On Riemannian warped-twisted product submersions

1.   Introduction

In the 1960's, B. O'Neill ^[1] introduced the notion of Riemannian submersion as a tool to study the geometry of a manifold in terms of the simpler components, namely, fibers and base space. A. L. Besse considered warped product Riemannian submersion ^[2]. Further, I. K. Erken and C. Murathan ^[3] studied warped product Riemannian submersion and obtained fundamental geometric properties.

J. F. Nash ^[4] started the study of warped product manifolds and proved that every warped product manifold can be embedded as a Riemannian submanifold in some Euclidean spaces. In 1969, B. O'Neill and R. L. Bishop ^[5] studied the warped product manifold as a fruitful generalization of the Riemannian product manifold.

Warped product manifolds play key roles in mathematical physics ^[6]. H. M. Tastan and S. B. Aydin ^[7,8] introduced the concept of a warped-twisted product as an extension of the twisted product and consequently, the warped product. The warped-twisted product, denoted as $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$, refers to the product manifold $\mathcal M_1 \times \mathcal M_2$ endowed with the metric tensor $g$, which is defined by

where, $\varphi_{i}: \mathcal{M}_1\times \mathcal{M}_2\longrightarrow \mathcal{M}_i$ is the natural projections, for $i \in \{1, 2\}$. The function $f_2 \in C^{\infty}\left(\mathcal M_2\right)$ is named a warping function, and the function $f_1 \in C^{\infty}\left(\mathcal M_1 \times \mathcal M_2\right)$ is named a twisting function of $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$. If the function $f_1$ solely depends on the points of $\mathcal M_2$ in this instance, the resulting warped-twisted product can be classified as a base conformal warped product ^[8]. A warped-twisted product is considered non-trivial if it does not fall into any categories of a doubly warped product, a warped product, or a base conformal warped product. For more details about the concerned studies, we refer the papers ^{[9,10,11,12,13,14,15,16,17,18,19,20,21,22]}.

The following is our definition of warped-twisted product submersions:

Definition 1.1. Suppose that $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ are warped-twisted product manifolds and $\varphi_i: \mathcal{M}_i \rightarrow \aleph_i, i\in \{1, 2\}$, are Riemannian submersion between the manifolds $\mathcal{M}_{i}$ and $\mathcal{N}_{i}$. Then the map

given by $\varphi\left(x_1, x_2\right) = \left(\varphi_1\left(x_1\right), \varphi_2\left(x_2\right)\right)$ is a Riemannian submersion, which is called warped-twisted product submersion.

Our primary objective of this paper is to investigate the fundamental geometric properties associated with warped-twisted product submersions. The notion of warped product generalizes usual products, which is further generalized by the twisted product and doubly warped product. Non-trivial wraped-twisted product is neither twisted nor base conformal nor direct product. The definition of warped product submersion and its geometrical properties was discussed by Murathan, C. in his article "Riemannian warped product submersions". These results, which are presented in that article, serve as our motivation.

The paper is organized in the following way. In Section 2, we recall definitions and some fundamental results of Riemannian submersions and warped-twisted product manifolds which are useful for this paper. In Section 3, we defined Riemannian warped-twisted product submersion and discuss some geometrical properties for this submersion. In Section 4, we obtain the Ricci tensors for Riemannian warped-twisted product submerion and discuss Einstein's condition on vertical and horizontal distributions of total manifold.

2.   Preliminaries

In this section, we recall some definitions, results and notations that are necessary for the paper.

2.1.   Riemannian submersion

Let $(\mathcal{M}, g_\mathcal{M})$ and $(\aleph, g_\aleph)$ be two Riemannian manifolds with $dim \mathcal{M} = m$ and $dim \aleph = n$, where $m > n$. A smooth map $\varphi:(\mathcal{M}, g_\mathcal{M})\longrightarrow(\aleph, g_\aleph)$ is said to be Riemannian submersion if the following axioms are satisfied:

$1)$ $\varphi_{\ast}$(derivative map of $\varphi$) is onto,

$2)$ $\varphi_{\ast}$ preserves the length of horizontal vectors, i.e.,

For each $p_2\in \aleph$, $\phi^{-1}(p_2)$ is a submanifold of dimension $(m-n)$ called fibers. If the fibers are orthogonal then a vector field on $\mathcal{M}$ is referred to as horizontal and it is referred to as vertical if the fibers are tangent. Let $\varphi:(\mathcal{M}, g_\mathcal{M})\longrightarrow(\aleph, g_\aleph)$ be a smooth map. Then $\Gamma(T\mathcal{M})$ has the following decomposition:

B. O'Neill ^[1] first introduced the fundamental tensors of submersions, and are defined by

where $E$ and $F$ are vector fields on $\mathcal{M}$; $\mathcal H$ and $\mathcal V$ are the projection morphism on the distribution $(ker\varphi{\ast})^{\perp}$ and $(ker\varphi_{\ast})$, respectively. We observe that the tensor fields $T$ and $A$ satisfy

$1)$ $T_UV = T_VU, \quad U, V\in \Gamma(ker\varphi_{\ast}),$

$2)$ $A_XY = -A_YX, \quad X, Y\in \Gamma(ker\varphi_{\ast})^{\perp}.$

Equations (2.1) and (2.2) give the following lemma.

Lemma 2.1. ^[1]. Let $X, Y\in \Gamma(ker\varphi_{\ast})^{\perp}$ and $U, V\in \Gamma(ker\varphi_{\ast})$; then we have

where $\nabla$ is the Levi-Civita connection of $(\mathcal{M}, g_\mathcal{M})$ and $\hat{\nabla}_UV = \mathcal{V}\nabla_UV$.

It is noted that if the tensor field $A$ (respectively $T$) vanishes, then the horizontal distribution $\mathcal H$ (respectively, vertical distribution $\mathcal V$ or fiber) is integrable. Also, any fiber of Riemannian submersion $\phi$ is totally umbilical if and only if

where $\mathcal H$ is the mean curvature vector field of the fiber given by

such that

and $\{U_{1}, U_{2}, \cdots, U_{s}\}$ denotes the orthonormal basis of vertical distribution and $s$ denotes the dimension of any fiber. It is easy to see that any fiber of Riemannian submersion $\phi$ is minimal if and only if the horizontal vector field $N$ vanishes.

2.2.   Warped-twisted product manifolds

Let $(\mathcal{M}_i, g_{\mathcal{M}_i})$ be two Riemannian manifolds of dimensions $m_1$ and $m_2$, respectively and $f_1$ and $f_2$ be two positive differentiable functions on ${\mathcal{M}_1}$ and $\mathcal{M}_1\times \mathcal{M}_2$, respectively. Let $\varphi_{i}: \mathcal{M}_1\times \mathcal{M}_2\longrightarrow \mathcal{M}_i$ be the natural projections from product manifold $\mathcal{M}_1\times \mathcal{M}_2$ to $\mathcal{M}_{i}$, $i\in\{1, 2\}$. Then the warped-twisted product manifold $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ is a product manifold $\mathcal{M} = \mathcal{M}_1\times \mathcal{M}_2$ endowed with the metric $g_{\mathcal{M}}$ such that

for any $X, Y\in \mathcal{M}$.

For any $X$ on $\mathcal{M}_1$, the lift of $X$ to ${}_{f_{2}}\mathcal{M}_{1} \times_{f_1} \mathcal{M}_2$ is the vector field $\tilde{X}$ whose value at each $(p, q)$ is the lift $X_p$ to $(p, q)$. Thus the lift of $X$ is the unique vector field on ${}_{f_{2}}\mathcal{M}_{1} \times_{f_1} \mathcal{M}_2$, that is, $\varphi_1$-related to $X$ and $\varphi_2$-related to the zero vector field on $\mathcal{M}_2$.

Let $\nabla$ and $\nabla^{i}$ be the Levi-Civita connections of ${}_{f_2} \mathcal{M}_1 \times{ }_{f_1} \mathcal{M}_2$ and $\mathcal{M}_i$, respectively for $i \in\{1, 2\}$. The lifts of vector fields on $\mathcal{M}_i$ is denoted by $\mathfrak{L}\left(\mathcal{M}_i\right)$.

Then, the covariant derivative formulae for a warped-twisted product manifold are given as ^[8]:

for $X, Y \in \mathfrak{L}\left(\mathcal{M}_1\right)$ and $U, V \in \mathfrak{L}\left(\mathcal{M}_2\right)$. Now, for any smooth function $\psi$ on a warped-twisted product $({}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}, g_{\mathcal{M}})$, we have

for any $X \in \mathfrak{L}\left(\mathcal{M}_1\right)$ and $U \in \mathfrak{L}\left(\mathcal{M}_2\right)$, where the definition of the Hessian tensor being used. Now let $\mathbb S$ and $\mathbb S^i$ be the Ricci tensors of $(\mathcal M, g)$ and $\left(\mathcal M_i, g_i\right)$, respectively. Then we have the following relations:

Lemma 2.2. ^[23] Let $\mathbb M = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ be a warped-twisted product manifold. Then we have

for $X, Y \in \mathcal{L}\left(\mathcal M_1\right)$ and $U, V \in \mathcal{L}\left(\mathcal M_2\right)$, where $\Delta$ is Laplacian operator and $\nabla$ is gradiant of the function.

3.   Riemannian warped-twisted product submersions

In this section, we define Riemannian warped-twisted product submersion and obtain some fruitful results.

Proposition 3.1. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be two warped-twisted product manifolds and let $\varphi_i: \mathcal{M}_i \rightarrow \aleph_i, i\in \{1, 2\}$ be Riemannian submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then the map

given by $\varphi\left(x_1, x_2\right) = \left(\varphi_1\left(x_1\right), \varphi_2\left(x_2\right)\right)$ is a Riemannian submersion, which is called warped-twisted product submersion.

Proof. Making use of Proposition 2 ^[3], it is easy to show that the map $\varphi$ is a Riemannian submersion. Now it is enough to show that the map $\varphi$ is a warped-twisted product. Since

It shows that $\varphi_{\ast}$ preserve the length of the horizontal vector field. Thus $\varphi$ is a warped-twisted product submersion.

Next, we obtain fundamental tensors for the Riemannian warped-twisted product submersion in the subsequent lemmas:

Lemma 3.1. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ are warped-twisted product manifolds and $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ is Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then we have

1) $T_{U_1}V_1 = T^{1}_{U_1}V_1-g_\mathcal M(U_1, V_1)\mathcal H\nabla\left(\ln \left(f_2\circ \varphi_2\right)\right),$

2) $T_{U_1}U_2 = 0,$

3) $T_{U_2}V_2 = T^{2}_{U_2}V_2-g_\mathcal M(U_2, V_2)\mathcal H\nabla\left(\ln \left(f_1\right)\right)$

for any $U_i, V_i\in \Gamma(\mathcal V_i)$, $i = \{1, 2\}$.

Proof. From Eq (2.3), we get

By using Eq (2.8), we obtain

Using Eq (2.3) in Eq (3.2) and combining the result with Eq (3.1), we get result 1).

By using Eq (2.3), we obtain

Making use of Eq (2.9), we have

Combining Eq (3.3) with Eq (3.4), we get

From Eq (2.1), we obtain

Using Eq (2.10), we get

From Eq (2.3), we know that

By using Eqs (3.6) and (3.7) in Eq (3.5), we get the desired result 3).

Lemma 3.2. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ are warped-twisted product manifolds and $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ is Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then we have

1) $\mathcal H\nabla_{X_1}Y_1 = \mathcal {H}\nabla^{1}_{X_1}Y_1-g_\mathcal M(X_1, Y_1)\mathcal{H}\nabla \left(\ln \left(f_2\circ \varphi_2\right)\right),$

$A_{X_1}Y_1 = A^{1}_{X_1}Y_1-g_\mathcal M(X_1, Y_1)\mathcal{V}\nabla \left(\ln \left(f_2\circ \varphi_2\right)\right),$

2) $\mathcal H\nabla_{X_1}X_2 = \mathcal H\nabla_{X_2}X_1 = X_2(\ln \left(f_2\circ \varphi_2)\right)X_1+X_1(\ln \left(f_1)\right)X_2,$

$A_{X_2}X_1 = 0 = A_{X_1}X_2,$

3) $A_{X_2}Y_2 = A^{2}_{X_2}Y_2 \quad and \quad \mathcal V \nabla \left(\ln \left(f_1\right)\right) = 0,$

$\mathcal H\nabla_{X_2}Y_2 = \mathcal H\nabla^2_{X_2}Y_2+X_2(\ln f_1)Y_2+Y_2(\ln f_1)X_2-g_\mathcal M(X_2, Y_2)\mathcal H\nabla \left(\ln \left(f_1\right)\right)$

for any $X_i, Y_i\in \Gamma(\mathcal H_i)$, $i = \{1, 2\}$.

Proof. From Eq (2.7), we have

From Eqs (2.8) and (3.8), we get

Separating the horizontal and vertical parts in Eq (3.9), we obtain

From Eq (2.7), we have

From Eq (2.9), we get

Combining Eqs (3.10)–(3.12), we obtain result 2). We know that

Using Eq (2.10) in the above equation, we get

and

Since $A$ and $A^2$ are skew-symmetric tensor fields and $g_\mathcal M$ is a symmetric tensor field, by using Eqs (3.13) and (3.14), we obtain the required result 3).

Lemma 3.3. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and let $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then we have

1) $\mathcal H\nabla_{V_1}X_1 = \mathcal H\nabla^{1}_{V_1}X_1 \quad and \quad T_{V_1}X_1 = T^{1}_{V_1}X_1,$

2) $T_{V_1}X_2 = X_2\left (\ln \left(f_2\circ \varphi_2\right)\right)V_1 = \mathcal V\nabla_{X_2}V_1 \quad and$

$A_{X_2}V_1 = V_1\left (\ln \left(f_1\right)\right)X_2 = \mathcal H\nabla_{V_2}X_1,$

3) $T_{V_2}X_1 = X_1\left (\ln \left(f_1\right)\right)V_2 = \mathcal V\nabla_{X_1}V_2 \quad and$

$A_{X_1}V_2 = V_2\left (\ln \left(f_2\circ \varphi_2\right)\right)X_1 = \mathcal H\nabla_{V_2}X_1,$

4) $T_{V_2}X_2 = T^{2}_{V_2}X_2+X_2(\ln f_1)V_2 \quad and$

$\mathcal H\nabla_{V_2}X_2 = \mathcal H\nabla^{2}_{V_2}X_2+V_2(\ln f_1)X_2$

for any $V_i \in \Gamma(\mathcal V_i)$, and $X_i\in \Gamma(\mathcal H_i)$ where $i = \{1, 2\}$.

Proof. For $V_1\in \Gamma (\mathcal V_i)$ and $X_1\in \Gamma(\mathcal H_i)$, by using Eq (2.4), we have

Making use of Eqs (2.8) and (2.4), we obtain

Combining Eqs (3.15) and (3.16) and comparing the vertical and the horizontal parts in the resulting expression, we get result 1).

From Eq (2.9), we have

From Eqs (2.4) and (2.5), we have

Combining Eqs (3.17)–(3.19) and comparing the vertical and the horizontal parts in the resulting expression, we obtain result 2). On a similar line, we get result 3).

Further, using Eqs (2.4) and (2.10), we obtain result 4).

Further, using the above lemmas, we obtained the fundamental geometric properties of Riemannian warped-twisted product submersion in the consequent theorems.

Theorem 3.1. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and let $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$ with $\operatorname{dim} \mathcal M_1 = m_1, \operatorname{dim}\mathcal M_2 = m_2, \operatorname{dim} \aleph_1 = n_1$ and $\operatorname{dim} \aleph_2 = n_2$. Then

(i) $\varphi$ has totally geodesic fibers if and only if $\varphi_1$ and $\varphi_2$ have totally geodesic fibers and $f_{1}$ and $f_{2}$ are constants,

(ii) The fundamental metric tensor $T$ satisfies the following inequality

with the equality holding if and only if $\varphi_1$ and $\varphi_2$ have totally geodesic fibers.

Proof. (ⅰ) Let $e_k \in \Gamma\left(\mathcal{V}_1\right)$ and $k = 1, \ldots, m_1-n_1$ and $e_c \in \Gamma\left(\mathcal{V}_2\right), c = m_1-n_1+$ $1, \ldots, m_1-n_1+m_2-n_2$ be orthonormal vectors of vertical spaces of submersion $\pi$. Then using Lemma (3.2), we have

(ⅱ) It follows from the above equation.

Theorem 3.2. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and let $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then $\varphi$ has totally umbilical fibers if and only if $\varphi_1$ and $\varphi_2$ have totally geodesic fibers and $\vec{\mathcal H}^\varphi = \mathcal{H}(\nabla \ln f_{1}) = \mathcal{H}(\nabla \ln f_{2})$, where $\vec{\mathcal H}^\varphi$ denotes the mean curvature of $\varphi$.

Proof. From Lemma (3.1) and the fact that $\varphi$ has totally umbilical fibers, we have

for any $U_i, V_i \in \Gamma\left(\mathcal{V}_i\right)$, $i\in \{1, 2\}$ which gives the following relation

Converse follow easily.

Theorem 3.3. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and let $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$. Then $\varphi$ has minimal fibers if and only if the mean curvature of $\varphi_1$ and $\varphi_2$ is given by $\vec{\mathcal H}_1 = \frac{m_2-n_2}{m_1-n_1} \mathcal{H}(\nabla^{1} \ln f_{1})$ and $\vec{\mathcal H}_2 = \frac{m_1-n_1}{m_2-n_2} \mathcal{H}(\nabla \ln f_{2})+\mathcal H(\nabla^{2} \ln f_1)$.

Proof. We suppose that $\varphi$ has minimal fibers for $\mathcal{M}$. Let $e_k \in \Gamma\left(\mathcal{V}_1\right)$ and $k = 1, \ldots, m_1-n_1$ and $e_c \in \Gamma\left(\mathcal{V}_2\right), c = m_1-n_1+1 \ldots, m_1-n_1+m_2-n_2$ be orthonormal frames of vertical spaces of submersion $\varphi$. Then using Eq (2.7) and Lemma (3.1), we have

Since, $\mathcal H(\nabla \ln f_2)\in \Gamma (\mathcal H_2)$ and $\mathcal H(\nabla \ln f_1)\in \Gamma (\mathcal H_1\times \mathcal H_2)$. So, we can write $\mathcal H(\nabla \ln f_1) = \mathcal H(\nabla^{1} \ln f_1)+\mathcal H(\nabla^{2} \ln f_1)$, where $\mathcal H(\nabla^{1} \ln f_1)\in \Gamma(\mathcal H_1)$ and $\mathcal H(\nabla^{2} \ln f_1)\in \Gamma (\mathcal H_2)$. Then, we obtain $\vec{\mathcal H}_1 = \frac{m_2-n_2}{m_1-n_1} \mathcal{H}(\nabla^{1} \ln f_{1})$ and $\vec{\mathcal H}_2 = \frac{m_1-n_1}{m_2-n_2} \mathcal{H}(\nabla \ln f_{2})+ \mathcal H(\nabla^{2} \ln f_1)$. Converse follows quickly from the above relation.

4.   Riemannian warped-twisted product submersion as an Einstein manifold

In this section, we obtain the Ricci tensor for Riemannian warped-twisted product submersion. Further, we discuss Einstein's condition on vertical and horizontal spaces of $\mathcal M_i$ for Riemannian warped-twisted product submersion.

Definition 4.1. ^[2]. A Riemannian manifold $(\mathcal M, g_\mathcal M)$ is called an Einstein manifold if

where $\lambda$ is a real constant, and $\mathbb S$ is the Ricci tensor on $\mathcal M$.

Making use of Lemma 2.2, we have

Lemma 4.1. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$ with $dim\mathcal M_i = m_i$ and $dim\aleph_i = n_i$; $i\in \{1, 2\}$. Let $\mathbb{S}^1$ and $\mathbb{S}^{2}$ be the lifts of the Ricci curvatures on $\mathcal{M}_1$ and $\mathcal{M}_2$, respectively. Then for any $X_i, Y_i\in \Gamma(\mathcal H_{i})$ and $U_i, V_i\in \Gamma(\mathcal V_{i})$, we have following relations:

1) $\mathbb S(X_1, U_1) = (1-m_2)X_1U_1(\ln f_1)+(m_1+m_2-2)X_1(\ln f_1)U_1(\ln f_2),$

2) $\mathbb S(X_1, X_2) = (1-m_2)X_1X_2(\ln f_1)+(m_1+m_2-2)X_1(\ln f_1)X_2(\ln f_2),$

3) $\mathbb S(X_1, Y_1) = \mathbb S^{1}(X_1, Y_1)+h^{\ln f_2}(X_1, Y_1)-m_2\{h_1^{\ln f_1}(X_1, Y_1)+X_1(\ln f_1)Y_1(\ln f_1)\}$ -$g_{\mathcal M}(X_1, Y_1)\{\Delta \ln f_2+g_{\mathcal M}(\nabla \ln f_2, \nabla \ln f_2)\},$

4) $\mathbb S(X_2, Y_2) = \mathbb S^{2}(X_2, Y_2)+h^{\ln f_1}(X_2, Y_2)+(1-m_2)h_{2}^{\ln f_1}(X_2, Y_2)+m_2X_2(\ln f_1)Y_2(\ln f_1)$-$g_{\mathcal M}(X_2, Y_2)\{\Delta \ln f_1+g_{\mathcal M}(\nabla \ln f_1, \nabla \ln f_1)\}-m_1\{h_{2}^{\ln f_2}(X_2, Y_2)+X_2(\ln f_2)Y_2(\ln f_2)-X_2(\ln f_2)Y_2(\ln f_1)-X_2(\ln f_1)Y_2(\ln f_2)\},$

5) $\mathbb S(X_1, U_2) = (1-m_2)X_1U_2(\ln f_1)+(m_1+m_2-2)X_1(\ln f_1)U_2(\ln f_2),$

6) $\mathbb S(X_2, U_1) = (1-m_2)U_1X_2(\ln f_1)+(m_1+m_2-2)U_1(\ln f_1)X_2(\ln f_2),$

7) $\mathbb S(X_2, U_2) = (1-m_2)X_2U_2(\ln f_1)+(m_1+m_2-2)X_2(\ln f_1)U_2(\ln f_2),$

8) $\mathbb S(U_1, V_1) = \mathbb S^{1}(U_1, V_1)+h^{\ln f_2}(U_1, V_1)-m_2\{h_1^{\ln f_1}(U_1, V_1)+U_1(\ln f_1)V_1(\ln f_1)\}$-$g_{\mathcal M}(U_1, V_1)\{\Delta \ln f_2+g_{\mathcal M}(\nabla \ln f_2, \nabla \ln f_2)\},$

9) $\mathbb S(U_1, U_2) = (1-m_2)U_1U_2(\ln f_1)+(m_1+m_2-2)U_1(\ln f_1)U_2(\ln f_2),$

10) $\mathbb S(U_2, V_2) = \mathbb S^{2}(U_2, V_2)+h^{\ln f_1}(U_2, V_2)+(1-m_2)h_{2}^{\ln f_1}(U_2, V_2)+m_2U_2(\ln f_1)V_2(\ln f_1)$-$g_{\mathcal M}(U_2, V_2)\{\Delta \ln f_1+g_{\mathcal M}(\nabla{\ln f_1}, \nabla \ln f_1)\}-m_1\{h_{2}^{\ln f_2}(U_2, V_2)+U_2(\ln f_2)V_2(\ln f_2)$-$U_2(\ln f_2)V_2(\ln f_1)-U_2(\ln f_1)V_2(\ln f_2)\}.$

Now, we study Einstein conditions for the horizontal and vertical distributions of Riemannian warped-twisted product submersion.

Theorem 4.1. Let $\mathcal{M} = {}_{f_{2}}\mathcal{M}_{1}\times _{f_{1}}\mathcal{M}_{2}$ and $\aleph = {}_{\rho_{2}}\aleph_{1}\times _{\rho_{1}}\aleph_{2}$ be warped-twisted product manifolds and $\varphi_{i}: \mathcal{M}_{i} \rightarrow \aleph_{i}$ be Riemannian warped-twisted product submersion between the manifolds $\mathcal{M}_{i}$ and $\aleph_{i}$ with $dim\mathcal M_i = m_i$ and $dim\aleph_i = n_i$; $i\in \{1, 2\}$. Then

$(i)$ If the vertical space $\mathcal V_1$ $($or horizontal space $\mathcal H_1$$)$ of $\mathcal M$ is Einstein, then vertical space of $\mathcal M_1$ $($resp. horizontal space $\mathcal H_1$ $)$ is Einstein assuming that $h^{\ln f_1}_1, h^{\ln f_2}$ and, $U_1(\ln f_1)V_1(\ln f_1)$ is proportional to constant times the metric $g_{\mathcal M}$ and $\Delta \ln f_2+g_{\mathcal M}(\nabla \ln f_2, \nabla \ln f_2)$ is constant,

$(ii)$ If the vertical space $\mathcal V_2$ $($or horizontal space $\mathcal H_2$$)$ of $\mathcal M$ is Einstein, then vertical space of $\mathcal M_2$ $($resp. horizontal space $\mathcal H_2$$)$ is Einstein assuming that $h^{\ln f_2}_2, h^{\ln f_1}_2, h^{\ln f_1}$ $U_2(\ln f_1)V_2(\ln f_1)$ and $U_2(\ln f_2)V_2(\ln f_2)-U_2(\ln f_2)V_2(\ln f_1)-U_2(\ln f_1)V_2(\ln f_2)\}$ are proportional to the metric $g_{\mathcal M}$ and, $\{\Delta \ln f_1+g_{\mathcal M}(\nabla \ln f_1, \nabla \ln f_1)\}$ is constant.

Proof. $(i)$ Suppose, $\mathcal V_1$ of $\mathcal M$ is Einstein's manifold. Then, from Eq (4.1) and Lemma (4.1) we have

Now if $h^{\ln f_1}_1, h^{\ln f_2}$ and $U_1(\ln f_1)V_1(\ln f_1)$ is proportional to constant times the metric $g_{\mathcal M}$ and $\Delta \ln f_2+g_{\mathcal M}(\nabla \ln f_2, \nabla \ln f_2)$ are constants, then $\mathcal{M}_{1}$ is Einstein's manifold.

$(ii)$ For vertical space $\mathcal V_2$ of $\mathcal M$, using definition of Einstein's manifold and relation $10$ of Lemma (4.1), we obtain the required result.

5.   Conclusions

The Einstein equations are of significant importance within the framework of the general theory of relativity, as they form the foundation for the gravitational and cosmological models. The Einstein equation can be expressed as $\mathbb {S} = \lambda g$, which is a non-linear second-order system of differential equations. In the context of this system, the symbol $\lambda$ is referred to as the Einstein constant, whereas physicists commonly refer to it as the cosmological constant ^[2]. The geometric properties of warped-twisted products encompass a broader class than both warped products and twisted products. This class exhibits a multitude of applications, not only within the realm of geometry but also in the field of theoretical physics. The exact solution of Einstein's equations exhibits a warped product structure. The technique of Riemannian submersion is commonly employed in the construction of Riemannian manifolds showing positive sectional curvature. Additionally, it is employed in the construction of widely recognized instances of Einstein manifolds. Riemannian submersion finds extensive application within the domains of Kaluza-Klein theory, Yang-Mills theory, superstring and supergravity theory in physics ^[24,25,26]. The objective of this study is to decompose the Einstein's equation for warped-twisted product into their constituent elements, specifically the base and fiber components.

6.   Remark

In ^[8,23], H. M. Tastan and S. G. Aydin considered Einstein like warped-twisted product manifolds and warped-twisted product semi-slant submanifolds. On the other hand, I. K. Erken and C. Murathan ^[3] studied Riemannian warped product submersion. In this paper, we investigated Riemannian warped-twisted product submersions. In light of these studies, it could be interesting to work on warped product semi-slant submersions and warped-twisted product semi-slant submersions.

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 are really thankful to the learned reviewers for their careful reading of our manuscript and their many insightful comments and suggestions that have improved the quality of our manuscript. The author Fatemah Mofarreh, expresses her gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.

Conflict of interest

The authors declare no conflicts of interest.