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Abbreviations:

ACM: Almost contact metric; CBSS: Conformal bi-slant submersion; RM: Riemannian manifold; KM: Kenmotsu manifold

1.   Introduction

O'Neill ^[26] and Gray ^[16] were the ones who first proposed and developed the concept of submersions and immersions. For Riemannian manifolds, they discovered certain Riemannian equations by studying the geometrical characteristics. Submersions theory is an important topic in differential geometry that discusses the properties between differentiable structures. Riemannian submersions is the subject of study throughout both mathematics and physics since it has numerous applications, most notably in the Kaluza-Klein theory and Yang-Mills theory (see ^{[10,20,24,40]}). Watson ^[39] investigated the Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds in the year 1976. Later, Şahin ^[32] studied geometric characteristics and Riemannian submersions geometry. Using an almost Hermitian manifold, he defined anti-invariant Riemannian submersions onto Riemannian manifolds. "He demonstrates that, under the almost complex structure of the total manifold, their vertical distribution is anti-invariant." Many authors looked into and expanded on this research by studying anti-invariant submersions ^[3,32], semi-invariant submersions ^[33], slant submersions ^[13,34] and semi-slant submersions ^[19,27], among other topics. Tastan et al. ^[37] defined and investigated hemi-slant submersions from almost Hermitian manifolds as a generalization case of semi-invariant and semi-slant submersions.

From almost Hermitian to almost contact metric manifolds, Chinea ^[11] expanded the notion of Riemannian submersions. He examined base space, total and fibre space from an intrinsic geometric perspective point. Prasad et al. extended the concept of hemi-slant submersions a step further, by defining quasi-bi-slant submersions from an almost contact metric manifold ^[28,29]. The results he obtained for submersions were interesting and he also discovered some decomposition theorems.

Fuglede ^[14] and Ishihara ^[21] introduced the concept of conformal submersion as a generalization of Riemannian submersions and talked about some of their geometric characteristics. If the positive function $\lambda = 1$, which is dilation, then the conformal submersions become Riemannian submersions. Gudmundsson and Wood ^[18] investigated conformal holomorphic submersion as a generalization of holomorphic submersion. They were able to obtain the necessary and sufficient conditions for harmonic morphisms of conformal holomorphic submersions. Later on, conformal anti-invariant submersions, ^[2,30], conformal semi-invariant submersions ^[4], conformal slant submersions ^[6] and conformal semi-slant submersions ^[5] were studied and defined by Akyol and Şahin. A number of researchers have recently explored the geometry of conformal hemi-slant submersions ^[1,23,35], conformal bi-slant submersions ^[7] and quasi bi-slant conformal submersions ^[8] and they have discussed some decomposition theorems. Additionally, they expanded the idea of pluriharmonicity from almost Hermitian manifolds to almost contact metric manifolds. The present paper is a complement of ^[7]. In ^[7], $\xi$ is horizontal and in the present paper it is vertical.

In this paper, we investigate conformal bi-slant submersions from a Kenmotsu manifold onto a Riemannian manifold with vertical vector field $\xi$. The structure of the paper is as follows. Section 2 introduces almost contact metric manifolds, precisely the Kenmotsu manifolds with the properties required for this study. Section 3 includes a definition of conformal bi-slant submersion as well as some noteworthy discoveries. Section 4, details the conditions needed for distribution integrability as well as the total geodesicness of its leaves. This section also discusses how a total space Kenmotsu manifold becomes a locally twisted product manifold. Finally, at the end of the study, the concept of $\phi$-pluriharmonicity is addressed.

2.   Preliminaries

We start off by providing a few definitions and findings that will be quite helpful for our research and will aid in exploring the central idea of the research paper.

Let $(\bar O_1, g_1)$ and $(\bar O_2, g_2)$ be Riemannian manifolds, where dim$(\bar O_1) = m$, dim$(\bar O_2) = n$ and $m > n$. A Riemannian submersion $\mathcal{J}$: $\bar O_1 \rightarrow \bar O_2$ is a surjective map of $\bar O_1$ onto $\bar O_2$ satisfying the following axioms:

(i) $\mathcal{J}$ has maximal rank.

(ii) The differential $\mathcal{J}_*$ preserves the lenghts of horizontal vectors.

For each $q \in \bar O_2$, $\mathcal{J}^{-1}(q)$ is an $(m - n)$ dimensional submanifold of $\bar{O_1}$. The submanifolds $\mathcal{J}^{-1}(q), q \in \bar O_2$ are called fibers. A vector field on $\bar O_1$ is called vertical if it is always tangent to fibers. A vector field on $\bar O_1$ is called horizontal if it is always orthogonal to fibers. A vector field $X$ on $\bar O_1$ is called basic if $X$ is horizontal and $\mathcal{J}$- related to a vector field $X_*$ on $\bar O_2$, i.e., $\mathcal{J}_*X_p = X_{*\mathcal{J}(p)}$, for all $p \in \bar O_1$. Note that we denote the projection morphisms on the distributions $\mathrm{ker \mathcal{J}_*}$ and $(\mathrm{ker \mathcal{J}_*})^\perp$ by $\mathcal{V}$ and $\mathcal{H}$, respectively.

Definition 2.1. ^[9] Let $\mathcal{J}$ be a Riemannian submersion from an ACM manifold $({\bar O_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar O_2}, g_2)$. Then ${\mathcal{J}}$ is called a horizontally conformal submersion, if there is a positive function $\lambda$ such that

for any $U_1, V_1 \in \Gamma(ker {\mathcal{J}_*})^\perp$. It is obvious that every Riemannian submersions is a particularly horizontally conformal submersion with $\lambda = 1$. This $\lambda$ is usually called the dilation function.

The formulae of $(1, 2)$ tensor fields $\mathcal{T}$ and $\mathcal{A}$ are

for all vector fields $L_1, L_2 \in \Gamma (T{\bar{O_1}})$ ^[15].

It is obvious that a Riemannian submersion ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ has totally geodesic fibers if and only if $\mathcal{T}$ vanishes identically. Taking account the fact from (2.2) and (2.3) we may have

for all ${\bar{W_1}}, {\bar{Z_1}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and ${\bar{X_1}}, {\bar{Y_1}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\perp$ where

Then we can easily see that $\mathcal{T}_{\bar{Z}}$ and $\mathcal{A}_{\bar{W}}$ are skew-symmetric, i.e.,

for all $F_1, F_2 \in \Gamma(T_p{\bar{O_1}})$.

Here, we recall the proposition as follows:

Proposition 2.1. ^[17] Let ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ be horizontally conformal submersion with dilation $\lambda$ and ${\bar{Z}}, {\bar{W}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$, then

Then the second fundamental form of ${\mathcal{J}}$ is given by

A map is said to be totally geodesic if $(\nabla {\mathcal{J}}_*)({\bar{W}}, {\bar{Z}}) = 0$ for all ${\bar{W}}, {\bar{Z}} \in \Gamma(T_p{\bar{O_1}})$, where Levi-Civita and pullback connections are $\nabla$ and $\nabla^{\mathcal{J}}$ ^[38].

Lemma 2.1. Let ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ be a horizontal conformal submersion. Then, we have

(i) $(\nabla {\mathcal{J}}_*)({\bar{W_1}}, {\bar{Z_1}}) = {\bar{W_1}}(\ln \lambda){\mathcal{J}}_*({\bar{Z_1}}) + {\bar{Z_1}} (\ln \lambda){\mathcal{J}}_*({\bar{W_1}})- {g_1}({\bar{W_1}}, {\bar{Z_1}}){\mathcal{J}}_*(\mathrm{grad} \ln \lambda),$

(ii) $(\nabla {\mathcal{J}}_*)({\bar{U_1}}, {\bar{V_1}}) = - {\mathcal{J}}_*(\mathcal{T}_{\bar{U_1}} {\bar{V_1}}),$

(iii) $(\nabla {\mathcal{J}}_*)({\bar{W_1}}, {\bar{U_1}}) = - {\mathcal{J}}_*(\nabla_{\bar{W_1}}^{\bar{J}} {\bar{U_1}}) = - {\mathcal{J}}_*(\mathcal{A}_{\bar{W_1}}{\bar{U_1}})$

for any ${\bar{W_1}}, {\bar{Z_1}} \in \Gamma(ker \mathcal{J_*})^\perp$ and ${\bar{U_1}}, {\bar{V_1}}\in \Gamma(ker \mathcal{J_*})$ ^[9].

Let $M$ be a differentiable manifold of dimension $n$, is said to be having an almost contact structure $(\phi, \xi, \eta)$ if, it carries a tensor field $\phi$, vector field $\xi$ and 1-form $\eta$ on $M$ satisfying

where, $I$ is identity tensor. The almost contact structure $(\phi, \xi, \eta)$ 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

for all $\bar{Z}, \bar{W} \in \Gamma(TM)$. Then $(\phi, \xi, \eta, {{g}})$-structure is called an ACM structure, Tanno ^[36] determined connected ACM manifolds with the largest automorphism groups. The sectional curvature of a plane section containing $\xi$ for such a manifold is constant $c$. The characterizing equations of these manifolds are

These spaces are referred to as Kenmotsu manifolds, since Kenmotsu investigated some of these manifolds' basic differential geometric features ^[22]. On a KM, we can deduce that

and the covariant derivative of $\phi$ is defined by

for any ${\bar{W}}, {\bar{Z}} \in \Gamma(T{M})$.

Definition 2.2. Suppose $\mathfrak{D}$ is a $k$-dimensional smooth distribution on $\bar O_1$. Then an immersed submanifold $i$: $\bar O_2 \hookrightarrow \bar O_2$ is called an integral manifold for $\mathfrak{D}$ if for every $x \in \bar O_2$, the image of $d_i {\bar O_2}$: $T_p{\bar O_2} \rightarrow T_p{\bar O_1}$ is $\mathfrak{D_p}$. We say the distribution $\mathfrak{D_p}$ is integrable if through each point of $\bar O_1$ there exists an integral manifold of $\mathfrak{D}$.

Further, a distribution $\mathfrak{D}$ is involutive if it satisfies the Frobenius condition such that if $X, Y \in \Gamma(T\bar O_1)$ belongs to $\mathfrak{D}$, so $[X, Y] \in \mathfrak{D}$. Frobenius theorem states that an involutive distribution is integrable.

Definition 2.3. Let $\bar O_1$ be $n$-dimensional smooth manifold. A foliation $\mathfrak{F}$ of $\bar O_1$ is a decomposition of $\bar O_1$ into a union of disjoint connected submanifolds $\bar O_1 = \cup_{L \in \mathfrak{F}} L$ called the leaves of the foliation, such that for each $m \in \bar O_1$, there is a neighborhood $U$ of $\bar O_1$ 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 2.4. Let $\bar O_1$ be a Riemannian manifold, and let $\mathfrak{F}$ be a foliation on $\bar O_1$. $\mathfrak{F}$ is totally geodesic if each leaf $L$ is a totally geodesic submanifold of $\bar O_1$; that is, any geodesic tangent to $L$ at some point must lie within $L$.

3.   Conformal bi-slant submersions

Definition 3.1. Let $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ be an ACM manifold and $({\bar{O_2}}, g_2)$ be a RM. A conformal submersion ${\mathcal{J}}$ is said to be a CBSS with vertical $\xi$ if $D_{\theta_1}$ and $D_{\theta_2}$ are slant distributions with slant angel $\theta_1$ and $\theta_2$ such that $\mathrm{ker} {\mathcal{J}}_* = D_{\theta_1} \oplus D_{\theta_2}\oplus < \xi >$, where $< \xi >$ is a $1$-dimensional distribution spanned by $\xi$ and ${\mathcal{J}}$ is called proper if $\theta_1, \theta_2 \neq 0, \frac{\pi}{2}$.

If $n_1, \; n_2\neq 0, \; 0 < \theta_1 < \frac{\pi}{2}$ and $0 < \theta_2 < \frac{\pi}{2}$, then, ${\mathcal{J}}$ is said to be a proper CBSS with vertical $\xi$, where $n_1, n_2$ are the dimensions of $D_{\theta_1}$ and $D_{\theta_2}$ respectively.

In this part, we provide a non-trivial example to support our research.

Example 3.1. Let $\left(x_i, y_i, z\right)$ be Cartesian coordinates on $\mathbb{R}^{2 n+1}$ for $i = \; 1, 2, 3, \cdots, n$. An ACM structure $(\phi, \xi, \eta, g)$ is defined as follows:

where $\xi = \frac{\partial}{\partial z}$ and $a_i, b_i, c$ are $C^{\infty}$- real valued functions in $R^{2 n+1}$. Let $\eta = d z, g$ is Euclidean metric and

is orthonormal base field of vectors on $\mathbb{R}^{2 n+1}$. Then, it can be easily seen that $(\phi, \xi, \eta, g_{\mathbb{R}^{2 n+1}})$ is a Kenmotsu structure on $\mathbb{R}^{2 n+1}$.

Define a conformal submersion ${\mathcal{J}}$: $\mathbb{R}^9 \rightarrow \mathbb{R}^4$ such that

where $\left(x_1, \ldots, x_4, y_1, \ldots, y_4, z\right)$ are natural coordinates of $\mathbb{R}^9$ and $(\mathbb{R}^9, g_{\mathbb{R}^9})$ is a KM with above defined structure and $\lambda = \pi^5$. Then it follows that

and

Thus, ${\mathcal{J}}$ is a CBSS with vertical $\xi$ and slant angle $\theta_1$ and $\theta_2$ with $\lambda = \pi^5$.

Suppose that ${\mathcal{J}}$ is a CBSS with vertical $\xi$ from KM $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar{O_2}}, g_2)$, then for any ${\bar{U}} \in \mathrm{ker} {\mathcal{J}}_*$

where ${\mathfrak{K}}{\bar{U}} \in \Gamma(D_{\theta_1})$ and ${\mathfrak{L}}{\bar{U}} \in \Gamma(D_{\theta_2})$.

Also, for ${\bar{U}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$

where $\psi {\bar{U}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and $\zeta {\bar{U}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$. For any ${\bar{X}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$, we have

where $t{\bar{X}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and $f{\bar{X}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$.

The horizontal distribution $(\mathrm{ker} {\mathcal{J}}_*)^\bot$ is decomposed as

such that $\mu$ is the complementary distribution to $\zeta D_{\theta_1}\oplus \zeta D_{\theta_2}$ in $\Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$.

Given that ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ is a CBSS with vertical $\xi$, let's present some insightful findings that will be used throughout the work.

Theorem 3.1. Let ${\mathcal{J}}$: $({\bar{O_1}}, \phi, \xi, \eta, g_1) \rightarrow ({\bar{O_2}}, g_2)$ be a CBSS with vertical $\xi$ from ACM manifold onto a RM with slant angles $\theta_1$ and $\theta_2$. Then we have

(i) $\psi^2 {\bar{U}} = -(\cos^2 \theta_i) {\bar{U}}$,

(ii) $g_1(\psi {\bar{U}}, \psi {\bar{V}}) = cos^2 \theta_i g_1({\bar{U}}, {\bar{V}})$,

(iii) $g_1(\zeta {\bar{U}}, \zeta {\bar{V}}) = sin^2 \theta_i g_1({\bar{U}}, {\bar{V}})$

for any vector fields ${\bar{U}}, {\bar{V}} \in \Gamma(D_{\theta_i})$, where $i = 1, 2.$

Due to similarities with the proof of [12, Theorem 3.4], we omit the proof of the aforementioned result.

Lemma 3.1. Let $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM. If ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ is a CBSS with vertical $\xi$, then we have

for any ${\bar{U}} \in \Gamma(\operatorname{ker {\mathcal{J}}_*})$ and ${\bar{X}} \in \Gamma{\operatorname(ker {\mathcal{J}}_*)}^\perp$.

Proof. Equations (2.14), (3.2) and (3.3) are used to obtain outcomes from simple calculations. □

Let $({\bar{O_2}}, g_2)$ is a RM and that $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ is a KM. Now, let us check how the Kenmotsu structure on ${\bar{O_1}}$ affects the tensor fields $\mathcal{T}$ and $\mathcal{A}$ of a $\mathcal{BSCS} \; {\mathcal{J}}$: $({\bar{O_1}}, \phi, \xi, \eta, g_1) \rightarrow ({\bar{O_2}}, g_2)$

Lemma 3.2. If ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ is a CBSS with vertical $\xi$ where $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM, then we have

(i) $\mathcal{A}_{\bar{X}} t{\bar{Y}}+\mathcal{H}\nabla_{\bar{X}}f{\bar{Y}} = f\mathcal{H}\nabla_{\bar{X}}{\bar{Y}}+\zeta\mathcal{A}_{\bar{X}}{\bar{Y}}$,

(ii) $\mathcal{V}\nabla_{\bar{X}}t{\bar{Y}}+\mathcal{A}_{\bar{X}}f{\bar{Y}} = t\mathcal{H}\nabla_{\bar{X}}{\bar{Y}}+\psi\mathcal{A}_{\bar{X}}{\bar{Y}}-g_1(\phi {\bar{X}}, {\bar{Y}})\xi$,

(iii) $\mathcal{V}\nabla_{\bar{X}}\psi {\bar{V}}+\mathcal{A}_{\bar{X}} \zeta {\bar{V}} = t\mathcal{A}_{\bar{X}}{\bar{V}}+\psi \mathcal{V}\nabla_{\bar{X}}{\bar{V}}+g_1(\phi\bar{X}, \bar{V})\xi-\eta(\bar{V})t\bar{X}$,

(iv) $\mathcal{A}_{\bar{X}}{\psi {\bar{V}}}+\mathcal{H}\nabla_{\bar{X}} \zeta {\bar{V}} = f\mathcal{A}_{\bar{X}}{\bar{V}}+\zeta\mathcal{V}\nabla_{\bar{X}}{\bar{V}}-\eta(\bar{V})f\bar{X}$,

(v) $\mathcal{V}\nabla_{\bar{V}}t{\bar{X}}+\mathcal{T}_{\bar{V}}f{\bar{X}} = \psi \mathcal{T}_{\bar{V}}f{\bar{X}}+t\mathcal{H}\nabla_{\bar{V}}{\bar{X}}+g_1(\phi\bar{X}, \bar{V})\xi$,

(vi) $\mathcal{T}_{\bar{V}}t{\bar{X}}+\mathcal{H}\nabla_{\bar{V}}f{\bar{X}} = \zeta \mathcal{T}_{\bar{V}}{\bar{X}}+f\mathcal{H}\nabla_{\bar{V}}{\bar{X}}$,

(vii) $\mathcal{V}\nabla_{\bar{U}}\psi {\bar{V}}+\mathcal{T}_{\bar{U}}\zeta {\bar{V}}+\eta({\bar{V}})\psi \bar{U} = t\mathcal{T}_{\bar{U}}{\bar{V}}+\psi\mathcal{V}\nabla_{\bar{U}}{\bar{V}}+g_1(\phi \bar{U}, {\bar{V}})\xi$,

(viii) $\mathcal{T}_{\bar{U}}\psi {\bar{V}}+\mathcal{H}\nabla_{\bar{U}}\zeta {\bar{V}}+\eta({\bar{V}})\zeta {\bar{U}} = f\mathcal{T}_{\bar{U}}{\bar{V}}+\zeta\mathcal{V}\nabla_{\bar{U}}{\bar{V}}$

for any ${\bar{U}}, {\bar{V}} \in \Gamma(\operatorname{ker }{\mathcal{J}}_*)$ and ${\bar{X}}, {\bar{Y}} \in \Gamma(\operatorname{{ker} {\mathcal{J}}_*})^\perp$.

Proof. By some simple steps of calculation with using (2.7), (2.15) and (3.3), $(i)$ and $(ii)$ are easily obtained. In the same manner, from Eqs (2.4)–(2.6), (2.15), (3.2) and (3.3), we will get the desired results. □

We will now go through some fundamental findings that can be used to investigate the conformal bi-slant submersions ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ geometry. Define the following in this regard:

$(a)\; (\nabla_{\bar{U}}\psi) {\bar{V}} = \mathcal{V}\nabla_{\bar{U}}\psi {\bar{V}}-\psi\mathcal{V}\nabla_{\bar{U}}{\bar{V}}$,

$(b)\; (\nabla_{\bar{U}}\zeta) {\bar{V}} = \mathcal{H}\nabla_{\bar{U}}\zeta {\bar{V}}-\zeta \mathcal{V}\nabla_{\bar{U}}{\bar{V}}$,

$(c)\; (\nabla_{\bar{X}}t){\bar{Y}} = \mathcal{V}\nabla_{\bar{X}}t{\bar{Y}}-t\mathcal{H}\nabla_{\bar{X}}{\bar{Y}}$,

$(d)\; (\nabla_{\bar{X}}f){\bar{Y}} = \mathcal{H}\nabla_{\bar{X}}f{\bar{Y}}-f\mathcal{H}\nabla_{\bar{X}}{\bar{Y}}$

for all ${\bar{U}}, {\bar{V}} \in \Gamma(\operatorname{ker } {\mathcal{J}}_*)$ and ${\bar{X}}, {\bar{Y}} \in \Gamma(\operatorname{ker }{\mathcal{J}}_*)^\perp$.

Lemma 3.3. Let $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM. If ${\mathcal{J}}$: ${\bar{O_1}} \rightarrow {\bar{O_2}}$ is a CBSS with vertical $\xi$, then we have

(i) $(\nabla_{\bar{U}}\psi){\bar{V}} = t\mathcal{T}_{\bar{U}}{\bar{V}}-\mathcal{T}_{\bar{U}}\zeta {\bar{V}}+g_1(\phi {\bar{U}}, {\bar{V}})\xi-\eta({\bar{V}})\psi {\bar{U}}$,

(ii) $(\nabla_{\bar{U}}\zeta){\bar{V}} = f\mathcal{T}_{\bar{U}}{\bar{V}}-\mathcal{T}_{\bar{U}}\psi {\bar{V}}-\eta({\bar{V}})\zeta {\bar{U}}$,

(iii) $(\nabla_{\bar{X}}t){\bar{Y}} = \psi\mathcal{A}_{\bar{X}}{\bar{Y}}-\mathcal{A}_{\bar{X}}f{\bar{Y}}$,

(iv) $(\nabla_{\bar{X}}f){\bar{Y}} = \zeta\mathcal{A}_{\bar{X}}{\bar{Y}}-\mathcal{A}_{\bar{X}}t{\bar{Y}}$

for any ${\bar{U}}, {\bar{V}} \in \Gamma(\operatorname{ker {\mathcal{J}}_*})$ and ${\bar{X}}, {\bar{Y}} \in \Gamma(\operatorname{ker {\mathcal{J}}_*})^\perp$.

Proof. By taking account the fact from (2.4)–(2.7), (2.13), $(i)$, $(ii)$ part from Lemma 3.2 and from part $(a)$–$(d)$, it is easy to get the proof of the lemma. □

It is given that $\nabla$ is the Levi-Civita connection of Kenmotsu manifolds ${\bar{O_1}}$. Let us suppose that the tenors $\psi$ and $\zeta$ are parallel, we can write

for any ${\bar{X}}, {\bar{Y}} \in\Gamma(T{\bar{O_1}})$.

4.   Geometry of leaves of distributions

We will talk about the geometry of distribution leaves and integrability conditions in this section. We begin with the prerequisites that must be met in order for the slant distributions to be integrable.

Theorem 4.1. Let ${\mathcal{J}}$: $({\bar{O_1}}, \phi, \xi, \eta, g_1) \rightarrow ({\bar{O_2}}, g_2)$ be a proper CBSS with vertical $\xi$ with slant angles $\theta_1$ and $\theta_2$, where $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ is a KM and $({\bar{O_2}}, g_2)$ is a RM. Then the distribution $D_{\theta_1}$ is integrable if and only if

for any ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(D_{\theta_1})$ and ${{\bar{U}}_2} \in \Gamma(D_{\theta_2}).$

Proof. For any ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(D_{\theta_1})$ and ${{\bar{U}}_2} \in \Gamma(D_{\theta_2})$ and on using (2.12), (2.13) and from (3.2), we have

Considering Theorem 3.1, we have

By using (2.5), we have

Now considering Lemma 2.1 and (2.10), we have

□

Studying distribution leaves will be significant since they are crucial to the geometry of conformal bi-slant submersions from the Kenmotsu manifold. In order to do this, we are determining the circumstances in which distributions define total geodesic foliation on M.

Theorem 4.2. Let $\mathcal{J}$: $(\bar O_1, \phi, \xi, \eta, g_1) \rightarrow ({\bar{O_2}}, g_2)$ be a CBSS with vertical $\xi$ from a KM onto a RM ${\bar{O_2}}$. Then $D_{\theta_1}$ is not totally geodesic on $\bar O_1$.

Proof. For any vector field $\bar{U}, {\bar{V}} \in \Gamma(D_\theta)$ with the fact that ${\bar{V}}$ and $\xi$ are orthogonal, we have

By considering the (2.14), we get

Since,

that is $g_1(\nabla_{\bar{U}}{\bar{V}}, \xi)\neq 0$. Hence, the distribution is not defines totally geodesic foliation on $\bar O_1$. □

Since the Reeb vector field $\xi$ is assumed to be vertical, the slant distribution $D_{\theta_1}$ does not define total geodesic foliation. In order to deal with this issue, the geometry of the leaves of the slant distribution $D_{\theta_1}{\oplus < \xi > }$ is being examined here.

Theorem 4.3. Let $(\bar O_1, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM such that $\mathcal{J}$ is a CBSS with vertical $\xi$ from $M$ onto ${\bar{O_2}}$. Then the distribution $D_{\theta_1}\oplus < \xi >$ defines totally geodesic foliation on $\bar O_1$ if and only if

and

for any ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(D_{\theta_1}\oplus < \xi > ), {{\bar{U}}_2} \in \Gamma(D_{\theta_2})$ and ${\bar{X}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$.

Proof. For any ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(D_{\theta_1}\oplus < \xi > )$ and ${{\bar{U}}_2} \in \Gamma(D_{\theta_2})$ with using (2.12), (2.13) and (3.2), we have

From Theorem 3.1, we can write

On using (2.5), we have

Considering (2.10) and Lemma 2.1, we obtain

this proves first part of theorem.

On the other hand, ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(D_{\theta_1})$ and ${\bar{X}} \in \Gamma (\mathrm{ker} {\mathcal{J}}_*)^\bot$ with using (2.12), (2.13) and (3.2), we can write

Considering Theorem 3.1, we obtained

On using (2.7), we have

Using Lemma 2.1, we yields

This completes the proof of the Theorem. □

It is obvious that the Theorems 4.1–4.3 hold for distribution $D_{\theta_2}$. Now, we look at certain circumstances that allow horizontal and vertical distributions to be totally geodesic. We commence by giving the findings for vertical distribution.

Theorem 4.4. Let $(\bar{O_1}, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM such that $\mathcal{J}$ is a CBSS with vertical $\xi$ from $\bar{O_1}$ onto ${\bar{O_2}}$. Then vertical distribution $(\mathrm{ker} {\mathcal{J}}_*)$ defines totally geodesic foliation on $\bar{O_1}$ if and only if

for ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and ${{\bar{X}}_1} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$.

Proof. On taking ${{\bar{U}}_1}, {{\bar{V}}_1} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and ${{\bar{X}}_1} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$ with using (2.12), (2.13) and (3.2), we have

On using decomposition (3.1) and Theorem 3.1, we get

On using (2.7), we can write

Using (2.10), we get

Considering Lemma 2.1, we have

This completes the proof of the Theorem. □

Similarly, we examined the totally geodesic prerequisite for horizontal distributions.

Theorem 4.5. Let $(\bar{O_1}, \phi, \xi, \eta, g_1)$ be a KM and $({\bar{O_2}}, g_2)$ be a RM such that $\mathcal{J}$ is a CBSS with vertical $\xi$ from $\bar{O_1}$ onto ${\bar{O_2}}$. Then horizontal distribution $(\mathrm{ker} {\mathcal{J}}_*)^\bot$ is totally geodesic if and only if

for any ${{\bar{X}}_1}, {{\bar{X}}_2} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot, {{\bar{U}}_1} \in \Gamma(\mathrm{Ker}J_*)$.

Proof. For any ${{\bar{X}}_1}, {{\bar{X}}_2} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)^\bot$ and ${{\bar{U}}_1} \in \Gamma(\mathrm{ker}J_*)$ with using (2.12), (2.13) and (3.2), we have

By using Theorem 3.1, we can write

From (2.7), we get

Considering Lemma 2.1, we have

□

It is now fascinating to investigate if the whole space $\bar{O_1}$ can become a locally twisted product manifold under specific circumstances. We find some criteria that make total space $\bar{O_1}$ a locally twisted product manifold in the following result. Here, we give the definition of the twisted product manifold defined by Ponge ^[31]. Let $g_B$ be a Riemannian metric tensor on the manifold $B = \bar O_1 \times \bar O_2$ and assume that the canonical foliations $D_{\bar O_1}$ and $D_{\bar O_2}$ intersect perpendicularly everywhere. The $g_B$ is a metric tensor of

(i) a twisted product if and only if $D_{\bar O_1}$ is totally geodesic foliation and $D_{\bar O_2}$ is totally umbilical foliation,

(ii) a usually product of Riemannian manifolds if and only if $D_{\bar O_1}$ and $D_{\bar O_2}$ are totally geodesic foliations,

(iii) a warped product if and only if $D_{\bar O_1}$ is totally geodesic foliation and $D_{\bar O_2}$ is a spheric foliation, i.e., it is umbilical and its mean curvature vector field is parallel.

Theorem 4.6. Let ${\mathcal{J}}$ be a CBSS with vertical $\xi$ from a KM $(\bar{O_1}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar{O_2}}, g_2)$. Then $\bar{O_1}$ is a locally twisted product manifold of the form ${\bar{O_1}}_{(\mathrm{ker} {\mathcal{J}}_*)} \times_{\lambda} {\bar{O_1}}_{(\mathrm{ker} {\mathcal{J}}_*)^\perp}$ if and only if

and

where $H$ is mean curvature vector and for any ${\bar{V}}, {\bar{W}} \in \Gamma(\mathrm{ker}{\mathcal{J}}_*)$ and ${\bar{X}}, {\bar{Y}} \in \Gamma(\mathrm{ker}{\mathcal{J}}_*)^\perp$.

Proof. For any ${\bar{V}}, {\bar{W}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$ and ${\bar{X}}, {\bar{Y}} \in \Gamma(\mathrm{ker}{\mathcal{J}}_*)^\perp$, we have

Since $\nabla$ is torsion free, $[{\bar{V}}, \phi {\bar{W}}] \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$, we have

Since ${\mathcal{J}}$ is CBSS with vertical $\xi$, by using Lemma 2.1 and from the fact that $g_1(f{\bar{X}}, \phi {\bar{V}}) = 0$ for ${\bar{X}} \in (\mathrm{ker} {\mathcal{J}}_*)^\perp$ and ${\bar{V}} \in (\mathrm{ker} {\mathcal{J}}_*)$, we have

It follows that ${\bar{O_1}}_{(ker {\mathcal{J}}_*)}$ is totally geodesic if and only if the (4.5) holds good. Now, for ${\bar{X}}, {\bar{Y}} \in\Gamma(\mathrm{ker} {\mathcal{J}}_*)^\perp, {\bar{V}} \in \Gamma(\mathrm{ker} {\mathcal{J}}_*)$, we have

From above equation, we get

Since ${\mathcal{J}}$ is a CBSS with vertical $\xi$, from (2.4) and on using Lemma 2.1, we get

Moreover using the fact that $g_1(f{\bar{X}}, \phi {\bar{V}}) = 0$, for ${\bar{X}} \in \Gamma(\operatorname{\mathrm{ker}} {\mathcal{J}}_*)^\perp, {\bar{V}} \in \Gamma(\operatorname{\mathrm{ker}} {\mathcal{J}}_*)$, we arrived at

With the last equation, we can say that ${\bar{O_1}}_{(\operatorname{\mathrm{ker}} {\mathcal{J}}_*)^\perp}$ is totally umbilical if and only if the (4.6) satisfied. This proves the theorem completely. □

5.   $\phi$-Pluriharmonicity of CBSS with vertical $\xi$

Now, we recall the concept of $\mathcal{J}$-pluriharmonicity which is defined by Ohnita ^[25] and extend the notion from a almost Hermitian manifold to ACM manifold.

Let ${{\mathcal{J}}}$ be a CBSS with vertical $\xi$ from KM $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ onto a RM $(\bar{O_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Then conformal bi-slant is $\phi$-pluriharmonic, ${D_{\theta_i}}$-$\phi$-pluriharmonic, $ker {\mathcal{J}}_*$-$\phi$-pluriharmonic, $(ker {\mathcal{J}}_*)^\perp$-$\phi$-pluriharmonic and $((ker {\mathcal{J}}_*)^\perp-ker {\mathcal{J}}_*)$-$\phi$-pluriharmonic if

for any ${\bar{W}}, {\bar{Z}} \in \Gamma(\mathfrak{D^{\theta_i)}}$, for any ${\bar{W}}, {\bar{Z}} \in \Gamma(ker {\mathcal{J}}_*)$, for any ${\bar{W}}, {\bar{Z}} \in \Gamma(ker {\mathcal{J}}_*)^\perp$ and for any ${\bar{W}}\in \Gamma(ker {\mathcal{J}}_*)^\perp, {\bar{Z}} \in \Gamma(ker {\mathcal{J}}_*)$.

Theorem 5.1. Let $\mathcal{J}$ be a CBSS with vertical $\xi$ from KM $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ onto a RM $(\bar{O_2}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that ${\mathcal{J}}$ is ${\mathfrak{D^{\theta_1}}}$-$\phi$-pluriharmonic. Then ${{D_{\theta_1}}}$ defines totally geodesic foliation ${\bar{O_1}}$ if and only if

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

Proof. For any ${{\bar{U}}}, {\bar{V}} \in\Gamma({{D_{\theta_1}}})$ and since, ${\mathcal{J}}$ is ${{D_{\theta_1}}}$-$\phi$-pluriharmonic, then by using (2.3) and (2.4), we have

By using (2.10), (3.1) with Theorem 3.1, the above equation finally takes the form

from which we get the desired result. □

Theorem 5.2. Let $\mathcal{J}$ be a CBSS with vertical $\xi$ from KM $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar{O_2}}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that ${\mathcal{J}}$ is ${{D_{\theta_2}}}$-$\phi$-pluriharmonic. Then ${{D_{\theta_2}}}$ defines totally geodesic foliation ${\bar{O_1}}$ if and only if

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

Proof. Due to the similarity of proof of above result to Theorem 5.1, we omit it. □

Theorem 5.3. Let $\mathcal{J}$ be a CBSS with vertical $\xi$ from KM $({\bar{O_1}}, \phi, \xi, \eta, g_1)$ onto a RM $({\bar{O_2}}, g_2)$ with slant angles $\theta_1$ and $\theta_2$. Suppose that ${\mathcal{J}}$ is $((ker {\mathcal{J}}_*)^\perp-ker {\mathcal{J}}_*)$-$\phi$-pluriharmonic. Then the horizontal distribution $(ker {\mathcal{J}}_*)^\perp$ defines totally geodesic foliation on $\bar{O_1}$ if and only if

for any ${\bar{X}} \in \Gamma(ker {\mathcal{J}}_*)^\perp$ and ${\bar{U}} \in \Gamma(ker {\mathcal{J}}_*)$

Proof. For any ${\bar{X}} \in \Gamma(ker {\mathcal{J}}_*)^\perp$ and ${{{\bar{U}}}} \in \Gamma(ker {\mathcal{J}}_*)$, since ${\mathcal{J}}$ is $((ker {\mathcal{J}}_*)^\perp-ker {\mathcal{J}}_*)$-$\phi$-pluriharmonic, then by using (2.4), (2.10) and (3.1), we get

By using (2.1), we have

Now on using decomposition (2.8), Theorem 3.1 with (2.10), we may yields

From (2.4)–(2.7) and after simple calculation, we may write

On using the conformality of ${\mathcal{J}}$ with (2.4) and from Lemma 2.1, we finally have

which completes the proof of theorem.□

Now, we are giving some definitions of integrablity and totally geodesic of leaves of distributions.

Definition 5.1. Suppose ${D}$ is a $k$-dimensional smooth distribution on $M$. Then An immersed submanifold $i$: $N \hookrightarrow M$ is called an integral manifold for ${D}$ if for every $x \in N$, the image of $d_i N$: $T_pN \rightarrow T_pM$ is ${D_p}$. We say the distribution ${D_p}$ is integrable if through each point of $M$ there exists an integral manifold of ${D}$.

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

Definition 5.2. 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 5.3. 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 $L$.

6.   Conclusions

The properties of submersions between Riemannian manifolds have emerged as an interesting area of study in contact as well as complex geometry. The geometry of conformal bi-slant submersion, whose base manifold is a Kenmotsu manifold, was examined in this study. We suppose that the Reeb vector field $\xi$ is vertical and establishes the condition of integrability of slant distributions because the Reeb vector field $\xi$ plays a crucial role in the geometry of leaves of the distributions. Additionally, the total geodesics of the leaves of distributions were determined. Furthermore, the idea of pluriharmonicity from the Kenmotsu manifold was explored.

Use of AI tools declaration

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

Acknowledgments

This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (grant number IMSIU-RP23003).

Conflict of interest

The authors have no conflicts of interest.