker {\Psi}_*)^\perp$ ^[12].

It is easily seen that $\mathcal{T}$ and $\mathcal{A}$ are skew-symmetric, that is

for any vector fields $E_1, F_1 \in \Gamma(T_p {\bar Q_1})$.

Definition 2.2. A horizontally conformally submersion $\Psi: {\bar Q_1} \rightarrow {\bar Q_2}$ is called horizontally homothetic if the gradient (G) of its dilation $\lambda$ is vertical, i.e.,

at $p \in {TM}_1$, where $H$ is the complement orthogonal distribution to $\nu = ker\; \Psi_*$ in $\Gamma(T_pM)$.

The second fundamental form of smooth map $\Psi$ is given by the formula

and the map be totally geodesic if $(\nabla \Psi_*)({{\widehat{U}}_1}, {{\widehat{V}}_1}) = 0$ for all ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma(T_pM)$ where $\nabla$ and $\nabla^{\Psi}$ are Levi-Civita and pullback connections.

Lemma 2.1. Let $\Psi : {\bar Q_1} \rightarrow {\bar Q_2}$ be a horizontal conformal submersion. Then, we have

(i) $(\nabla {\Psi}_*)({\widehat{X}}_1, {\widehat{Y}}_1) = {\widehat{X}}_1(ln \lambda){\Psi}_*({\widehat{Y}}_1) + {\widehat{Y}}_1 (ln \lambda){\Psi}_*({\widehat{X}}_1)- g_1({\widehat{X}}_1, {\widehat{Y}}_1){\Psi}_*(grad\; ln \lambda),$

(ii) $(\nabla {\Psi}_*)({\widehat{U}}_1, {\widehat{V}}_1) = -{\Psi}_*(\mathcal{T}_{{\widehat{U}}_1} {\widehat{V}}_1),$

(iii) $(\nabla {\Psi}_*)({\widehat{X}}_1, {\widehat{U}}_1) = -{\Psi}_*(\nabla_{{\widehat{X}}_1} {\widehat{U}}_1) = -{\Psi}_*(\mathcal{A}_{{\widehat{X}}_1}{\widehat{U}}_1),$

for any horizontal vector fields ${\widehat{X}}_1, {\widehat{Y}}_1$ and vertical vector fields ${\widehat{U}}_1, {\widehat{V}}_1$ ^[7].

Definition 2.3. Suppose $\mathfrak{D}$ is a $k$-dimensional smooth distribution on $M$. Then An immersed submanifold $i : N \hookrightarrow M$ is called an integral manifold for $\mathfrak{D}$ if for every $x \in N$, the image of $d_i N : T_pN \rightarrow T_pM$ is $\mathfrak{D_p}$. We say the distribution $\mathfrak{D_p}$ is integrable if through each point of $M$ there exists an integral manifold of $\mathfrak{D}$.

Further, A distribution $\mathfrak{D}$ is involutive if it satisfies the Frobenius condition such that if $X, Y \in \Gamma(TM)$ belongs to $\mathfrak{D}$, so $[X, Y] \in \mathfrak{D}$. Frobenius theorem state that an involutive distribution is integrable.

Definition 2.4. Let $M$ be $n$-dimensional smooth manifold. A foliation $\mathfrak{F}$ of $M$ is a decomposition of $M$ into a union of disjoint connected submanifolds $M = \cup_{L \in \mathfrak{F}} L$, called the leaves of the foliation, such that for each $m \in M$, there is a neighborhood $U$ of $M$ and a smooth submersion $f_U : U \rightarrow R^k$ with $f_U^{-1}(x)$ a leaf of $\mathfrak{F}|_U$ the restriction of the foliation to $U$, for each $x \in R^k$.

Definition 2.5. Let $M$ be a Riemannian manifold, and let $\mathfrak{F}$ be a foliation on $M$. $\mathfrak{F}$ is totally geodesic if each leaf $L$ is a totally geodesic submanifold of $M$; that is, any geodesic tangent to $L$ at some point must lie within $L$.

3.   Quasi bi-slant conformal $\xi^{\perp}$-submersions

Definition 3.1. Let $({\bar Q_1}, \phi, \xi, \eta, g_1)$ be a ACM manifold and $({\bar Q_2}, g_2)$ a Riemannian manifold. A RS $\Psi : {\bar Q_1} \rightarrow {\bar Q_2}$ where $\xi\in \Gamma(ker \Psi_*)^\bot$ is called quasi bi-slant conformal $\xi^{\perp}$-submersion ($\mathcal{QBSC}\; \xi^{\perp}$-submersion) if there exists three mutually orthogonal distributions ${\mathfrak{D}}, \; {\mathfrak{D_{\theta_1}}}$ and ${\mathfrak{D_{\theta_2}}}$ such that

(i) $ker {\Psi}_* = {\mathfrak{D}}\oplus {\mathfrak{D_{\theta_1}}}\oplus {\mathfrak{D_{\theta_2}}}$,

(ii) ${\mathfrak{D}}$ is invariant. i.e., $\phi {\mathfrak{D}} = {\mathfrak{D}}$,

(iii) $\phi {\mathfrak{D_{\theta_1}}}\perp {\mathfrak{D_{\theta_2}}}$ and $\phi {\mathfrak{D_{\theta_2}}}\perp {\mathfrak{D_{\theta_1}}}$,

(iv) for any non-zero vector field ${\widehat{V}}_1 \in ({\mathfrak{D_{\theta_1}}})_p, p \in {\bar Q_1}$ the angle $\theta_1$ between $({\mathfrak{D_{\theta_1}}})_p$ and $\phi {\widehat{V}}_1$ is constant and independent of the choice of the point $p$ and ${\widehat{V}}_1 \in ({\mathfrak{D_{\theta_1}}})_p$,

(v) for any non-zero vector field ${\widehat{V}}_1 \in ({\mathfrak{D_{\theta_2}}})_q, q \in {\bar Q_1}$ the angle $\theta_2$ between $({\mathfrak{D_{\theta_2}}})_q$ and $\phi {\widehat{V}}_1$ is constant and independent of the choice of the point $q$ and ${\widehat{V}}_1 \in ({\mathfrak{D_{\theta_2}}})_q$,

where $\theta_1$ and $\theta_2$ are called the slant angles of submersion.

If we suppose $m_1$, $m_2$ and $m_3$ are the dimensions of ${\mathfrak{D}}$, ${\mathfrak{D_{\theta_1}}}$ and ${\mathfrak{D_{\theta_2}}}$ respectively, then we have the following:

$({\rm{i}})$ If $m_1\neq 0$, $m_2 = 0$ and $m_3 = 0$, then ${\Psi}$ is an invariant submersion.

$({\rm{ii}})$ If $m_1\neq 0$, $m_2\neq 0, 0 < \theta_1 < \frac{\pi}{2}$ and $m_3 = 0$, then $\Psi$ is a proper semi-slant submersion.

$({\rm{iii}})$ If $m_1 = 0$, $m_2 = 0$ and $m_3 \neq 0, 0 < \frac{\pi}{2}$, then ${\Psi}$ is a slant submersion with slant angle $\theta_2$.

$({\rm{iv}})$ If $m_1 = 0, m_2 \neq 0, 0 < \theta_1 < \frac{\pi}{2}$ and $m_3\neq 0, \theta_2 = \frac{\pi}{2}$, then ${\Psi}$ proper hemi-slant submersion.

$({\rm{v}})$ If $m_1 = 0, m_2\neq 0, 0 < \theta_1 < \frac{\pi}{2}$ and $m_3\neq 0, 0 < , \theta_2 < \frac{\pi}{2}$, then ${\Psi}$ is proper bi-slant submersion with slant angles $\theta_1$ and $\theta_2$.

$({\rm{vi}})$ If $m_1\neq 0, m_2\neq 0, 0 < \theta_1 < \frac{\pi}{2}$ and $m_3\neq 0, 0 < \theta_2 < \frac{\pi}{2}$, then ${\Psi}$ is proper quasi bi-slant submersion with slant angles $\theta_1$ and $\theta_2$.

We construct an example of $\mathcal{QBSC}\; {\xi}^\perp$-submersions from Sasakian manifold. Let $\left(R^{2 k+1}, g_{2 k+1}, \phi, \xi, \eta\right)$ denote the manifold with its usual Sasakian structure given by

where $x_1, \ldots, x_k, y_1 \ldots., y_k, z$ are the cartesian coordinates. Its Riemannian metric $g$ is defined as $g = \eta \otimes \eta+\frac{1}{4}\left(\sum_{i = 1}^k\left[d x_i \otimes d x_i+d y_i \otimes d y_i\right]\right)$, where $\eta$ is its usual contact form and given as $\eta$ = $\frac{1}{2}\left(d z-\sum_{i = 1}^k Y_i d x_i\right)$. The characteristic vector field $\xi$ is given by $2 \dfrac{\partial}{\partial z}$. The vector fields $E_i = 2 \dfrac{\partial}{\partial y_i}, E_{k+i} = 2\left(\dfrac{\partial}{\partial x_i}+y_i \dfrac{\partial}{\partial z}\right)$ and $\xi$ form a $\phi$-basis for the Sasakian structure and $\left(R^{2 k+1}, \phi, \xi, \eta, g\right)$ is a Sasakian manifold. Throughout this section we will use the notation.

Example. Let $\left(R^{15}, \phi, \xi, \eta, g\right)$ be Sasakian manifold. Let $\Psi: R^{15} \rightarrow R^7$ be map defined by $\Psi\left(x_1, \ldots, x_7, y_1, \ldots, y_7, z\right) = \sqrt{\pi}\left(\cos \theta_1 x_3+\right.$ $\left.\sin \theta_1 x_4, x_5, x_7, y_3, \cos \theta_2 y_5+\sin \theta_2 y_6, y_7, z\right)$, which is a quasi bi-slant conformal submersion with dilation $\lambda = \sqrt{\pi}$ such that

$\operatorname{ker} \Psi_* = \mathfrak{D} \oplus \mathfrak{D_{\theta_1}} \oplus \mathfrak{D_{\theta_2}}$, where

and

where $\mathfrak{\theta_1}$ and $\mathfrak{\theta_2}$ are the slant angles of the submersion for the distribution $\mathfrak{D_{\theta_1}}$ and $\mathfrak{D_{\theta_2}}$, respectively.

Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from an ACM manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$. Then, for any $U \in (ker {\Psi}_*)$, we have

where ${\mathfrak{P_1}}, {\mathfrak{P_2}}$ and ${\mathfrak{P_3}}$ are the projections morphism onto ${\mathfrak{D}}, {\mathfrak{D_{\theta_1}}},$ and ${\mathfrak{D_{\theta_2}}}$. Now, for any ${\widehat{U}} \in (ker {\Psi}_*)$, we have

where $\alpha {\widehat{U}} \in \Gamma(ker {\Psi}_*)$ and $\beta {\widehat{U}} \in \Gamma(ker {\Psi}_*)^\perp$. From Eqs (3.1) and (3.2), we have

Since $\phi {\mathfrak{D}} = {\mathfrak{D}}$ and $\beta ({\mathfrak{P_1}}{\widehat{U}}) = 0$, we have

Hence we have the decomposition as

From Eq (3.3), we have the following decomposition:

where $\mu$ is the orthogonal complement to $\beta {\mathfrak{D_{\theta_1}}}\oplus \beta {\mathfrak{D_{\theta_2}}}$ in $({ker {\Psi}_*})^\perp$ such that $\mu = (\phi\mu)\oplus < \xi >$ and $\mu$ is invariant with respect to $\phi$. Now, for any ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$, we have

where ${\mathbb{C}}{\widehat{X}} \in \Gamma(ker {\Psi}_*)$ and ${\mathbb{B}}{\widehat{X}}\in \Gamma(ker \Psi_*)^\perp$.

Lemma 3.1. Let $({\bar Q_1}, \phi, \xi, \eta, g_1)$ be an ACM manifold and $({\bar Q_2}, g_2)$ be a RM. If ${\Psi} : {\bar Q_1} \rightarrow {\bar Q_2}$ is a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, then we have

for ${\widehat{U}} \in \Gamma({ker {\Psi}_*})$ and ${\widehat{X}} \in \Gamma{(ker {\Psi}_*})^\perp$.

Proof. On using Eqs (2.1), (3.2) and (3.5), we get the desired results. □

Since $\Psi: {\bar Q_1} \rightarrow {\bar Q_2}$ is a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, Then let us provide some helpful findings that will be utilise throughout the paper.

Lemma 3.2. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from an ACM manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$, then we have

(i) $\alpha^2 {\widehat{U}} = -\operatorname{cos^2 \theta_1} {\widehat{U}}$,

(ii) $g_1(\alpha {\widehat{U}}, \alpha {\widehat{V}}) = \operatorname{\cos^2 \theta_1} g_1({\widehat{U}}, {\widehat{V}})$,

(iii) $g(\beta {\widehat{U}}, \beta {\widehat{V}}) = \operatorname{\sin^2 \theta_1}g_1({\widehat{U}}, {\widehat{V}}),$

for any vector fields ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D_{\theta_1}}})$.

Lemma 3.3. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from an ACM manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$, then we have

(i) $\alpha^2 {\widehat{Z}} = -\operatorname{cos^2 \theta_2} {\widehat{Z}}$,

(ii) $g_1(\alpha {\widehat{Z}}, \alpha {\widehat{W}}) = \operatorname{\cos^2 \theta_2} g_1({\widehat{Z}}, {\widehat{W}})$,

(iii) $g_1(\beta {\widehat{Z}}, \beta {\widehat{W}}) = \operatorname{\sin^2 \theta_2}g_1({\widehat{Z}}, {\widehat{W}}),$

for any vector fields ${\widehat{Z}}, {\widehat{W}} \in \Gamma({\mathfrak{D_{\theta_2}}})$.

Proof. The proof of above Lemmas is similar to the proof of the Theorem 2.2 of ^[10]. □

Let $(\bar Q_2, g_2)$ be a Riemannian manifold and $(\bar Q_1, \phi, \xi, \eta, g_1)$ be a Sasakian manifold. We now consider how the Sasakian structure on $\bar Q_1$ affects the tensor fields $\mathcal{T}$ and $\mathcal{A}$ of a $\mathcal{QBSC}{\xi}^\perp$-submersion $\Psi: (\bar Q_1, \phi, \xi, \eta, g_1) \rightarrow (\bar Q_2, g_2)$.

Lemma 3.4. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$, then we have

for any vector fields ${\widehat{U}}, {\widehat{V}} \in \Gamma(\operatorname{ker }{\Psi}_*)$ and ${\widehat{X}}, {\widehat{Y}} \in \Gamma(\operatorname{ker} {\Psi}_*)^\perp$.

Proof. From (2.5), (2.12) and (3.5), we obtained the conditions (3.6) and (3.7). Again using Eqs (3.2), (3.5), (2.9)–(2.12) and (2.5), finish he result. □

Now we will go through some fundamental results that can be used to investigate the geometry of $\mathcal{QBSC}\; \xi^{\perp}$-submersion $\Psi : {\bar Q_1} \rightarrow {\bar Q_2}$. For this, define the following:

for any vector fields ${\widehat{U}}, {\widehat{V}} \in \Gamma(\operatorname{ker } {\Psi}_*)$ and ${\widehat{X}}, {\widehat{Y}} \in \Gamma(\operatorname{ker }{\Psi}_*)^\perp$.

Lemma 3.5. Let $({\bar Q_1}, \phi, \xi, \eta, g_1)$ be a Sasakian manifold and $({\bar Q_2}, g_2)$ be a RM. If ${\Psi} : {\bar Q_1} \rightarrow {\bar Q_2}$ is a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, then we have

for all vector fields ${\widehat{U}}, {\widehat{V}} \in \Gamma({ker {\Psi}_*})$ and ${\widehat{X}}, {\widehat{Y}} \in \Gamma({ker {\Psi}_*})^\perp$.

Proof. On using Eqs (2.5), (2.9)–(2.12) and (3.14)–(3.17), we get the desired result. □

If $\alpha$ and $\beta$, the tensor fields, are parallel with respect to the connection $\nabla$ of ${\bar Q_1}$ then, we have

for any vector fields ${\widehat{U}}, {\widehat{V}} \in\Gamma(T{\bar Q_1})$.

4.   Integrability and totally geodesicness of distributions

Since $\Psi: {\bar Q_1} \rightarrow {\bar Q_2}$ is a $\mathcal{QBSC}$ ${\xi}^\perp$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ represents a Kenmotsu manifold and $({\bar Q_2}, g_2)$ a Riemannian manifold. The definition of $\mathcal{QBSC}$ ${\xi}^\perp$-submersion ensures the existence of three mutually orthogonal distributions, which include an invariant distribution $\mathfrak{D}$, a pair of slant distributions $\mathfrak{D^{\theta_1}}$ and $\mathfrak{D^{\theta_2}}$. We start the discussion on the integrability of distributions by determining the integrability of the slant distribution in the manner described below:

Theorem 4.1. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$. Then slant distribution ${\mathfrak{D_{\theta_1}}}$ is integrable if and only if

for any ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma({\mathfrak{D_{\theta_1}}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_2}}})$.

Proof. For all ${\widehat{U}}_1, {\widehat{V}}_1 \in \Gamma({\mathfrak{D_{\theta_1}}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_2}}})$ with using Eqs (2.2), (2.5), (2.14) and (3.2), we get

By using Eqs (2.5), (2.14) and (3.2), we have

Taking account the fact of Lemma 3.2 with Eq (2.10), we get

On using Eq (2.6), formula (2.15) with Lemma 2.1, we finally get

□

In a same manner, we can obained the condition of integrability for ${\mathfrak{D_{\theta_2}}}$ as follows:

Theorem 4.2. Let ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ a Sasakian manifold and $({\bar Q_2}, g_2)$ a RM. Then slant distribution ${\mathfrak{D_{\theta_2}}}$ is integrable if and only if

for any ${\widehat{U}}_2, {\widehat{V}}_2 \in \Gamma({\mathfrak{D_{\theta_2}}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_1}}})$.

Proof. On using Eqs (2.2), (2.5), (2.14) and (3.2), we have

for any ${\widehat{U}}_2, {\widehat{V}}_2 \in \Gamma({\mathfrak{D_{\theta_2}}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_1}}})$. From Eq (2.10) and Lemma 3.3, we get

Since ${\Psi}$ is $\mathcal{QBSC}\; \xi^{\perp}$-submersion, using conformality condition with Eqs (2.6) and (2.15), we finally get

This completes the proof of the theorem. □

Since, the invariant distribution is mutually orthogonal to the slant distributions in accordance with the concept of $\mathcal{QBSC}\; \xi^{\perp}$-submersion, this led us to investigate the necessary and sufficient condition for the invariant distribution to be integrable.

Theorem 4.3. Let ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ a Sasakian manifold and $({\bar Q_2}, g_2)$ a RM. Then the invariant distribution ${\mathfrak{D}}$ is integrable if and only if

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D_{\theta_1}}}\oplus {\mathfrak{D_{\theta_2}}})$.

Proof. For all ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D_{\theta_1}}}\oplus {\mathfrak{D_{\theta_2}}})$ with using Eqs (2.2), (2.9), (2.14) and decomposition (3.1), we have

On using Eq (3.2), we finally have

This completes the proof of theorem. □

The necessary and sufficient prerequisites that must also exist in order for distributions to be totally geodesic will now be discussed after the necessary conditions for distributions integrability. We start with investigating the necessary and sufficient conditions for distributions to be totally geodesic.

Theorem 4.4. Let ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ a Sasakian manifold and $({\bar Q_2}, g_2)$ a RM. Then invariant distribution ${\mathfrak{D}}$ defines totally geodesic foliation on ${\bar Q_1}$ if and only if

(i) $\lambda^{-2}g_2\{((\nabla {\Psi}_*)({\widehat{U}}, \phi {\widehat{V}}), {\Psi}_*\beta {\widehat{Z}})\} = g_1(\mathcal{V}\nabla_{\widehat{U}} \phi {\widehat{V}}, \alpha {\widehat{Z}})$,

(ii) $\lambda^{-2}\{g_2((\nabla {\Psi}_*)({\widehat{U}}, \phi {\widehat{V}}), {\Psi}_* {\mathbb{B}}{\widehat{X}})\} = g_1(\mathcal{V} \nabla_{\widehat{U}} \phi {\widehat{V}}, {\mathbb{C}}{\widehat{X}})+g_1(\phi \widehat{U}, \widehat{V})\eta(\widehat{X})$,

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D_{\theta_1}}}\oplus {\mathfrak{D_{\theta_2}}})$.

Proof. For any ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D}})$ and ${\widehat{Z}} \in \Gamma({\mathfrak{D_{\theta_1}}}\oplus {\mathfrak{D_{\theta_2}}})$ with using Eqs (2.2), (2.5), (2.14) and (3.2), we may write

On using the conformality of ${\Psi}$ with Eqs (2.6) and (2.15), we get

On the other hand, using Eqs (2.2), (2.5) and (2.14) with conformality of ${\Psi}$, we finally have

from which we get the desired result. □

In same manner, we can examine the geometry of leaves of ${\mathfrak{D_{\theta_1}}}$ as follows:

Theorem 4.5. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$. Then slant distribution ${\mathfrak{D_{\theta_1}}}$ defines totally geodesic foliation on ${\bar Q_1}$ if and only if

and

for any ${\widehat{Z}}, {\widehat{W}} \in \Gamma({\mathfrak{D_{\theta_1}}}), {\widehat{U}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_2}}})$ and ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$.

Proof. By using Eqs (2.2), (2.5), (2.14) and (3.2), we get

for ${\widehat{Z}}, {\widehat{W}} \in \Gamma({\mathfrak{D_{\theta_1}}})$ and ${\widehat{U}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_2}}})$. Again using Eqs (2.2), (2.5), (2.10), (2.14) and (3.2) with Lemma 3.2, we may write

Since, ${\Psi}$ is conformal, using Lemma 2.1 with Eqs (2.6) and (2.15), we have

On the other hand, for ${\widehat{Z}}, {\widehat{W}} \in \Gamma({\mathfrak{D_{\theta_1}}})$ and ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$, with using Eqs (2.2), (2.5), (2.14) and (3.2), we get

From Lemma 3.2 with Eqs (2.10) and (3.5), the above equation takes the form

Since ${\Psi}$ is conformal and from Eqs (2.6) and (2.15), we have

from which we get the result. □

In the following theorem, we study the necessary and sufficient conditions for slant distribution ${\mathfrak{D_{\theta_2}}}$ to be totally geodesic.

Theorem 4.6. Let ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ a Sasakian manifold and $({\bar Q_2}, g_2)$ a RM. Then slant distribution ${\mathfrak{D_{\theta_2}}}$ defines totally geodesic foliation on ${\bar Q_1}$ if and only if

and

for any ${\widehat{Z}}, {\widehat{W}} \in \Gamma({\mathfrak{D_{\theta_2}}}), {\widehat{V}} \in \Gamma({\mathfrak{D}}\oplus {\mathfrak{D_{\theta_1}}})$ and ${\widehat{Y}} \in \Gamma(ker {\Psi}_*)^\perp$.

Proof. The proof of above theorem is similar to the proof of Theorem 4.5. □

Since, ${{\Psi}}$ is $\mathcal{QBSC}\; \xi^{\perp}$-submersion, its vertical and horizontal distribution are $(ker {{\Psi}} _*)$ and $(ker {{\Psi}} _*)^\perp$, respectively. Now, we examine the conditions under which distributions defines totally geodesic foliation on ${\bar Q_1}$. With regards to the totally geodesicness of vertical distribution, we have

Theorem 4.7. Let ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion, where $({\bar Q_1}, \phi, \xi, \eta, g_1)$ a Sasakian manifold and $({\bar Q_2}, g_2)$ a RM. Then $ker {\Psi}_*$ defines totally geodesic foliation on ${\bar Q_1}$ if and only if

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$.

Proof. For any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$ with using Eqs (2.2), (2.5), (2.14) with decomposition (3.1), we get

On using Eq (3.2) with Lemmas 3.2 and 3.3, we have

From Eqs (2.9), (2.10) and (3.5), we may yields

From decomposition (3.1), the above equation takes the form

Using the conformality of ${\Psi}$ with Eqs (2.6) and (2.15), we have

This completes the proof of the theorem. □

We can now talk about the geometry of leaves of horizontal distribution. The following theorem presents the necessary and sufficient condition under which horizontal distribution defines totally geodesic foliation on ${\bar Q_1}$.

Theorem 4.8. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$, . Then $(ker {\Psi}_*)^\perp$ defines totally geodesic foliation on ${\bar Q_1}$ if and only if

for any ${\widehat{X}}, {\widehat{Y}} \in \Gamma(ker {\Psi}_*)^\perp$ and ${\widehat{Z}} \in \Gamma(ker {\Psi}_*)$.

Proof. For any ${\widehat{X}}, {\widehat{Y}} \in \Gamma(ker {\Psi}_*)^\perp$ and ${\widehat{Z}} \in \Gamma(ker {\Psi}_*)$ with using Eqs (2.2), (2.5) and (2.14) with decomposition (3.1), we get

From Eqs (2.11) and (3.2) with Lemma 3.2, we have

Since $\beta \mathfrak{P_2}Z+\beta\mathfrak{P_3}Z = \beta Z$ and with using the Eqs (2.12) and (3.2), we get

From formula (2.6) and (2.15), we yields that

Since ${\Psi}$ is conformal submersion, then we finally get

This completes the proof of theorem. □

We currently have a few prerequisites that must be met in order for $\mathcal{QBSC}\; \xi^{\perp}$-submersion $\Psi: \bar Q_1 \rightarrow \bar Q_2$ to be a totally geodesic map. In this regard, we offer the subsequent finding.

Theorem 4.9. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$. Then ${\Psi}: ({\bar Q_1}, \phi, \xi, \eta, g_1)\rightarrow ({\bar Q_1}, g_2)$ is totally geodesic map if and only if

and

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}}, {\widehat{Y}} \in \Gamma(ker {\Psi}_*)^\perp$.

Proof. Now, using Eqs (2.1), (2.5), (2.14) and (2.15). we can write

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker {\Psi}_*)$. From decomposition (3.1) and Eq (3.2), we have

By using Eqs (2.9) and (2.10), the above equation takes the form

Since ${\Psi}$ is conformal submersion, by using Lemmas 3.2 and 3.3 with Eq (3.2), we finally get

From this, the $(i)$ part of theorem proved. On the other hand, for any ${\widehat{U}} \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}} \in \Gamma(ker {\Psi}_*)^\perp$ with using Eqs (2.1), (2.5), (2.14) and (2.15), we can write

On using decomposition (3.1) with Eq (3.2), we have

By taking account the fact from Eqs (2.11) and (2.12), we get

Finally, from conformality of RS ${\Psi}$ and Lemmas 3.2 and 3.3, we can write

From which we obtain $(ii)$ part of theorem. This completes the proof of theorem. □

5.   Decomposition theorems

In this section, we recall the following result from ^[28] and discuss some decomposition theorems by using prior theorems. Let us suppose that $g$ be a Riemannian metric on the manifold $M = {\bar Q_1} \times {\bar Q_2}$, then

$({\rm{i}})$ $M = {\bar Q_1}\times_\lambda {\bar Q_2}$ is a locally product if and only if ${{\bar Q_1}}$ and ${{\bar Q_2}}$ are totally geodesic foliations,

$({\rm{ii}})$ a warped product ${\bar Q_1}\times_{\lambda} {\bar Q_2}$ if and only if ${\bar Q_1}$ is a totally geodesic foliation and ${\bar Q_2}$ is a spherics foliation, i.e., it is umbilic and its mean curvature vector field is parallel,

$({\rm{ii}})$ $M = {\bar Q_1}\times_\lambda {\bar Q_2}$ is a twisted product if and only if ${{\bar Q_1}}$ is a totally geodesic foliation and ${{\bar Q_2}}$ is a totally umbilic foliation.

The presence of three orthogonal complementary distributions $\mathfrak{D}$, $\mathfrak{D^{\theta_1}}$, and $\mathfrak{{\mathfrak{D^{\theta_2}}}}$, which satisfy some conditions of integrable and totally geodesic that we have stated previously, is ensured by the fact that ${{\Psi}} : ({\bar Q_1}, \phi, \xi, \eta, g_1) \rightarrow ({\bar Q_2}, g_2)$ is $\mathcal{QBSC}\; \xi^{\perp}$-submersion. It makes sense to now look for the conditions in which the total space ${\bar Q_1}$ converts into locally twisted product manifolds. Now, we are giving the following result.

Theorem 5.1. Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $(M, \phi, \xi, \eta, g_1)$ onto a RM $(M_2, g_2)$. Then ${\bar Q_1}$ is locally twisted product of the form ${\bar Q_1}{_{(ker {\Psi}_*)}}\times {\bar Q_1}{_{(ker {\Psi}_*)^\perp}}$ if and only if

and

where $H$ is a mean curvature vector and for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker {\Psi}_*)$ and ${\widehat{X}}_1, {\widehat{X}}_2 \in \Gamma(ker {\Psi}_*)^\perp$.

Proof. For any ${\widehat{X}}_1, {\widehat{X}}_2 \in\Gamma(ker {\Psi}_*)^\perp$ and ${\widehat{U}} \in \Gamma(ker {\Psi}_*)$ and using Eqs (2.2), (2.5), (2.11), (2.12) and (2.14), we have

From using formula (2.6), (2.15) and with conformality of RS ${\Psi}$, the above equation finally takes the form

It follows that the Eq (5.1) satisfies if and only if ${\bar Q_1}{_{(ker {\Psi}_*)}}$ is totally geodesic. On the other hand, for ${\widehat{U}}\in \Gamma(ker{\Psi}_*)$ and $\widetilde{X}, {\widehat{Y}} \in \Gamma(ker {\Psi}_*)^\perp$ with using Eqs (2.2), (2.5), (2.14) and (3.5), we get

By using the Eq (2.15) with definition of conformality of ${{{\Psi}} }$, we deduce that

Considering the $(i)$ part of Lemma 2.1, above equation turns in to

By direct calculation, finally we get

From the above equation we conclude that ${\bar Q_1}{_{(ker {\Psi}_*)^\perp}}$ is totally umbilical if and only if Eq (5.2) satisfied. □

6.   $\phi$-pluriharmonicity of quasi bi-slant conformal $\xi^{\perp}$-submersion

Y. Ohnita established $J$-pluriharminicity from a almost hermitian manifold in ^[22]. In this section, we extend the concept of $\phi$-pluriharmonicity to almost contact metric manifolds.

Let ${\Psi}$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Then $\mathcal{QBSC}$ submersion is $\phi$-pluriharmonic, ${\mathfrak{D}}$-$\phi$-pluriharmonic, $\mathfrak{D^{\theta_i}}$-$\phi$-pluriharmonic, $\mathfrak{(D}-\mathfrak{D^{\theta_i})}$-$\phi$ pluriharmonic (where $i = 1, 2$), $ker \Psi_*$-$\phi$-pluriharmonic, $(ker \Psi_*)^\perp$-$\phi$-pluriharmonic and $((ker \Psi_*)^\perp-ker \Psi_*)$-$\phi$-pluriharmonic if

for any ${\widehat{U}}, {\widehat{V}} \in \Gamma({\mathfrak{D}})$, for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(\mathfrak{D^{\theta_i)}}$, for any ${\widehat{U}}\in \Gamma({\mathfrak{D}}), {\widehat{V}} \in \Gamma(\mathfrak{D^{\theta_i})}$ (where $i = 1, 2$), for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker \Psi_*)$, for any ${\widehat{U}}, {\widehat{V}} \in \Gamma(ker \Psi_*)^\perp$ and for any ${\widehat{U}}\in \Gamma(ker \Psi_*)^\perp, {\widehat{V}} \in \Gamma(ker \Psi_*)$.

Theorem 6.1. Let $\Psi$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that $\Psi$ is ${\mathfrak{D_{\theta_1}}}$-$\phi$-pluriharmonic. Then ${\mathfrak{D_{\theta_1}}}$ defines totally geodesic foliation ${\bar Q_1}$ if and only if

for any ${{\widehat{U}}}, {\widehat{V}} \in\Gamma({\mathfrak{D_{\theta_1}}})$.

Proof. For any ${{\widehat{U}}}, {\widehat{V}} \in\Gamma({\mathfrak{D_{\theta_1}}})$ and since, $\Psi$ is ${\mathfrak{D_{\theta_1}}}$-$\phi$-pluriharmonic, then by using Eqs (2.9) and (2.15), we have

On using Eqs (3.2) and (3.5) with Lemmas 2.1 and 3.2, the above equation finally takes the form

from which we get the desired result. □

Theorem 6.2. Let $\vec f$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that $\Psi$ is ${\mathfrak{D_{\theta_2}}}$-$\phi$-pluriharmonic. Then ${\mathfrak{D_{\theta_2}}}$ defines totally geodesic foliation ${\bar Q_1}$ if and only if

for any ${\widehat{Z}}, {\widehat{W}} \in\Gamma({\mathfrak{D_{\theta_2}}})$.

Proof. The proof of the theorem is similar to the proof of Theorem 6.1. □

Theorem 6.3. Let $\vec f$ be a $\mathcal{QBSC}\; \xi^{\perp}$-submersion from Sasakian manifold $({\bar Q_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar Q_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that $\Psi$ is $((ker \Psi_*)^\perp-ker \Psi_*)$-$\phi$-pluriharmonic. Then the following assertion are equivalent.

(i) The horizontal distribution $(ker \Psi_*)^\perp$ defines totally geodesic foliation on ${\bar Q_1}$.

(ii) $(cos^2\theta_1+cos^2\theta_2){\Psi}_*\{{\mathbb{B}}\mathcal{T}_{{\mathbb{C}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+\beta\mathcal{V}\nabla_{{\mathbb{C}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+{\mathbb{B}}\mathcal{A}_{{\mathbb{B}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+\beta\mathcal{V}\nabla_{{\mathbb{B}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}\}$

$+{\Psi}_*\{\beta\mathcal{A}_{{\mathbb{B}}{\widehat{X}}}\beta\alpha {\mathfrak{P_2}}{\widehat{U}}+\beta\mathcal{A}_{{\mathbb{B}}{\widehat{X}}}\beta\alpha {\mathfrak{P_3}}{\widehat{U}}-\mathcal{H}\nabla_{{\mathbb{C}}{\widehat{X}}}\beta {\widehat{U}}\} +\nabla_{{\mathbb{B}}{\widehat{X}}}^{\Psi}{\Psi}_*\beta\alpha {\mathfrak{P_2}}{\widehat{U}}+\nabla_{{\mathbb{B}}{\widehat{X}}}^{\Psi}{\Psi}_*\beta\alpha {\mathfrak{P_3}}{\widehat{U}}$

$= {\Psi}_*\{{\mathbb{B}}\mathcal{T}_{{\mathbb{C}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+\beta\mathcal{V}\nabla_{{\mathbb{C}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+{\mathbb{B}}\mathcal{A}_{{\mathbb{B}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}+\beta\mathcal{H}{\nabla_{{\mathbb{B}}{\widehat{X}}}\alpha {\mathfrak{P_1}}{\widehat{U}}}\}$

$-{\Psi}_*\{\beta\mathcal{T}_{{\mathbb{C}}{\widehat{X}}}\beta\alpha {\mathfrak{P_2}}{\widehat{U}}+{\mathbb{B}}\mathcal{H}\nabla_{{\mathbb{C}}{\widehat{X}}}\beta\alpha {\mathfrak{P_2}}{\widehat{U}}+\beta\mathcal{T}_{{\mathbb{C}}{\widehat{X}}}\beta\alpha {\mathfrak{P_3}}{\widehat{U}}+{\mathbb{B}}\mathcal{H}\nabla_{{\mathbb{C}}{\widehat{X}}}\beta\alpha {\mathfrak{P_3}}{\widehat{U}}\}$

$+{\mathbb{B}}{\widehat{X}}(\ln\lambda){\Psi}_*\beta\alpha {\mathfrak{P_2}}{\widehat{U}}+\beta\alpha {\mathfrak{P_2}}{\widehat{U}}(\ln\lambda){\Psi}_*{\mathbb{B}}{\widehat{X}}-g_1({\mathbb{B}}{\widehat{X}}, \beta\alpha {\mathfrak{P_2}}{\widehat{U}}){\Psi}_*(grad\ln\lambda)$

$+{\mathbb{B}}{\widehat{X}}(\ln\lambda){\Psi}_*\beta\alpha {\mathfrak{P_3}}{\widehat{U}}+\beta\alpha {\mathfrak{P_3}}{\widehat{U}}(\ln\lambda){\Psi}_*{\mathbb{B}}{\widehat{X}}-g_1({\mathbb{B}}{\widehat{X}}, \beta\alpha {\mathfrak{P_3}}{\widehat{U}}){\Psi}_*(grad\ln\lambda)$

$+{\Psi}_*(\nabla_{\widehat{X}}{\widehat{U}})+\nabla_{\phi {\widehat{X}}}^{\Psi}{\Psi}_*\beta {\widehat{U}} +g_1(P\widehat{X}, \alpha\widehat{U})\Psi_* \xi,$

for any ${\widehat{X}} \in \Gamma(ker \Psi_*)^\perp$ and ${\widehat{U}} \in \Gamma(ker \Psi_*).$

Proof. For any ${\widehat{X}} \in \Gamma(ker \Psi_*)^\perp$ and ${{{\widehat{U}}}} \in \Gamma(ker \Psi_*)$, since $\Psi$ is $((ker \Psi_*)^\perp-ker \Psi_*)$-$\phi$-pluriharmonic, then by using (2.15), (3.2) and (3.5), we get

Taking account the fact from (2.1) and (2.10), we have

Now on using decomposition (3.1), Lemmas 3.2 and 3.3 with Eq (3.2), we may yields

From Eqs (2.9)–(2.12) and after simple calculation, we may write

Since $\Psi$ is conformal Riemannian submersion, the by using Eq (2.15) and from Lemma 2.1, we finally have

which completes the proof of theorem. □

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

The authors extend their appreciation to the Deanship of Scientific research, Imam Mohammad Ibn Saud Islamic University (IMSIU), Saudi Arabia, for funding this research work through Grant No. 221412008.

Conflict of interest

The authors declare that there is no conflict of interest.