Geometry of Bi-Warped Product Submanifolds of Nearly Trans-Sasakian Manifolds

Abstract

In the present work, we consider two types of bi-warped product submanifolds, $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ and $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.

1. Introduction

Let $\mathbf{M}={\mathbf{M}}_1\times {\mathbf{M}}_2\times {\mathbf{M}}_3\times \cdots \times {\mathbf{M}}_k$ be the Cartesian product of Riemannian manifolds ${\mathbf{M}}_1,{\mathbf{M}}_2,\cdots ,{\mathbf{M}}_k$ and let ${\pi }_i:\mathbf{M}⟶{\mathbf{M}}_k$ denote the canonical projection maps for $i=1,\cdots ,k$. If the positive-valued functions $f_1,\cdots ,f_k$ are defined such that $f_1,\cdots ,f_k:{\mathbf{M}}_1⟶(0,\infty )$, then the Riemannian metric g is defined as

$$\begin{array}{c}\hfill g(X,Y)=g({\pi }_{1*}X,{\pi }_{1*}Y)+\sum _{i=1}^k\left(f_i\circ \pi \right)g({\pi }_{i*}X,{\pi }_{i*}Y),\end{array}$$

where * is the symbol for tangent map, for any $X,Y$ tangent to $\mathbf{M}$, then $\mathbf{M}$ is called a multiple warped product manifold [1,2]. If we choose two fibers of a multiple warped product ${\mathbf{M}}^1{\times }_{f_1}{\mathbf{M}}^2\times \cdots {\times }_{f_k}{\mathbf{M}}^k$, such that $\mathbf{M}={\mathbf{M}}_1{\times }_{f_1}{\mathbf{M}}_2{\times }_{f_2}{\mathbf{M}}_3$, then $\mathbf{M}$ is defined as a bi-warped product submanifold, which satisfies the following:

$$\begin{array}{c}\hfill {\nabla }_{{\mathbb{U}}_1}\mathbb{Z}=\sum _{i=2}^3({\mathbb{U}}_{1}ln{f}_i){\mathbb{Z}}_i,\end{array}$$

(1)

where ${\mathbb{U}}_1\in \mathsf{\Gamma }\left(T{\mathbf{M}}_1\right)$, $\mathbb{Z}\in \mathsf{\Gamma }\left(T({\mathbf{M}}_2\times {\mathbf{M}}_3)\right)$ and ${\mathbb{Z}}_i$ tangent to ${\mathbf{M}}_i$, for each $i=2,3$. Moreover, ∇ is the Levi–Civita connection and for more details, see [3,4,5]. An odd-dimensional analog of the nearly Kähler metric is the nearly Sasakian metric. The nearly Kähler cone over a nearly Sasakian Einstein manifold has applications in physics. The Sasakian geometry has been extensively studied, due to its recently perceived relevance in string theory. Sasakian Einstein metrics have received a lot of attention in physics—for example, related to p-brane solutions in superstring theory and the Maldacena conjecture [6]. On the other hand, the vanishing of Dirichlet energies is equivalent to the Dirichlet condition with the unique solution of Poisson’s equation ${\nabla }^{2}f=-4f\rho .$ This implies Neumann or Dirichlet boundary conditions, as classified by the electrostatic problem (see, e.g., [7]). In the present paper, we consider the bi-warped product submanifolds in a nearly trans-Sasakian manifold, inspired by the publication of Taştan’s seminal work [5], and obtained some inequalities for the Dirichlet energy and the second fundamental form. The study of bi-warped product submanifolds with two distinct fibers has been a topic of great interest; see, e.g., Naghi et al. [4], Ali et al. [8], Siraj et al. [9,10,11], and Awatif et al. [12]. It has been noted that the class of bi-warped product submanifolds is a generalization of several classes, such as CR-warped products, warped product semi-slant submanifolds, and warped product pseudo-slant submanifolds. On the other hand, as a generalization of nearly cosymplectic, nearly Sasakian [13], nearly Kenmotsu [8,14], nearly $\alpha$-Sasakian, and nearly $\beta$-Kenmotsu manifolds, nearly trans-Sasakian manifolds have been studied on a large scale; see [15,16,17,18,19]. Therefore, our objective was to remove the gap in the nearly trans-Sasakian manifold literature, as they are an interesting structure of the almost contact manifolds that have generalized many others structures. The main goal of this paper was to discuss the geometry of bi-warped product submanifolds of the types ${\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ and $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ in a nearly trans-Sasakian manifold. Some inequalities for the length of the second fundamental form are obtained, including the length of warping functions and slant immersions. Various inequalities, which have been derived in [13,14,18,19,20,21,22,23,24,25,26,27], can be recovered from our inequalities under some conditions. Therefore, our results may find applications in mathematical physics.

2. Preliminaries

An odd-dimensional $C^{\infty }$-manifold $(\tilde{\mathbf{M}},g)$ associated with an almost contact structure $(\psi ,\zeta ,\eta )$ is referred to as an almost contact metric manifold if there exist a $(1,1)$ tensor field $\psi$, a vector field $\zeta$ (called a characteristic or Reeb vector field), and a 1-form $\eta$ satisfying the following conditions:

$$\begin{array}{c}\hfill {\psi }^2=-I+\eta \otimes \zeta ,\eta \left(\zeta \right)=1,\psi \left(\zeta \right)=0,\eta \circ \psi =0,\end{array}$$

(2)

$$\begin{array}{c}\hfill g(\psi {\mathbb{U}}_1,\psi {\mathbb{W}}_2)=g({\mathbb{U}}_1,{\mathbb{W}}_2)-\eta \left({\mathbb{U}}_1\right)\eta \left({\mathbb{W}}_2\right),\eta \left({\mathbb{U}}_1\right)=g({\mathbb{U}}_1,\zeta ),\end{array}$$

(3)

∀${\mathbb{U}}_1,{\mathbb{W}}_2\in \mathsf{\Gamma }\left(T\tilde{\mathbf{M}}\right)$. The above structure $(\psi ,\eta ,\zeta )$ can be reduced to a nearly trans-Sasakian manifold (cf. [18,19]) if the following holds:

$$\begin{array}{cc}\hfill ({\tilde{\nabla }}_{{\mathbb{U}}_1}\psi ){\mathbb{V}}_1+({\tilde{\nabla }}_{{\mathbb{V}}_1}\psi ){\mathbb{U}}_1=& \alpha \left(2g({\mathbb{U}}_1,{\mathbb{V}}_1)\xi -\eta \left({\mathbb{U}}_1\right){\mathbb{V}}_1-\eta \left({\mathbb{V}}_1\right){\mathbb{U}}_1\right)\hfill \\ \hfill & -\beta \left(\eta \left({\mathbb{V}}_1\right)\psi {\mathbb{U}}_1+\eta \left({\mathbb{U}}_1\right)\psi {\mathbb{V}}_1\right),\hfill \end{array}$$

(4)

for any ${\mathbb{U}}_1,{\mathbb{V}}_1\in \mathsf{\Gamma }\left(T\tilde{\mathbf{M}}\right)$, where is the Riemannian connection associated with the metric g on $\tilde{\mathbf{M}}$. According to the structure, we have the following classifications:

(i)

A nearly trans-Sasakian $\tilde{\mathbf{M}}$ is a nearly cosymplectic if $\alpha =0and,\beta =0$ in (4).

(ii)

If $\alpha =1$ and $\beta =0$ in (4), then $\tilde{\mathbf{M}}$ is a nearly Sasakian manifold under the condition

$$\begin{array}{c}\hfill ({\tilde{\nabla }}_{{\mathbb{U}}_1}\psi ){\mathbb{V}}_1+({\tilde{\nabla }}_{{\mathbb{V}}_1}\psi ){\mathbb{U}}_1=2g({\mathbb{U}}_1,{\mathbb{V}}_1)\xi -\eta \left({\mathbb{V}}_1\right){\mathbb{U}}_1-\eta \left({\mathbb{U}}_1\right){\mathbb{V}}_1.\end{array}$$

(5)

(iii)

A nearly trans-Sasakian $\tilde{\mathbf{M}}$ is a nearly Kenmotsu manifold if $\alpha =0$ and $\beta =1$ in (4).

(iv)

Similarly, nearly α-Sasakian and nearly β-Kenmotsu manifolds can be defined from a nearly trans-Sasakian manifold by substituting $\beta =0$ and $\alpha =0$ in (4), respectively.

The Gauss and Weingarten formulas, which specify the relation between Levi–Civita connections ∇ on a submanifold $\mathbf{M}$ and $\tilde{\nabla }$ on an ambient manifold $\tilde{\mathbf{M}}$, are given by (for more detail, see [28]):

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathbb{U}}_1}{\mathbb{W}}_2& ={\nabla }_{{\mathbb{U}}_1}{\mathbb{W}}_2+\mathbf{B}({\mathbb{U}}_1,{\mathbb{W}}_2),\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{{\mathbb{U}}_1}\xi & =-A_{\xi }{\mathbb{U}}_1+{\nabla }_{{\mathbb{U}}_1}^{\perp }\xi ,\hfill \end{array}$$

(7)

for every ${\mathbb{U}}_1,{\mathbb{W}}_2\in \mathsf{\Gamma }\left(T\mathbf{M}\right)$ and $\xi \in \mathsf{\Gamma }\left(T^{\perp }\mathbf{M}\right)$. In addition, $\mathbf{B}$ and $A_{\xi }$ are the second fundamental form and shape operator, respectively, having the relation $g(\mathbf{B}({\mathbb{U}}_1,{\mathbb{W}}_2),\xi )=g(A_{\xi }{\mathbb{U}}_1,{\mathbb{W}}_2).$ If we replace ${\mathbb{U}}_1=\xi ,{\mathbb{V}}_1=\xi$ in (4), we find that $\left({\tilde{\nabla }}_{\xi }\psi \right)\xi =0$, which implies that $\psi {\tilde{\nabla }}_{\xi }\xi =0$. Applying $\psi$ and using (4), we get ${\tilde{\nabla }}_{\xi }\xi =0$. From the Gauss formula, we get ${\nabla }_{\xi }\xi =0$ and $h(\xi ,\xi )=0$. For more classification, see [18,19]. We also have:

$$\begin{array}{c}\hfill \psi {\mathbb{U}}_1=\mathbf{T}{\mathbb{U}}_1+\mathbf{F}{\mathbb{U}}_1,\end{array}$$

(8)

in which $\mathbf{F}{\mathbb{U}}_1$ and $\mathbf{T}{\mathbb{U}}_1$ are normal and tangential elements of $\psi {\mathbb{U}}_1$, respectively. If $\mathbf{M}$ is invariant and/or anti-invariant, then $\mathbf{F}{\mathbb{U}}_1$ and/or $\mathbf{T}{\mathbb{U}}_1$ are zero, respectively. Similarly, we have

$$\begin{array}{c}\hfill \psi \xi =t\xi +f\xi ,\end{array}$$

(9)

where $t\xi$ (respectively $f\xi$) are tangential (respectively normal) components of $\psi \xi$. The covariant derivative of the endomorphism $\psi$ is defined by

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_5={\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_5-\psi {\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_5,\forall {\mathbb{X}}_0,{\mathbb{X}}_5\in \mathsf{\Gamma }\left(T\mathbf{M}\right).\end{array}$$

(10)

There is a motivating class of submanifolds, presented as slant submanifolds. For any non-zero vector $U_1$ tangential to $\mathbf{M}$ at point p, such that $U_1$ is not proportional to ${\zeta }_p$, $0\le \varphi \left(U_1\right)\le \pi /2$ refers to the angle between $\psi U_1$ and $T_p\mathbf{M}$, which is named the Wirtinger angle. If $\varphi \left(U_1\right)$ is constant for any $U_1\in T_p\mathbf{M}-<\zeta >$ at point $p\in \mathbf{M}$, then $\mathbf{M}$ is referred to as the slant submanifold [29] and $\varphi$ is the slant angle of $\mathbf{M}$. We consider the following necessary and sufficient for a submanifold $\mathbf{M}$ to be a slant submanifold [29,30]:

$$\begin{array}{c}\hfill {\mathbf{T}}^2={cos}^2\varphi (-I+\eta \otimes \zeta ),\end{array}$$

(11)

for $0\le \varphi \le \frac{\pi }{2}$ and $\mathbf{T}$ is an endomorphism defined in (8). The following result is from Equation (11):

$$\begin{array}{c}\hfill g(\mathbf{T}{\mathbb{U}}_1,\mathbf{T}{\mathbb{W}}_2)={cos}^2\varphi \left\{g({\mathbb{U}}_1,{\mathbb{W}}_2)-\eta \left({\mathbb{U}}_1\right)\eta \left({\mathbb{U}}_1\right)\right\},\end{array}$$

(12)

$$\begin{array}{c}\hfill g(\mathbf{F}{\mathbb{U}}_1,\mathbf{F}{\mathbb{W}}_2)={sin}^2\varphi \left\{g({\mathbb{U}}_1,{\mathbb{W}}_2)-\eta \left({\mathbb{U}}_1\right)\eta \left({\mathbb{W}}_2\right)\right\},\end{array}$$

(13)

∀${\mathbb{U}}_1,{\mathbb{W}}_2\in \mathsf{\Gamma }\left(T\mathbf{M}\right)$. The following are the definitions of semi-slant and psuedo-slant submanifolds.

Definition 1.

A submanifold $\mathbf{M}$ of an almost contact metric manifold $\tilde{\mathbf{M}}$ is said to be a semi-slant submanifold if there exists a pair of orthogonal distributions, $\mathbf{D}$ and ${\mathbf{D}}^{\varphi }$, on M such that

(i)

the tangent bundle TM admits the orthogonal direct decomposition $T\mathbf{M}=\mathbf{D}\oplus {\mathbf{D}}^{\varphi }\oplus \zeta$;

(ii)

the distribution $\mathbf{D}$ is invariant under ψ (i.e., $\psi \mathbf{D}=\mathbf{D}$); and

(iii)

the distribution ${\mathbf{D}}^{\varphi }$ is slant, with slant angle ϕ.

Definition 2.

A submanifold $\mathbf{M}$ of an almost contact metric manifold $\tilde{\mathbf{M}}$ is said to be a pseudo-slant submanifold if there exists a pair of orthogonal distributions, $\mathbf{D}$ and ${\mathbf{D}}^{\varphi }$, on $\mathbf{M}$ such that

(i)

the tangent bundle TM admits the orthogonal direct decomposition $T\mathbf{M}={\mathbf{D}}^{\perp }\oplus {\mathbf{D}}^{\varphi }\oplus \zeta$;

(ii)

the distribution ${\mathbf{D}}^{\perp }$ is totally real under ψ (i.e., $\psi {\mathbf{D}}^{\perp }\subset T^{\perp }\mathbf{M}$); and

(iii)

the distribution ${\mathbf{D}}^{\varphi }$ is slant, with slant angle ϕ.

The mean curvature vector, $\mathcal{H}$, for an orthonormal frame $\{e_1,e_2,\cdots ,e_n\}$ of the tangent space $T\mathbf{M}$ on ${\mathbf{M}}^n$ is defined by

$$\begin{array}{c}\hfill \mathcal{H}=\frac{1}{n}trace\left(\mathbf{B}\right)=\frac{1}{n}\sum _{i=1}^n\mathbf{B}\left(e_i,e_i\right),\end{array}$$

(14)

where $n=dimM$. In addition, we set

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2=\sum _{\alpha ,\beta =1}^{n}g(\mathbf{B}(e_{\alpha },e_{\beta }),\mathbf{B}(e_{\alpha },e_{\beta })),\end{array}$$

(15)

where ${\left|\right|\mathbf{B}\left|\right|}^2$ is the length of the second fundamental form.

For a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, a bi-warped product submanifold $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ in $\tilde{\mathbf{M}}$, where ${\mathbf{M}}_T,{\mathbf{M}}_{\perp },{\mathbf{M}}_{\varphi }$ refer to holomorphic, totally real, and proper slant submanifolds of $\tilde{\mathbf{M}}$, respectively. Suppose that $T\mathbf{M}=\mathbf{D}\oplus {\mathbf{D}}^{\perp }\oplus {\mathbf{D}}^{\varphi }$ and $T^{\perp }\mathbf{M}=\psi {\mathbf{D}}^{\perp }\oplus \mathbf{F}{\mathbf{D}}^{\varphi }\oplus \mu$, where $\mu$ is classified as a $\psi$-invariant normal sub-bundle of $T^{\perp }\mathbf{M}$. The next practical consequence will be used later in this paper.

Proposition 1.

Let $\mathbf{M}={\mathbf{M}}_1{\times }_{f_1}{\mathbf{M}}_2{\times }_{f_2}{\mathbf{M}}_3$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}.$ Then, $\mathbf{M}$ is only a single warped product if the structure vector field ζ is tangent to the fiber; that is, either $\zeta \in \mathsf{\Gamma }\left({\mathbf{D}}_{\mathbf{2}}\right)$ or $\zeta \in \mathsf{\Gamma }\left({\mathbf{D}}_3\right).$

Proof. 

If we take $\zeta \in \mathsf{\Gamma }\left({\mathbf{D}}_{\mathbf{2}}\right),$ for any $\mathbb{U}\in \mathsf{\Gamma }\left({\mathbf{D}}_{\mathbf{2}}\right),$ we then get

Making use of (1), we obtain

$$\begin{array}{c}\hfill \mathbb{U}(ln{f}_1)\xi =\beta U.\end{array}$$

By taking the inner product with $\xi$ and using the fact that $\xi \in \mathsf{\Gamma }\left({\mathbf{D}}_{\mathbf{2}}\right),$ we have

$$\begin{array}{c}\hfill \mathbb{U}(ln{f}_1)=0,\end{array}$$

where $f_1$ is a constant. Similarly, we can easily obtain that

$$\begin{array}{c}\hfill \mathbb{U}(ln{f}_2)=0,\end{array}$$

which implies that $f_2$ is constant. □

Proposition 2.

Let $\mathbf{M}={\mathbf{M}}_1{\times }_{f_1}{\mathbf{M}}_2{\times }_{f_2}{\mathbf{M}}_3$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}.$ Then,

$$\begin{array}{c}\hfill \zeta (ln{f}_a)=\beta ,\forall a=1,2,\end{array}$$

(16)

for each structure vector field ζ is tangent to ${\mathbf{M}}_1.$

Proof. 

For any ${\mathbb{X}}_0\in \mathsf{\Gamma }\left({\mathbf{D}}_1\right),$ we expand the following:

$$\begin{array}{c}\hfill {\tilde{\nabla }}_{{\mathbb{X}}_0}\zeta =\beta {\mathbb{X}}_0.\end{array}$$

Using (1) and (8), we obtain

$$\begin{array}{c}\hfill \zeta (ln{f}_1){\mathbb{X}}_0=\beta {\mathbb{X}}_0.\end{array}$$

Taking the inner product with ${\mathbb{X}}_0$, we obtain

$$\begin{array}{c}\hfill \zeta (ln{f}_1)=\beta .\end{array}$$

Similarly, we can find that

$$\begin{array}{c}\hfill \zeta (ln{f}_2)=\beta .\end{array}$$

Therefore, the statement is proved. □

Remark 1.

It can be noticed that, if the structure vector field ζ is tangent to any fiber, then the warped product will be trivial due to Proposition 1. On the other hand, Proposition 2 assures that the warped product is always non-trivial if the structure vector field ζ is tangent to any base manifold of a nearly trans-Sasakian manifold.

Lemma 1.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$. Therefore, we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)& =0,\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =0,\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)& =-\left(\psi {\mathbb{X}}_0(ln{f}_1)+\alpha \eta \left({\mathbb{X}}_0\right)\right)g({\mathbb{X}}_2,{\mathbb{X}}_3),\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)& =\left({\mathbb{X}}_0(ln{f}_1)-\eta \left({\mathbb{X}}_0\right)\right)g({\mathbb{X}}_2,{\mathbb{X}}_3),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\zeta ,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)& =-\alpha g({\mathbb{X}}_2,{\mathbb{X}}_3),\hfill \end{array}$$

(21)

for any ${\mathbb{X}}_0,{\mathbb{X}}_1\in \mathsf{\Gamma }\left(\mathbf{D}\right)$, ${\mathbb{X}}_2,{\mathbb{X}}_3\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$ and ${\mathbb{X}}_4\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right).$

Proof. 

For all ${\mathbb{X}}_0,{\mathbb{X}}_1\in \mathsf{\Gamma }\left(\mathbf{D}\right)$ and ${\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$, we have

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)=g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_1,\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_1,{\mathbb{X}}_2)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_1,{\mathbb{X}}_2).\end{array}$$

Equation (1) gives the following:

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_1,{\mathbb{X}}_2)+{\mathbb{X}}_0(ln{f}_1)g(\psi {\mathbb{X}}_1,{\mathbb{X}}_2).\end{array}$$

The normality of vector fields implies that

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_1,{\mathbb{X}}_2).\end{array}$$

(22)

Replacing ${\mathbb{X}}_0$ with ${\mathbb{X}}_1$ in (22) gives

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_1}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_2).\end{array}$$

(23)

Combining Equations (22) and (23) results in

$$\begin{array}{c}\hfill 2g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_1+\left({\tilde{\nabla }}_{{\mathbb{X}}_1}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_2).\end{array}$$

Then, using Equation (4), we arrive at the first part (17). The second part, (18), can be obtained through a similar process as the first part. For the next part, we have:

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)=g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_0,\psi {\mathbb{X}}_3)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_3)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi {\mathbb{X}}_0,{\mathbb{X}}_3)\end{array}$$

for all ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right)$ and ${\mathbb{X}}_2,{\mathbb{X}}_3\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$. From (4), (6), and (1), we obtain

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\psi W)=& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_2,{\mathbb{X}}_3)+\alpha \{g(\zeta ,{\mathbb{X}}_3)g({\mathbb{X}}_0,{\mathbb{X}}_2)\hfill \\ \hfill & -\eta \left({\mathbb{X}}_2\right)g({\mathbb{X}}_0,{\mathbb{X}}_3)-\eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_2,{\mathbb{X}}_3)\}\hfill \\ \hfill & -\beta \left\{\eta \left({\mathbb{X}}_0\right)g(\psi {\mathbb{X}}_2,{\mathbb{X}}_3)+\eta \left({\mathbb{X}}_2\right)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_3)\right\}-\psi {\mathbb{X}}_0(ln{f}_1)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill =& -g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_2,{\mathbb{X}}_3)+g(\psi {\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_2,{\mathbb{X}}_3)-\psi {\mathbb{X}}_0(ln{f}_1)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_2,{\mathbb{X}}_3).\hfill \end{array}$$

This implies the following:

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)=& g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)-\psi {\mathbb{X}}_0(ln{f}_1)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_2,{\mathbb{X}}_3).\hfill \end{array}$$

(24)

Replacing ${\mathbb{X}}_2$ with ${\mathbb{X}}_3$ in (24) results in

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)=& g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)-\psi {\mathbb{X}}_0(ln{f}_1)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_2,{\mathbb{X}}_3),\hfill \end{array}$$

(25)

which implies Equation (19) from (24) and (25). Now, replacing ${\mathbb{X}}_0$ with $\psi {\mathbb{X}}_0$ in (19) and using (2), we obtain (20). If we substitute ${\mathbb{X}}_0=\zeta$ in (19), we finally reach (21). This completes the proof of the lemma. □

Lemma 2.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$. Then, we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_4)& =\frac{1}{2}g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)=0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =\frac{1}{3}\left\{({\mathbb{X}}_{0}ln{f}_2)-\beta \eta \left({\mathbb{X}}_0\right)\right\}g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -\left\{\psi {\mathbb{X}}_0(ln{f}_2)+\alpha \eta \left({\mathbb{X}}_0\right)\right\}g({\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =\frac{1}{3}(\psi {\mathbb{X}}_{0}ln{f}_2)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)+\{{\mathbb{X}}_0(ln{f}_2)-\eta \left({\mathbb{X}}_0\right)\}g({\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}\mathbf{T}{\mathbb{X}}_5)& =\frac{1}{3}\left\{({\mathbb{X}}_{0}ln{f}_2)-\beta \eta \left({\mathbb{X}}_0\right)\right\}{cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -\left\{\psi {\mathbb{X}}_0(ln{f}_2)+\alpha \eta \left({\mathbb{X}}_0\right)\right\}g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5),\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}\mathbf{T}{\mathbb{X}}_5)& =\frac{1}{3}(\psi {\mathbb{X}}_{0}ln{f}_2){cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & +\left\{{\mathbb{X}}_0(ln{f}_2)-\eta \left({\mathbb{X}}_0\right)\right\}g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5),\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =-\frac{1}{3}\left\{({\mathbb{X}}_{0}ln{f}_2)-\beta \eta \left({\mathbb{X}}_0\right)\right\}{cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -\left\{\psi {\mathbb{X}}_0(ln{f}_2)+\alpha \eta \left({\mathbb{X}}_0\right)\right\}g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =-\frac{1}{3}(\psi {\mathbb{X}}_{0}ln{f}_2){cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & +\left\{{\mathbb{X}}_0(ln{f}_2)-\eta \left({\mathbb{X}}_0\right)\right\}g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\mathbf{F}\mathbf{T}{\mathbb{X}}_5)& =-\frac{1}{3}\left\{{\mathbb{X}}_0(ln{f}_2)-\beta \eta \left({\mathbb{X}}_0\right)\right\}{cos}^2\varphi g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill & -\left\{\psi {\mathbb{X}}_0(ln{f}_2)+\alpha \eta \left({\mathbb{X}}_0\right)\right\}{cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\mathbf{F}\mathbf{T}{\mathbb{X}}_5)& =-\frac{1}{3}(ln{f}_2){cos}^2\varphi g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill & +\left\{{\mathbb{X}}_0(ln{f}_2)-\eta \left({\mathbb{X}}_0\right)\right\}{cos}^2\varphi g({\mathbb{X}}_4,{\mathbb{X}}_5),\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\zeta ,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =\frac{1}{3}(\zeta ln{f}_2)g(\mathbb{T}{\mathbb{X}}_4,{\mathbb{X}}_5)-\alpha g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \end{array}$$

(35)

for all ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right),{\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$ and ${\mathbb{X}}_4,{\mathbb{X}}_5\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right).$

Proof. 

For all ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right),{\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$ and ${\mathbb{X}}_4\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right)$, from which the following can be derived:

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_4)=& g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_0,\psi {\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4)\hfill \\ \hfill =& g(\left({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi {\mathbb{X}}_0,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4)\hfill \\ \hfill =& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_2,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_0\right)g(\psi {\mathbb{X}}_2,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_2\right)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_4)\hfill \\ \hfill & -\psi {\mathbb{X}}_{0}ln{f}_{1}g({\mathbb{X}}_2,{\mathbb{X}}_4)-{\mathbb{X}}_{0}ln{f}_{1}g({\mathbb{X}}_2,\mathbf{T}{\mathbb{X}}_4).\hfill \end{array}$$

By the use of (4), (10), and the normality of vector fields, we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_4)=& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_2,{\mathbb{X}}_4)\hfill \\ \hfill =& -g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_2,{\mathbb{X}}_4)+g(\psi {\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_2,{\mathbb{X}}_4)\hfill \\ \hfill =& g(A_{\psi {\mathbb{X}}_2}{\mathbb{X}}_0,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_2,\mathbf{T}{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_2,\mathbf{F}{\mathbb{X}}_4)\hfill \\ \hfill =& g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)-{\mathbb{X}}_{0}ln{f}_{1}g({\mathbb{X}}_2,\mathbf{T}{\mathbb{X}}_4)-g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),F{\mathbb{X}}_4).\hfill \end{array}$$

Therefore, using (8)–(10) results in

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_4)=\frac{1}{2}g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2),\end{array}$$

(36)

and we reached (i), as desired. Additionally, we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)=& g\left({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0\right),\psi {\mathbb{X}}_2)\hfill \\ \hfill =& g(\left({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_2)-g({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi {\mathbb{X}}_0,{\mathbb{X}}_2)\hfill \\ \hfill =& g(\left({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_2)-(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_2).\hfill \end{array}$$

By the use of (4) and (8), we arrive at

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)=& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_4,{\mathbb{X}}_2)\hfill \\ \hfill =& g(\left({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_2)+\eta \left({\mathbb{X}}_0\right)g(\psi {\mathbb{X}}_4,{\mathbb{X}}_2)+\eta \left({\mathbb{X}}_4\right)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_2)\hfill \\ \hfill =& g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_4,{\mathbb{X}}_2)+g(\psi {\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0,{\mathbb{X}}_2)\hfill \\ \hfill =& g({\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_2)+g({\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{F}{\mathbb{X}}_4,{\mathbb{X}}_2)-g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0,\psi {\mathbb{X}}_2).\hfill \end{array}$$

Therefore, it follows from (6), (7), and (1) that

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)=({\mathbb{X}}_{0}ln{f}_2)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_2)+g(A_{\mathbf{F}{\mathbb{X}}_4}{\mathbb{X}}_0,{\mathbb{X}}_2)-g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2).\end{array}$$

Once more, by the use of (10)–(11), we construct

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\psi {\mathbb{X}}_2)=\frac{1}{2}g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_4).\end{array}$$

(37)

From (36) and (37), we obtain the second part of (i). For the third part, we calculate the following:

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)=& g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0,\psi {\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill =& -g(\psi {\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_0,{\mathbb{X}}_5)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill =& g(\left({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi {\mathbb{X}}_0,{\mathbb{X}}_5)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill =& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_4,{\mathbb{X}}_5)+\alpha \{2g({\mathbb{X}}_0,{\mathbb{X}}_4)g(\zeta ,{\mathbb{X}}_4)\hfill \\ \hfill & -\eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_0,{\mathbb{X}}_5)-\eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_4,{\mathbb{X}}_5)\}\hfill \\ \hfill & -\beta \left\{\eta \left({\mathbb{X}}_0\right)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)-\eta \left({\mathbb{X}}_4\right)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_5)\right\}\hfill \\ \hfill & -(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_5)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill =& g(\psi ({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_4,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_4,{\mathbb{X}}_5)-\beta \eta \left({\mathbb{X}}_0\right)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_5)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)-\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill =& -g({\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{F}{\mathbb{X}}_4,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)-\beta \eta \left({\mathbb{X}}_0\right)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_4,\mathbf{F}{\mathbb{X}}_5)-(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_5)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_4,{\mathbb{X}}_5).\hfill \end{array}$$

By the use of (6)–(7) and (8), we have

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\mathbf{F}{\mathbb{X}}_4)-({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill & -(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_5)-\beta \eta \left({\mathbb{X}}_0\right)g(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_5)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_4,{\mathbb{X}}_5).\hfill \end{array}$$

(38)

Replacing ${\mathbb{X}}_4$ with ${\mathbb{X}}_5$ in (38), we have

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)& =g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_4),\mathbf{F}{\mathbb{X}}_5)+({\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,\mathbf{T}{\mathbb{X}}_5)\hfill \\ \hfill & -(\psi {\mathbb{X}}_{0}ln{f}_2)g({\mathbb{X}}_4,{\mathbb{X}}_5)-\beta \eta \left({\mathbb{X}}_0\right)g(\mathbf{T}{\mathbb{X}}_5,{\mathbb{X}}_4)\hfill \\ \hfill & -\alpha \eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_4,{\mathbb{X}}_5).\hfill \end{array}$$

(39)

Equations (38) and (39) give the second part. The rest of the terms are derived by interchanging ${\mathbb{X}}_0$ with $\psi {\mathbb{X}}_0$, as well as ${\mathbb{X}}_4$ and ${\mathbb{X}}_5$ with $\mathbf{T}{\mathbb{X}}_4$ and $\mathbf{T}{\mathbb{X}}_5$, respectively. Thus, the proof is completed. □

Remark 2.

If $\mathbf{B}({\mathbb{U}}_1,{\mathbb{X}}_2)=0\forall {\mathbb{U}}_1\in \mathsf{\Gamma }\left(\mathbf{D}\right),\mathrm{and}\forall {\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right)$, $\mathbf{B}({\mathbb{U}}_1,{\mathbb{X}}_3)=0\forall {\mathbb{U}}_1\in \mathsf{\Gamma }\left(\mathbf{D}\right),$ and $\forall {\mathbb{X}}_3\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right)$, then the bi-warped product submanifold $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ in a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$ refers to a $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic (respectively, $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }-$ mixed totally geodesic).

Remark 3.

It easily proved that a proper bi-warped product submanifold $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$ is trivial, by using the $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic in (19) and (26).

Now, we are in the position to give the proof of our main result. More precisely, we give the following inequality theorem for bi-warped product submanifolds of the type ${\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$.

Theorem 1.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$. Then, the following inequality is satisfied for the second fundamental form $\mathbf{B}$:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2+{\alpha }^2-{\beta }^2\right\}\hfill \\ \hfill & +n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2-{\beta }^2\right\}+{\alpha }^2\right\},\hfill \end{array}$$

(40)

where $n_2=dim{\mathbf{M}}_{\perp },n_3=dim{\mathbf{M}}_{\varphi }.$ Moreover, $\nabla (ln{f}_a)$ is the gradient of $ln{f}_a$ for $a=1,2$, while $\zeta \in {\mathbf{M}}_T$, ${\mathbf{M}}_{\varphi }$, and ${\mathbf{M}}_{\perp }$ are holomorphic, proper slant, and totally real submanifolds of $\tilde{\mathbf{M}}$, respectively. If the inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is geodesic totally in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

Proof. 

Assume that $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ is a n-dimensional proper bi-warped product submanifold of the nearly trans-Sasakian manifold ${\tilde{\mathbf{M}}}^{2m+1}$. In addition, let the local orthonormal vector fields $v_1,\cdots ,v_n$ of $T\mathbf{M}$ be as follows:

$$\begin{array}{cc}\hfill \mathbf{D}=& span\{v_1,\cdots ,v_t,v_{t+1}=\psi v_1,\cdots ,v_{2t}=\psi v_t\}\hfill \\ \hfill {\mathbf{D}}^{\perp }=& span\{v_{2t+1}={\widehat{v}}_1,\cdots ,v_{2t+l}=\widehat{v_l}\}\hfill \\ \hfill {\mathbf{D}}^{\varphi }=& span\{v_{2t+l+1}=v_1^{*},\cdots v_{2t+l+k}=v_k^{*},v_{2t+l+k+1}=sec\varphi \mathbf{T}{v}_1^{*},\hfill \\ & \cdots v_n=sec\varphi \mathbf{T}{v}_k^{*}\}.\hfill \end{array}$$

Therefore, $dim{\mathbf{M}}_T=n_1=2t+1$, $dim{\mathbf{M}}_{\perp }=n_2=l$, and $dim{\mathbf{M}}_{\varphi }=n_3=2k$. Furthermore, the orthonormal frame fields $v_1,\cdots ,v_{2m+1-n-l-2k}$ of the normal sub-bundle $T^{\perp }\mathbf{M}$ are as follows:

$$\begin{array}{cc}\hfill \psi {\mathbf{D}}^{\perp }=& \{v_1=\psi {\widehat{v}}_1,\cdots ,v_l=\psi {\widehat{v}}_l\}\hfill \\ \hfill \mathbf{F}{\mathbf{D}}^{\varphi }=& span\{v_{l+1}={\tilde{v}}_1=csc\varphi \mathbf{F}{v}_1^{*},\cdots ,v_{l+k}={\tilde{v}}_k=csc\varphi \mathbf{F}{v}_k^{*},\hfill \\ \hfill & v_{l+k+1}=csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_1^{*},\cdots ,v_{l+2k}={\tilde{v}}_{2k}=csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_k^{*}\}\hfill \\ \hfill \mu =& span\{v_{l+2k+1},\cdots ,v_{2m+1-n-l-2k}\}.\hfill \end{array}$$

According the definition $\mathbf{B}$, we have

$$\begin{array}{c}\hfill {\parallel \mathbf{B}\parallel }^2=\sum _{a,b=1}^{n}g\left(\mathbf{B}(v_a,v_b),\mathbf{B}(v_a,v_b)\right).\end{array}$$

The above equation can be broken into the components of submanifolds of ${\mathbf{M}}_T$, ${\mathbf{M}}_{\varphi }$, and ${\mathbf{M}}_{\perp }$ as follows:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& =\sum _{r=1}^l\sum _{a,b=1}^n{g}^2(\mathbf{B}(v_a,v_b),\psi {\widehat{v}}_r)+\sum _{r=l+1}^{l+2k}\sum _{a,b=1}^n{g}^2(\mathbf{B}(v_a,v_b),v_r)\hfill \\ \hfill & +\sum _{r=l+2k+1}^{2m+1-n-l-2k}\sum _{a,b=1}^n{g}^2(\mathbf{B}(v_a,v_b),v_r),\hfill \end{array}$$

(41)

where clearly $dim\mu =2m+1-n-l-2k.$ Expanding the above equation, according to the orthonormal bases of ${\mathbf{D}}^{\varphi }$, $\mathbf{D}$, and ${\mathbf{D}}^{\perp }$ (except for the last term), we arrive at

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge \sum _{r=1}^l\sum _{a,b=1}^{2t+1}{g}^2(\mathbf{B}(v_a,v_b),\psi {\widehat{v}}_r)+\sum _{r=1}^l\sum _{a,b=1}^{l}g(\mathbf{B}({\widehat{v}}_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)\hfill \\ \hfill & +\sum _{r=1}^l\sum _{a,b=1}^{2k}g(\mathbf{B}(v_a^{*},v_b^{*}),\psi {\widehat{v}}_r)+\sum _{r=1}^{2k}\sum _{a,b=1}^{2t+1}{g}^2(\mathbf{B}(v_a,v_b),{\tilde{v}}_r)\hfill \\ \hfill & +\sum _{r=1}^{2k}\sum _{a,b=1}^l{g}^2(\mathbf{B}({\widehat{v}}_a,{\widehat{v}}_b),{\tilde{v}}_r)+\sum _{r=1}^{2k}\sum _{a,b=1}^{2k}{g}^2(\mathbf{B}(v_a^{*},v_b^{*}),{\tilde{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^l\sum _{a=1}^{2t+1}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^l\sum _{a=1}^{2t+1}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(v_a,v_b^{*}),\psi {\widehat{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^l\sum _{a=1}^{2k}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a^{*},{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^{2k}\sum _{a=1}^l\sum _{b=1}^{2k}{g}^2(\mathbf{B}({\widehat{v}}_a,v_b^{*}),{\tilde{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^{2k}\sum _{a=1}^{2t+1}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(v_a,v_b^{*}),{\tilde{v}}_r)+2\sum _{r=1}^{2k}\sum _{a=1}^{2t+1}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a,{\widehat{v}}_b),{\tilde{v}}_r).\hfill \end{array}$$

Utilizing Equations (17), (18), and (26) in the preceding equation, we obtain

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge \sum _{r=1}^l\sum _{a,b=1}^{l}g(\mathbf{B}({\widehat{v}}_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)+\sum _{r=1}^l\sum _{a,b=1}^{2k}g(\mathbf{B}(v_a^{*},v_b^{*}),\psi {\widehat{v}}_r)\hfill \\ \hfill & +\sum _{r=1}^{2k}\sum _{a,b=1}^{2k}{g}^2(\mathbf{B}(v_a^{*},v_b^{*}),{\tilde{v}}_r)+\sum _{r=1}^{2k}\sum _{a,b=1}^l{g}^2(\mathbf{B}({\widehat{v}}_a,{\widehat{v}}_b),{\tilde{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^{2k}\sum _{a=1}^l\sum _{b=1}^{2k}{g}^2(\mathbf{B}({\widehat{v}}_a,v_b^{*}),{\tilde{v}}_r)+2\sum _{r=1}^l\sum _{a=1}^{2k}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a^{*},{\widehat{v}}_b),\psi {\widehat{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^l\sum _{a=1}^{2t+1}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^{2k}\sum _{a=1}^{2t+1}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(v_a,v_b^{*}),{\tilde{v}}_r).\hfill \end{array}$$

(42)

Leaving out all terms except for the last two relations in the above equation, we find

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge 2\sum _{r=1}^l\sum _{a=1}^{2t}\sum _{b=1}^l{g}^2(\mathbf{B}(v_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^l\sum _{b=1}^l{g}^2(\mathbf{B}(\zeta ,{\widehat{v}}_b),\psi {\widehat{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^{2k}\sum _{a=1}^{2t}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(v_a,v_b^{*}),{\tilde{v}}_r)+2\sum _{r=1}^{2k}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(\zeta ,v_b^{*}),{\tilde{v}}_r).\hfill \end{array}$$

From the frame fields of tangent and normal sub-bundles of $\mathbf{M}$, after ignoring the $\mu$-components part in (41), we derive

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge 2\sum _{r=1}^l\sum _{a=1}^t\sum _{b=1}^l{g}^2(\mathbf{B}(v_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^l\sum _{a=1}^t\sum _{b=1}^l{g}^2(\mathbf{B}(\psi v_a,{\widehat{v}}_b),\psi {\widehat{v}}_r)\hfill \\ \hfill & +2\sum _{r=1}^l\sum _{b=1}^l{g}^2(\mathbf{B}(\zeta ,{\widehat{v}}_b),\psi {\widehat{v}}_r)+2\sum _{r=1}^{2k}\sum _{b=1}^{2k}{g}^2(\mathbf{B}(\zeta ,v_b^{*}),{\tilde{v}}_r)\hfill \\ \hfill & +2{csc}^2\varphi \sum _{r,b=1}^k\sum _{a=1}^t\left\{g^2(\mathbf{B}(v_a,v_b^{*}),\mathbf{F}{v}_r^{*})+g^2(\mathbf{B}(\psi v_a,v_b^{*}),\mathbf{F}{v}_r^{*})\right\}\hfill \\ \hfill & +2{csc}^2\varphi {sec}^2\varphi \sum _{r,b=1}^k\sum _{a=1}^t\left\{g^2(\mathbf{B}(v_a,\mathbf{T}{v}_b^{*}),\mathbf{F}{v}_r^{*})+g^2(\mathbf{B}(\psi v_a,\mathbf{T}{v}_b^{*}),\mathbf{F}{v}_r^{*})\right\}\hfill \\ \hfill & +2{csc}^2\varphi {sec}^2\varphi \sum _{r,b=1}^k\sum _{b=1}^t\left\{g^2(\mathbf{B}({\widehat{v}}_a,v_b^{*}),\mathbf{F}\mathbf{T}{v}_r^{*})+g^2(\mathbf{B}(\psi v_a,v_b^{*}),\mathbf{F}\mathbf{T}{v}_r^{*})\right\}\hfill \\ \hfill & +2{csc}^2\varphi {sec}^4\varphi \sum _{r,b=1}^k\sum _{a=1}^t\left\{g^2(\mathbf{B}({\widehat{v}}_a,\mathbf{T}{v}_b^{*}),\mathbf{F}\mathbf{T}{v}_r^{*})+g^2(\mathbf{B}(\psi v_a,\mathbf{T}{v}_b),\mathbf{F}\mathbf{T}{v}_r^{*})\right\}.\hfill \end{array}$$

Utilizing Equations (19), (20), and (21) in the first terms, we substitute (26)–(35) to obtain

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge 2l\sum _{a=1}^t\left\{{\left(-\psi v_{a}ln{f}_1+\alpha \eta \left(v_a\right)\right)}^2+{\left(v_{a}ln{f}_1-\eta \left(v_a\right)\right)}^2\right\}\hfill \\ \hfill & +2k{csc}^2\varphi \sum _{a=1}^t{\left\{{\left(-\psi v_{a}ln{f}_2-\alpha \eta \left(v_a\right)\right)}^2+(v_{a}ln{f}_2-\eta \left(v_a\right)\right)}^2\}\hfill \\ \hfill & +\frac{2k}{9}{csc}^2\varphi {sec}^2\varphi {cos}^4\varphi \sum _{a=1}^t\left\{{\left(v_{a}ln{f}_2-\beta \eta \left(v_a\right)\right)}^2+{(-\psi v_{a}ln{f}_2)}^2\right\}\hfill \\ \hfill & +\frac{2k}{9}{csc}^2\varphi {sec}^2\varphi {cos}^4\varphi \sum _{a=1}^t\left\{{\left(v_{a}ln{f}_2-\beta \eta \left(v_a\right)\right)}^2+{(-\psi v_{a}ln{f}_2)}^2\right\}\hfill \\ \hfill & +2k{csc}^2\varphi {sec}^4\varphi {cos}^4\varphi \sum _{a=1}^t{\left\{{\left(-\psi v_{a}ln{f}_2-\alpha \eta \left(v_a\right)\right)}^2+(v_{a}ln{f}_2-\eta \left(v_a\right)\right)}^2\}\hfill \\ \hfill & +2l{\alpha }^2+4k{\alpha }^2.\hfill \end{array}$$

From the orthonormal frame $\eta \left(v_a\right)=0$ for $1\le i\le 2t$, and with some rearrangement in the remaining terms, we arrive at

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge 2l\sum _{a=1}^{2t}{(v_{a}ln{f}_1)}^2+2l{\alpha }^2+4k{csc}^2\varphi \sum _{a=1}^{2t}{(v_{a}ln{f}_1)}^2+\frac{4k}{9}{cot}^2\varphi \sum _{a=1}^{2t}{(v_{a}ln{f}_1)}^2+4k{\alpha }^2.\hfill \end{array}$$

Using trigonometric identities and adding and subtracting some terms, we obtain

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2& \ge 2l\parallel \nabla ln{f}_1{\parallel }^2-2l{(v_{2t+1}ln{f}_1)}^2+2l{\alpha }^2+\frac{4k}{9}\left(10{csc}^2\varphi -1\right){\parallel \nabla ln{f}_2\parallel }^2\hfill \\ \hfill & -\frac{4k}{9}\left(10{csc}^2\varphi -1\right){(v_{2t+1}ln{f}_2)}^2+4k{\alpha }^2.\hfill \end{array}$$

For a nearly trans-Sasakian manifold, two conditions are satisfied: $\zeta ln{f}_1=\beta$ and $\zeta ln{f}_2=\beta$ for the structure vector field $\zeta$ tangent to the base ${\mathbf{M}}_T$. The following inequality is obtained by substituting into the proceeding equation:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2l\parallel \nabla ln{f}_1{\parallel }^2-2l{\beta }^2+2l{\alpha }^2\hfill \\ \hfill & +\frac{4k}{9}\left(10{csc}^2\varphi -1\right){\parallel \nabla ln{f}_2\parallel }^2-\frac{4k}{9}\left(10{csc}^2\varphi -1\right){\beta }^2+4k{\alpha }^2,\hfill \end{array}$$

from which we reach the final result, (40), which we wanted to prove.

If the inequality (40) becomes an equality, the missing third part in (41) gives:

$$\begin{array}{c}\hfill \mathbf{B}(T\mathbf{M},T\mathbf{M})\perp \mu .\end{array}$$

(43)

Using terms (i) and (ii), which were not considered in (17), and (42), we can derive $\mathbf{B}(\mathbf{D},\mathbf{D})\perp \psi {\mathbf{D}}^{\perp }$ and $\mathbf{B}(\mathbf{D},\mathbf{D})\perp \mathbf{F}{\mathbf{D}}^{\varphi }$. It is obtained that $\mathbf{B}(\mathbf{D},\mathbf{D})=0$. The missing first and fourth terms in (42) lead to the following:

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })\perp \psi {\mathbf{D}}^{\perp }\&\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })\perp \mathbf{F}{\mathbf{D}}^{\varphi }.\end{array}$$

(44)

It can easily be seen that $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })=0$, by using the missing terms in (43) and (44). In addition, the second and third terms that were left in (42) give $\mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp }$ and $\mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })\perp \mathbf{F}{\mathbf{D}}^{\varphi }.$ This leads to $\mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })=0.$ In addition, term numbers five and six that were missed in (42) imply that $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp }$ and $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \mathbf{F}{\mathbf{D}}^{\varphi }$. These give us $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })=0.$ We obtain $\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\perp })\subset \psi {\mathbf{D}}^{\perp }$ by considering the second term in (26), along with (43). From the first term in (26) along with (43), we obtain $\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\varphi })\subset \mathbf{F}{\mathbf{D}}^{\varphi }.$ As ${\mathbf{M}}_T$ is totally geodesic in $\tilde{\mathbf{M}}$, by the use of this point along with $\mathbf{B}(\mathbf{D},\mathbf{D})=0$, $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })=0$, and $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })=0$, we find that ${\mathbf{M}}_T$ is totally geodesic in $\tilde{\mathbf{M}}$. As ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ are totally umbilical in $\mathbf{M}$ and we have $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })=0$, $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })=0,$ $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp }$, and $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \mathbf{F}{\mathbf{D}}^{\varphi }$, it can be concluded that ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ together are totally umbilical in $\tilde{\mathbf{M}}$. Furthermore, $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp }$ and $\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })\perp \mathbf{F}{\mathbf{D}}^{\varphi }$ imply that $\mathbf{M}$ is neither $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic nor $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic at $\tilde{\mathbf{M}}$, in light of Remark 2. This completes the proof of the main theorem. □

Some Geometric Consequences

If we consider $dim{\mathbf{M}}_{\varphi }=2k=0$, then Theorem 1 gives the following:

Theorem 2.

(Theorem 4.1 of [18]) Let $\mathbf{M}={\mathbf{M}}_T{\times }_f{\mathbf{M}}_{\perp }$ be a CR-warped product submanifold in a nearly trans-Sasakian manifold. Then, we have:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2+{\alpha }^2-{\beta }^2\right\},\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp }$.

If ${\mathbf{M}}_{\perp }$ vanishes, then Theorem 1 implies the following:

Theorem 3.

(Theorem 4.1 of [19]) Let $\mathbf{M}={\mathbf{M}}_T{\times }_f{\mathbf{M}}_{\varphi }$ be a warped product semi-slant submanifold of a nearly trans-Sasakian $\tilde{\mathbf{M}}$, then inequality (40) implies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2-{\beta }^2\right\}++{\alpha }^2\right\},\hfill \end{array}$$

where $n_3=dim{\mathbf{M}}_{\varphi }$.

Therefore, Theorem 1 is an extension of Theorem 4.1 of [19]. Similarly, for $\alpha =0$, we have:

Theorem 4.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly β-Kenmotsu manifold $\tilde{\mathbf{M}}$. Then, the second fundamental form $\mathbf{B}$ satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2l\left\{\parallel \nabla (ln{f}_1){\parallel }^2-{\beta }^2\right\}+2k\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2-{\beta }^2\right\}\right\},\hfill \end{array}$$

where $l=dim{\mathbf{M}}_{\perp },k=\frac{1}{2}dim{\mathbf{M}}_{\varphi }.$ In addition, $\nabla (ln{f}_a)$ is the gradient of $ln{f}_a$, while $\zeta \in {\mathbf{M}}_T$, ${\mathbf{M}}_{\varphi }$, and ${\mathbf{M}}_{\perp }$ are holomorphic, proper slant, and totally real submanifolds of $\tilde{\mathbf{M}}$, respectively. If inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is geodesic totally in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

Inserting $\alpha =0$ and $\beta =1$ into Theorem 1, we have

Theorem 5.

(Theorem 1 of [8]) Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly Kenmotsu manifold $\tilde{\mathbf{M}}$. Then, the second fundamental form $\mathbf{B}$ satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2-1\right\}+n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2-1\right\}\right\},\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp },n_3=\frac{1}{2}dim{\mathbf{M}}_{\varphi }.$ In addition, $\nabla (ln{f}_a)$ is the gradient of $ln{f}_a$, while $\zeta \in {\mathbf{M}}_T$, ${\mathbf{M}}_{\varphi }$, and ${\mathbf{M}}_{\perp }$ are holomorphic, proper slant, and totally real submanifolds of $\tilde{\mathbf{M}}$, respectively. If inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is geodesic totally in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

If we put $\alpha =0,\beta =0$ into Theorem 1, the following is obtained for the bi-warped product submanifold of a nearly cosymplectic manifold:

Theorem 6.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly cosymplectic manifold $\tilde{\mathbf{M}}$. Then, the second fundamental form $\mathbf{B}$ satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2\right\}+n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right){\parallel \nabla (ln{f}_2)\parallel }^2\right\},\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp },n_3=\frac{1}{2}dim{\mathbf{M}}_{\varphi }.$ Moreover, $\nabla (ln{f}_a)$ is the gradient of $ln{f}_a$, while $\zeta \in {\mathbf{M}}_T$, ${\mathbf{M}}_{\varphi }$, and ${\mathbf{M}}_{\perp }$ are holomorphic, proper slant, and totally real submanifolds of $\tilde{\mathbf{M}}$, respectively. If inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is totally geodesic in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

Again, we get the following by using $\alpha =0,\beta =0$, and $dim{\mathbf{M}}_{\varphi }=0$.

Theorem 7.

(Theorem 3.2 of [31]) Let $\mathbf{M}={\mathbf{M}}_T{\times }_f{\mathbf{M}}_{\perp }$ be a CR-warped product submanifold of a nearly cosymplectic manifold. Then, we have:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2{\parallel \nabla (ln{f}_1)\parallel }^2,\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp }$.

Inserting $\alpha =0,\beta =0$ and $dim{\mathbf{M}}_{\perp }=0$ into Theorem 1, we obtain the following:

Theorem 8.

(Theorem 3.5 of [32]) Let $\mathbf{M}={\mathbf{M}}_T{\times }_f{\mathbf{M}}_{\varphi }$ be a warped product semi-slant submanifold of a nearly cosymplectic $\tilde{\mathbf{M}}$. Then, inequality (40) implies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 4k\left\{\left(\frac{1}{9}{cot}^2\varphi +1{csc}^2\varphi \right){\parallel \nabla (ln{f}_2)\parallel }^2\right\},\hfill \end{array}$$

where $2k=dim{\mathbf{M}}_{\varphi }$.

By inserting $\beta =0$ and $\alpha =1,\beta =0$ into Theorem 1, the following results for the bi-warped product submanifolds of a nearly $\alpha$-Sasakian manifold and a nearly Sasakian manifold, respectively, can be obtained:

Theorem 9.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly α-Sasakian manifold $\tilde{\mathbf{M}}$. Then, the second fundamental form $\mathbf{B}$ satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2+{\alpha }^2\right\}\hfill \\ \hfill & +n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2\right\}+{\alpha }^2\right\},\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp },n_3=\frac{1}{2}dim{\mathbf{M}}_{\varphi }.$ If the above inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp },{\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is geodesic totally in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

Theorem 10.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly Sasakian manifold $\tilde{\mathbf{M}}$. Then, the second fundamental form $\mathbf{B}$ satisfies the following inequality:

$$\begin{array}{cc}\hfill {\parallel \mathbf{B}\parallel }^2\ge & 2n_2\left\{\parallel \nabla (ln{f}_1){\parallel }^2+1\right\}+n_3\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{\parallel \nabla (ln{f}_2){\parallel }^2\right\}+1\right\},\hfill \end{array}$$

where $n_2=dim{\mathbf{M}}_{\perp },n_3=\frac{1}{2}dim{\mathbf{M}}_{\varphi }.$ If inequality (40) becomes an equality, then ${\mathbf{M}}_{\perp },{\mathbf{M}}_{\varphi }$ are totally umbilical and ${\mathbf{M}}_T$ is geodesic totally in $\tilde{\mathbf{M}}$. The $\mathbf{D}\oplus {\mathbf{D}}^{\perp }$-mixed totally geodesic and $\mathbf{D}\oplus {\mathbf{D}}^{\varphi }$-mixed totally geodesic do not exist in $\mathbf{M}$ of $\tilde{\mathbf{M}}$.

Remark 4.

It can be easily seen that Theorem 4.1 of [18], Theorem 4.1 of [19], Theorem 3.2 of [31], Theorem 3.5 of [32], and Theorem 1 of [8] are special cases of our main Theorem 1.

3. Inequality for Bi-Warped Product Submanifold of the Type $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_T$

In this section, we consider the bi-warped product submanifold of type $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ in a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, with respect to the tangent spaces of ${\mathbf{M}}_T,{\mathbf{M}}_{\perp }$ and ${\mathbf{M}}_{\varphi }$, which are integral manifolds of $\mathbf{D},{\mathbf{D}}^{\perp }$, and ${\mathbf{D}}^{\varphi }$, respectively.

Lemma 3.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ be a bi-warped product submanifold of a trans-Sasakian manifold $\tilde{\mathbf{M}}$. Then, we construct the following:

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =\left\{(\mathbf{T}{\mathbb{X}}_{4}ln{f}_1)+\alpha \eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_2,{\mathbb{X}}_3),\mathbf{F}{\mathbb{X}}_4)& =g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)+g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)\hfill \\ \hfill & +2\left\{\mathbf{T}{\mathbb{X}}_{4}ln{f}_2+\alpha \eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)& =g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5),\hfill \\ \hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\mathbf{F}{\mathbb{X}}_4)& =0.\hfill \end{array}$$

Proof. 

For any ${\mathbb{X}}_0,{\mathbb{X}}_1\in \mathsf{\Gamma }\left(\mathbf{D}\right)$ and ${\mathbb{X}}_4,{\mathbb{X}}_5\in \mathsf{\Gamma }({\mathbf{D}}^{\varphi }\oplus \zeta ),$ we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_1,\mathbf{F}{\mathbb{X}}_4)=g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_1,\psi {\mathbb{X}}_4)-({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_1,\mathbf{T}{\mathbb{X}}_4)\hfill \\ \hfill & =g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_1,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_1,{\mathbb{X}}_4)+g({\mathbb{X}}_1,{\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{T}{\mathbb{X}}_4).\hfill \end{array}$$

Using (4), (8), (10), and (1), we get

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)=& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_1}\right){\mathbb{X}}_0,{\mathbb{X}}_4)+\left\{2g({\mathbb{X}}_0,{\mathbb{X}}_1)g(\zeta ,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_0\right)g({\mathbb{X}}_1,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_1\right)g({\mathbb{X}}_0,{\mathbb{X}}_4)\right\}\hfill \\ \hfill & -\beta \left\{\eta \left({\mathbb{X}}_0\right)g(\psi {\mathbb{X}}_1,{\mathbb{X}}_4)+\eta \left({\mathbb{X}}_1\right)(\psi ,{\mathbb{X}}_0,{\mathbb{X}}_4)\right\}+\mathbf{T}{\mathbb{X}}_4(ln{f}_1)g({\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \\ \hfill =& -g({\tilde{\nabla }}_{{\mathbb{X}}_1}\psi {\mathbb{X}}_0,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_1}{\mathbb{X}}_0,\mathbf{F}{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_1}{\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4)+2\alpha \eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \\ \hfill & +\mathbf{T}{\mathbb{X}}_4(ln{f}_1)g({\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \\ \hfill =& ({\mathbb{X}}_{4}ln{f}_1)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_1)-g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)+2\alpha \eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \\ \hfill & +2\mathbf{T}{\mathbb{X}}_4(ln{f}_1)g({\mathbb{X}}_0,{\mathbb{X}}_1)-({\mathbb{X}}_{4}ln{f}_1)g(\psi {\mathbb{X}}_0,{\mathbb{X}}_1).\hfill \end{array}$$

This is implies the first part (45) of the lemma. For the next part, we have, from (8),

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_2,{\mathbb{X}}_3),\mathbf{F}{\mathbb{X}}_4)=g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_3,\psi {\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_3,\mathbf{T}{\mathbb{X}}_4).\end{array}$$

Using (10) and (1), we obtain

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_2,{\mathbb{X}}_3),\mathbf{F}{\mathbb{X}}_4)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi \right){\mathbb{X}}_3,{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi {\mathbb{X}}_3,{\mathbb{X}}_4)+\mathbf{T}{\mathbb{X}}_{4}ln{f}_{2}g({\mathbb{X}}_2,{\mathbb{X}}_3).\end{array}$$

By use of (4), (8), and (10), we get

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_2,{\mathbb{X}}_3),\mathbf{F}{\mathbb{X}}_4)=& -g(\left({\tilde{\nabla }}_{{\mathbb{X}}_3}\psi \right){\mathbb{X}}_2,{\mathbb{X}}_4)+\alpha \{2\eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & -\eta \left({\mathbb{X}}_2\right)g({\mathbb{X}}_3,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_3\right)g({\mathbb{X}}_2,{\mathbb{X}}_4)\}\hfill \\ \hfill & -\beta \left\{\eta \left({\mathbb{X}}_2\right)g(\psi {\mathbb{X}}_3,{\mathbb{X}}_4)-\eta \left({\mathbb{X}}_3\right)g(\psi {\mathbb{X}}_2,{\mathbb{X}}_4)\right\}+\mathbf{T}{\mathbb{X}}_{4}ln{f}_{2}g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & +g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)\hfill \\ \hfill =& g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)+g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)\hfill \\ \hfill & +\mathbf{T}{\mathbb{X}}_{4}ln{f}_{2}g({\mathbb{X}}_2,{\mathbb{X}}_3)+2\alpha \eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_2,{\mathbb{X}}_3)\hfill \\ \hfill & -g({\tilde{\nabla }}_{{\mathbb{X}}_3}{\mathbb{X}}_2,\mathbf{F}{\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_3}{\mathbb{X}}_2,\mathbf{T}{\mathbb{X}}_4)\hfill \\ \hfill =& g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)+g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)\hfill \\ \hfill & +2\mathbf{T}{\mathbb{X}}_{4}ln{f}_{2}g({\mathbb{X}}_2,{\mathbb{X}}_3)+2\alpha \eta \left({\mathbb{X}}_4\right)g({\mathbb{X}}_2,{\mathbb{X}}_3).\hfill \end{array}$$

This is equivalent to (46). For any ${\mathbb{X}}_4,{\mathbb{X}}_5\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right),{\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right),$

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)=g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_5,\psi {\mathbb{X}}_2)=g(\left({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi \right){\mathbb{X}}_5,{\mathbb{X}}_2)-g({\tilde{\nabla }}_{{\mathbb{X}}_4}\psi {\mathbb{X}}_5,{\mathbb{X}}_2).\end{array}$$

Utilizing (4), we get the following:

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)=g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_2,\psi {\mathbb{X}}_5)=g({\nabla }_{{\mathbb{X}}_4}{\mathbb{X}}_2,\mathbf{F}{\mathbb{X}}_5)+g({\tilde{\nabla }}_{{\mathbb{X}}_4}{\mathbb{X}}_2,\mathbf{F}{\mathbb{X}}_5).\end{array}$$

In view of (6) and (7), we get

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)=g(\mathbf{B}({\mathbb{X}}_4,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5).\end{array}$$

For the last part, we have

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\mathbf{F}{\mathbb{X}}_4)& =g({\tilde{\nabla }}_{{\mathbb{X}}_5}{\mathbb{X}}_0,\psi {\mathbb{X}}_4)-g({\tilde{\nabla }}_{{\mathbb{X}}_5}{\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\hfill \\ \hfill & =-g({\tilde{\nabla }}_{{\mathbb{X}}_5}\psi {\mathbb{X}}_0,{\mathbb{X}}_4)+g(\left({\tilde{\nabla }}_{{\mathbb{X}}_5}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_4)-{\mathbb{X}}_5(ln{f}_1)g({\mathbb{X}}_0,\mathbf{T}{\mathbb{X}}_4),\hfill \end{array}$$

for any ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right)$ and ${\mathbb{X}}_4,{\mathbb{X}}_5\in \mathsf{\Gamma }\left({\mathbf{D}}^{\varphi }\right).$ Using (4), (7), and the normality of vector fields, we arrive at

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\mathbf{F}{\mathbb{X}}_4)=0.\end{array}$$

Thus, the required results are obtained. □

The following equalities can be derived by exchanging ${\mathbb{X}}_0$ with $\psi {\mathbb{X}}_0$ and ${\mathbb{X}}_1$ with $\psi {\mathbb{X}}_1$ in Lemma 3:

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =\left\{\mathbf{T}{\mathbb{X}}_4(ln{f}_1)+\alpha \eta \left({\mathbb{X}}_4\right)\right\}g(\psi {\mathbb{X}}_0,{\mathbb{X}}_1),\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,\psi {\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =\left\{\mathbf{T}{\mathbb{X}}_4(ln{f}_1)+\alpha \eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,\psi {\mathbb{X}}_1),\hfill \end{array}$$

(48)

and

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,\psi {\mathbb{X}}_1),\mathbf{F}{\mathbb{X}}_4)& =\left\{(\mathbf{T}{\mathbb{X}}_{4}ln{f}_1)+\alpha \eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,{\mathbb{X}}_1).\hfill \end{array}$$

(49)

Similarly, if we interchange ${\mathbb{X}}_4$ with $\mathbf{T}{\mathbb{X}}_4$ in Lemma 3, (47)–(49), and (16), we can derive

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}\mathbf{T}{\mathbb{X}}_4)& =-{cos}^2\varphi \left\{({\mathbb{X}}_{4}ln{f}_1)-\eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,{\mathbb{X}}_1),\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,{\mathbb{X}}_1),\mathbf{F}\mathbf{T}{\mathbb{X}}_4)& =-{cos}^2\varphi \left\{({\mathbb{X}}_{4}ln{f}_1)-\eta \left({\mathbb{X}}_4\right)\right\}g(\psi {\mathbb{X}}_0,{\mathbb{X}}_1)\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,\psi {\mathbb{X}}_1),\mathbf{F}\mathbf{T}{\mathbb{X}}_4)& =-{cos}^2\varphi \left\{({\mathbb{X}}_{4}ln{f}_1)-\eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,\psi {\mathbb{X}}_1),\hfill \end{array}$$

(52)

and

$$\begin{array}{cc}\hfill g(\mathbf{B}(\psi {\mathbb{X}}_0,\psi {\mathbb{X}}_1),\mathbf{F}\mathbf{T}{\mathbb{X}}_4)& =-{cos}^2\varphi \left\{({\mathbb{X}}_{4}ln{f}_1)-\eta \left({\mathbb{X}}_4\right)\right\}g({\mathbb{X}}_0,{\mathbb{X}}_1).\hfill \end{array}$$

(53)

Again, interchanging ${\mathbb{X}}_4$ with $\mathbf{T}{\mathbb{X}}_4$ in (46), Lemma 3, and (16), we obtain

$$\begin{array}{cc}\hfill 2g(\mathbf{B}({\mathbb{X}}_2,{\mathbb{X}}_3),\mathbf{F}\mathbf{T}{\mathbb{X}}_4)& =g(\mathbf{B}(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_2),\psi {\mathbb{X}}_3)+g(\mathbf{B}(\mathbf{T}{\mathbb{X}}_4,{\mathbb{X}}_3),\psi {\mathbb{X}}_2)\hfill \\ \hfill & -2{cos}^2\varphi \left(-\eta \left({\mathbb{X}}_4\right)+({\mathbb{X}}_{4}ln{f}_2)\right)g({\mathbb{X}}_2,{\mathbb{X}}_3).\hfill \end{array}$$

(54)

Lemma 4.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{M}$. Then, we have

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)=g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5)=0,\end{array}$$

(55)

for any ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right),{\mathbb{X}}_5\in \mathsf{\Gamma }({\mathbf{D}}^{\varphi }\oplus \zeta )$ and ${\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right).$

Proof. 

For any ${\mathbb{X}}_0\in \mathsf{\Gamma }\left(\mathbf{D}\right),{\mathbb{X}}_5\in \mathsf{\Gamma }({\mathbf{D}}^{\varphi }\oplus \zeta )$, and ${\mathbb{X}}_2\in \mathsf{\Gamma }\left({\mathbf{D}}^{\perp }\right),$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5)& =g({\tilde{\nabla }}_{{\mathbb{X}}_2}{\mathbb{X}}_0,\psi {\mathbb{X}}_5)+g({\tilde{\nabla }}_{{\mathbb{X}}_2}\mathbf{T}{\mathbb{X}}_5,{\mathbb{X}}_0),\hfill \\ \hfill & =g(\left({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi \right){\mathbb{X}}_0,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_2}\psi {\mathbb{X}}_0,{\mathbb{X}}_5)+\mathbf{T}{\mathbb{X}}_5(ln{f}_2)g({\mathbb{X}}_0,{\mathbb{X}}_2).\hfill \end{array}\end{array}$$

Equations (4) and (10) imply $g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5)=0.$ For the next equality, we compute

$$\begin{array}{cc}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5)& =g({\tilde{\nabla }}_{{\mathbb{X}}_0}{\mathbb{X}}_2,\psi {\mathbb{X}}_5)+g({\tilde{\nabla }}_{{\mathbb{X}}_0}\mathbf{T}{\mathbb{X}}_5,{\mathbb{X}}_2),\hfill \\ \hfill & =g(\left({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi \right){\mathbb{X}}_2,{\mathbb{X}}_5)-g({\tilde{\nabla }}_{{\mathbb{X}}_0}\psi {\mathbb{X}}_2,{\mathbb{X}}_5)+\mathbf{T}{\mathbb{X}}_5(ln{f}_1)g({\mathbb{X}}_0,{\mathbb{X}}_2).\hfill \end{array}$$

Again making use of (4) and (1), we find the following:

$$\begin{array}{c}\hfill g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_5),\psi {\mathbb{X}}_2)=g(\mathbf{B}({\mathbb{X}}_0,{\mathbb{X}}_2),\mathbf{F}{\mathbb{X}}_5),\end{array}$$

which is the first equality. Therefore, the result is proved. □

In this direction, we provide a relationship between the squared norm of the second fundamental form and the warping function for the bi-warped product. Before giving the next relationship, we define an orthonormal frame. Taking $\zeta$ tangent to the base manifold ${\mathbf{M}}_{\varphi }$ of an n-dimensional bi-warped product submanifold $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ in a $(2n+1)$-dimensional nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, we consider the dimensions $dim\left({\mathbf{M}}_T\right)=n_1,dim\left({\mathbf{M}}_{\perp }\right)=n_2$, and $dim\left({\mathbf{M}}_{\varphi }\right)=n_3.$ We provide proof of the main theorem as follows.

For the second type of bi-warped product submanifold, ${\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$, we prove the following result:

Theorem 11.

Assume that $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ is a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}.$ If ${\mathbf{D}}^{\perp }-{\mathbf{D}}^{\varphi }$ is mixed totally geodesic and ζ is tangent to ${\mathbf{M}}_{\varphi },$ then the length of the second fundamental form $\mathbf{B}$ is defined as

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2\ge n_1{csc}^2\varphi (1+{cos}^2\varphi )\left(\left|\right|\overrightarrow{\nabla }ln{f}_1{\left)\right||}^2-{\beta }^2\right)+n_2{cot}^2\varphi \left(\left|\right|\overrightarrow{\nabla }ln{f}_2{\left)\right||}^2-{\beta }^2\right),\end{array}$$

(56)

where $n_1=dim\left({\mathbf{M}}_T\right)$ and $n_2=dim\left({\mathbf{M}}_{\perp }\right)$. The gradients $\overrightarrow{\nabla }(ln{f}_1)$ and $\overrightarrow{\nabla }(ln{f}_2)$ of $ln{f}_1$ and $ln{f}_2$ are along ${\mathbf{M}}_T$ and ${\mathbf{M}}_{\perp }$, respectively. If the inequality (56) becomes an equality, then ${\mathbf{M}}_T$ and ${\mathbf{M}}_{\perp }$ are totally umbilical submanifolds, and ${\mathbf{M}}_{\varphi }$ is a totally geodesic submanifold in $\tilde{\mathbf{M}}$. Furthermore, $\mathbf{M}$ is a ${\mathbf{D}}^{\varphi }$-totally geodesic submanifold of $\tilde{\mathbf{M}}.$

Proof. 

Suppose the orthogonal frames of the corresponding tangent spaces of $\mathbf{D},{\mathbf{D}}^{\perp }$, and ${\mathbf{D}}^{\varphi }$, are as follows:

$$\begin{array}{cc}\hfill \mathbf{D}=& span\{v_1,\cdots ,v_p,v_{p+1}=\psi v_1,\cdots ,v_{n_1}=v_{2p}=\psi v_p\},\hfill \\ \hfill {\mathbf{D}}^{\perp }=& span\{v_{n_1+1}={\overline{v}}_1,\cdots ,v_{n_1+n_2}={\overline{v}}_{n_2}\},\hfill \\ \hfill {\mathbf{D}}^{\varphi }=& span\{v_{n_1+n_2+1}=v_1^{*},\cdots ,v_{n_1+n_2+q}=v_q^{*},v_{n_1+n_2+q+1}=v_{q+1}^{*}=sec\varphi \mathbf{T}{v}_1^{*}\hfill \\ \hfill & ,\cdots ,v_{n_1+n_2+2q}=v_{2q}^{*}=sec\varphi \mathbf{T}{v}_q^{*},v_m=v_{n_3}^{*}=v_{2q+1}^{*}=\xi \}.\hfill \end{array}$$

Then, the orthonormal frame fields of the normal sub-bundles of $\psi {\mathbf{D}}^{\perp },\mathbf{F}{\mathbf{D}}^{\varphi }$, and $\mu$, respectively, are as follows:

$$\begin{array}{cc}\hfill \psi {\mathbf{D}}^{\perp }=& span\{v_{m+1}={\tilde{v}}_1=\psi {\overline{v}}_1,...,v_{m+n_2}={\tilde{v}}_{n_2}=\psi {\overline{v}}_{n_2}\},\hfill \\ \hfill \mathbf{F}{\mathbf{D}}^{\varphi }=& span\{v_{m+n_2+1}={\tilde{v}}_{n_2+1}=csc\varphi \mathbf{F}{v}_1^{*},...,v_{m+n_2+q}={\tilde{v}}_{n_2+q}=csc\varphi \mathbf{F}{v}_q^{*},\hfill \\ \hfill & v_{m+n_2+q+1}={\tilde{v}}_{n_2+q+1}=csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_1^{*},\cdots ,v_{m+n_2+n_3-1}={\tilde{v}}_{n_2+n_3-1}=csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_q^{*}\},\hfill \\ \hfill \mu =& span\{v_{m+n_2+n_3}={\tilde{v}}_{n_2+n3},\cdots ,v_{2n+1}={\tilde{v}}_{2_{(n-n_2-n_3+1)-n_1}}\}.\hfill \end{array}$$

From the definition of the second fundamental form, we have

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2=\sum _{a,b=1}^{m}g(\mathbf{B}(v_a,v_b),\mathbf{B}(v_a,v_b))=\sum _{r=m+1}^{2n+1}\sum _{a,b=1}^{m}g{(\mathbf{B}(v_a,v_b),v_r)}^2.\end{array}$$

The above expression can be expanded, according the frame vector fields, as

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& =\sum _{r=1}^{n_2}\left(\sum _{a,b=1}^{m}g{(\mathbf{B}(v_a,v_b),{\tilde{v}}_r)}^2\right)+\sum _{r=n_2+1}^{n_2+n_3-1}\hfill \\ \hfill & +\sum _{r=n_2+n_3}^{2(n-n_2-n_3+1)-n_1}\left(\sum _{a,b=1}^{m}g{(\mathbf{B}(v_a,v_b),{\tilde{v}}_r)}^2\right).\hfill \end{array}$$

(57)

Ignoring the $\mu$-components and implementing the constructed frame fields of $\mathbf{D},{\mathbf{D}}^{\perp }$, and ${\mathbf{D}}^{\varphi },$ we obtain

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& \ge \sum _{r=1}^{n_2}\sum _{a,b=1}^{n_1}g{(\mathbf{B}(v_a,v_b),\psi {\overline{v}}_r)}^2+\sum _{r=1}^{n_2}\sum _{a,b=1}^{n_2}g{(\mathbf{B}({\overline{v}}_a,{\overline{v}}_b),\psi {\overline{v}}_r)}^2\hfill \\ \hfill & +\sum _{r=1}^{n_2}\sum _{a,b=1}^{n_3}g{(\mathbf{B}(v_a^{*},v_b^{*}),\psi {\overline{v}}_r)}^2+2\sum _{r=1}^{n_2}\sum _{a=1}^{n_1}\sum _{b=1}^{n_2}g{(\mathbf{B}(v_a,{\overline{v}}_b),\psi {\overline{v}}_r)}^2\hfill \\ \hfill & +2\sum _{r=1}^{n_2}\sum _{a=1}^{n_1}\sum _{b=1}^{n_3}g{(\mathbf{B}(v_a,v_b^{*}),\psi {\overline{v}}_r)}^2+2\sum _{r=1}^{n_2}\sum _{a=1}^{n_1}\sum _{b=1}^{n_3}g{(\mathbf{B}({\overline{v}}_a,v_b^{*}),\psi {\overline{v}}_r)}^2\hfill \\ \hfill & +\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a,b=1}^{n_1}g{(\mathbf{B}(v_a,v_b),{\tilde{v}}_r)}^2+\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a,b=1}^{n_2}g{(\mathbf{B}({\overline{v}}_a,{\overline{v}}_b),{\tilde{v}}_r)}^2\hfill \\ \hfill & +\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a,b=1}^{n_3}g{(\mathbf{B}(v_a^{*},v_b^{*}),{\tilde{v}}_r)}^2+2\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a=1}^{n_1}\sum _{b=1}^{n_2}g{(\mathbf{B}(v_a,{\overline{v}}_b),{\tilde{v}}_r)}^2\hfill \\ \hfill & +2\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a=1}^{n_1}\sum _{b=1}^{n_3}g{(\mathbf{B}(v_a^{*},v_b^{*}),{\tilde{v}}_r)}^2+2\sum _{r=n_2+1}^{n_2+n_3-1}\sum _{a=1}^{n_2}\sum _{b=1}^{n_3}g{(\mathbf{B}({\overline{v}}_a,v_b^{*}),{\tilde{v}}_r)}^2.\hfill \end{array}$$

(58)

As we assumed that $\mathbf{M}$ is a ${\mathbf{D}}^{\perp }-{\mathbf{D}}^{\varphi }$ mixed totally geodesic, this forces the third, sixth, and twelfth terms to vanish. In view of Lemma 4, the fifth and tenth terms are zero. Similarly, we ignore the first, second, fourth, and ninth terms, due to the lack of connections for warping functions. Now, the rest of the terms (i.e., the seventh and eighth) can be written as

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& \ge \sum _{r=1}^q\sum _{a,b=1}^{n_1}g{(\mathbf{B}(v_a,v_b),csc\varphi \mathbf{F}{v}_r^{*})}^2+\sum _{r=1}^q\sum _{a,b=1}^{n_1}g{(\mathbf{B}(v_a,v_b),csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_r^{*})}^2\hfill \\ \hfill & +\sum _{r=1}^q\sum _{a,b=1}^{n_2}g{(\mathbf{B}({\overline{v}}_a,{\overline{v}}_b),csc\varphi \mathbf{F}{v}_r^{*})}^2+\sum _{r=1}^q\sum _{a,b=1}^{n_2}g{(\mathbf{B}({\overline{v}}_a,{\overline{v}}_b),csc\varphi sec\varphi \mathbf{F}\mathbf{T}{v}_r^{*})}^2.\hfill \end{array}$$

Inserting the equations from Lemma 3 with (47)–(54), we derive

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& \ge n_1{csc}^2\varphi (1+{sec}^2\varphi )\sum _{r=1}^q{\left(\mathbf{T}{v}_r^{*}(ln{f}_1)+\alpha \eta \left(v_r^{*}\right)\right)}^2\hfill \\ \hfill & +n_1{csc}^2\varphi (1+{cos}^2\varphi )\sum _{r=1}^q{\left(v_r^{*}(ln{f}_1)-\eta \left(v_r^{*}\right)\right)}^2\hfill \\ \hfill & +n_2{csc}^2\varphi \sum _{r=1}^q{\left(\mathbf{T}{v}_r^{*}(ln{f}_1)+\alpha \eta \left(v_r^{*}\right)\right)}^2+n_2{csc}^2\varphi {cos}^2\varphi \sum _{r=1}^q{\left(v_r^{*}(ln{f}_2)-\eta \left(v_r^{*}\right)\right)}^2.\hfill \end{array}$$

From the orthornomal frame $\eta \left(e_r^{*}\right)=0$, for $1\le r\le q$, we have

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& \ge n_1{csc}^2\varphi (1+{sec}^2\varphi )\sum _{r=1}^{2q+1}{\left(\mathbf{T}{v}_r^{*}(ln{f}_1)\right)}^2-n_1{csc}^2\varphi (1+{sec}^2\varphi )\sum _{r=q+1}^{2q}g{(v_r^{*},\mathbf{T}\overrightarrow{\nabla }(ln{f}_1))}^2\hfill \\ \hfill & -n_1{csc}^2\varphi (1+{sec}^2\varphi ){\left(\mathbf{T}{v}_{2q+1}^{*}(ln{f}_1)\right)}^2+n_1{csc}^2\varphi (1+{cos}^2\varphi )\sum _{r=1}^q(v_r^{*}{(ln{f}_1)}^2\hfill \\ \hfill & +n_2{csc}^2\varphi \sum _{r=1}^{2q+1}{\left(\mathbf{T}{v}_r^{*}(ln{f}_2)\right)}^2-n_2{csc}^2\varphi \sum _{r=q+1}^{2q}g{(v_r^{*},\mathbf{T}\overrightarrow{\nabla }(ln{f}_2))}^2\hfill \\ \hfill & -n_2{csc}^2\varphi {\left(\mathbf{T}{v}_{2q+1}^{*}(ln{f}_2)\right)}^2+n_2{csc}^2\varphi {cos}^2\varphi \sum _{r=1}^q{\left(v_r^{*}(ln{f}_2)\right)}^2.\hfill \end{array}$$

The third and seventh terms are equal to zero by $e_{2q+1}^{*}=\zeta$ and $T\zeta =0$. Thus, the preceding inequality takes the following form:

$$\begin{array}{cc}\hfill {\left|\right|\mathbf{B}\left|\right|}^2& \ge n_1{csc}^2\varphi (1+{sec}^2\varphi )\left|\right|\mathbf{T}\overrightarrow{\nabla }(ln{f}_1){\left|\right|}^2-n_1{csc}^2\varphi (1+{sec}^2\varphi ){sec}^2\varphi \sum _{r=1}^{q}g{(\mathbf{T}{v}_r^{*},\mathbf{T}\overrightarrow{\nabla }(ln{f}_1))}^2\hfill \\ \hfill & +n_1{csc}^2\varphi (1+{cos}^2\varphi )\sum _{r=1}^q{\left(v_r^{*}(ln{f}_1)\right)}^2+n_2{csc}^2\varphi \left|\right|\mathbf{T}\overrightarrow{\nabla }(ln{f}_2){\left|\right|}^2\hfill \\ \hfill & -n_2{csc}^2\varphi {sec}^2\varphi \sum _{r=1}^{q}g{(\mathbf{T}{v}_r^{*},\mathbf{T}\overrightarrow{\nabla }(ln{f}_2))}^2\hfill \\ \hfill & +n_2{csc}^2\varphi {cos}^2\varphi \sum _{r=1}^q{\left(v_r^{*}(ln{f}_2)\right)}^2.\hfill \end{array}$$

Using (11) and the fact that $\xi (ln{f}_a)=\beta ,i=1,2$ from Proposition 2, we finally obtain

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2\ge n_1{csc}^2\varphi (1+{cos}^2\varphi )\left(\left|\right|\overrightarrow{\nabla }(ln{f}_1){\left|\right|}^2-{\beta }^2\right)+n_2{cot}^2\varphi \left(\right||\overrightarrow{\nabla }(ln{f}_2){\left|\right|}^2-{\beta }^2).\end{array}$$

For the equality case in (56), considering the third $\mu$-component term in (57), we derive

$$\begin{array}{c}\hfill \mathbf{B}(T\mathbf{M},T\mathbf{M})\perp \mu .\end{array}$$

(59)

By using the missing first, second, and fourth terms in (58), we find that

$$\begin{array}{c}\hfill \mathbf{B}(\mathbf{D},\mathbf{D})\perp \psi {\mathbf{D}}^{\perp },\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })\perp \psi {\mathbf{D}}^{\perp },\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\perp })\perp \psi {\mathbf{D}}^{\perp }.\end{array}$$

(60)

Evaluating the ninth term of (58), we get

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })\perp F{\mathbf{D}}^{\varphi }.\end{array}$$

(61)

Then, from (59) and (60), we obtain

$$\begin{array}{c}\hfill \mathbf{B}(\mathbf{D},\mathbf{D})\in \mathbf{F}{\mathbf{D}}^{\varphi },\mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\perp })\in \mathbf{F}{\mathbf{D}}^{\varphi },\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\perp })\in \mathbf{F}{\mathbf{D}}^{\varphi }.\end{array}$$

(62)

As $\mathbf{M}$ is ${\mathbf{D}}^{\perp }-{\mathbf{D}}^{\varphi }$ mixed totally geodesic, we can conclude the following:

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{D}}^{\perp },{\mathbf{D}}^{\varphi })=0.\end{array}$$

(63)

From the vanishing third term of (58), we get

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp }.\end{array}$$

(64)

From (59), (61), and (64), we get

$$\begin{array}{c}\hfill \mathbf{B}({\mathbf{D}}^{\varphi },{\mathbf{D}}^{\varphi })=0.\end{array}$$

(65)

Evaluating the fifth, tenth, and eleventh terms of (58), we can derive

$$\begin{array}{c}\hfill \mathbf{B}(\mathbf{D},{\mathbf{D}}^{\varphi })\perp \psi {\mathbf{D}}^{\perp },\mathbf{F}(\mathbf{D},{\mathbf{D}}^{\perp })\perp \mathbf{F}{\mathbf{D}}^{\varphi },\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\varphi })\perp \mathbf{F}{\mathbf{D}}^{\varphi }.\end{array}$$

(66)

Therefore, with (59), (60), and (66), we obtain

$$\begin{array}{c}\hfill \mathbf{B}(\mathbf{D},{\mathbf{D}}^{\varphi })=0,\mathbf{B}(\mathbf{D},{\mathbf{D}}^{\perp })=0.\end{array}$$

(67)

It is well-known that ${\mathbf{M}}_{\varphi }$ is a totally geodesic submanifold in $\mathbf{M}$ and we can analyze that ${\mathbf{M}}_{\varphi }$ is totally geodesic in $\tilde{\mathbf{M}}$ by (63), (65), and (67). Additionally, ${\mathbf{M}}_T$ and ${\mathbf{M}}_{\perp }$ are totally umbilical submanifolds in $\tilde{\mathbf{M}}$ by (62), as ${\mathbf{M}}_T$ and ${\mathbf{M}}_{\perp }$ are totally umbilical in $\mathbf{M}$. Equations (60)–(67) show that $\mathbf{M}$ is a ${\mathbf{D}}^{\varphi }$-totally geodesic submanifold in $\tilde{\mathbf{M}}$. This completes the proof of the theorem. □

3.1. Geometric Applications

In this section, we find a particular case of our main results. Particularizing $\beta =0$ and $\alpha =1$ or $\alpha =0$, along with $n_1=dim{\mathbf{M}}_T=0$, in Theorem 11, we get the following:

Theorem 12.

(Theorem 3.1 of [13] and Theorem 3.1 of [33]) Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_f\mathbf{M}\perp$ be a mixed totally geodesic warped product pseudo-slant submanifold of a nearly Sasakian manifold or a nearly cosymplectic manifold. Then, we have

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2\ge n_2{cot}^2\varphi \left|\right|\overrightarrow{\nabla }ln{f}_2{\left|\right|}^2,\end{array}$$

where $dim{\mathbf{M}}_{\perp }=n_2$.

If we assume that $\alpha =0,\beta =1$ with $dim{\mathbf{M}}_{\perp }=0$, we find another theorem:

Theorem 13.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_f{\mathbf{M}}_T$ be a warped product semi-slant submanifold of nearly Kenmotsu manifold. Then, we have

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2\ge n_1{csc}^2\varphi (1+{cos}^2\varphi )\left(\left|\right|\overrightarrow{\nabla }ln{f}_1{\left)\right||}^2-1\right).\end{array}$$

Remark 5.

It can be noted that only on a nearly Kenmotsu manifold does the warped product semi-slant submanifold of the type $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_f{\mathbf{M}}_T$ exist; in other structures, it becomes a trivial case (see [22]).

Similarly, if $\alpha =0,\beta =1$ with $dim{\mathbf{M}}_T=0$, then we have:

Theorem 14.

(Theorem 4.2 of [14]) Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_f\mathbf{M}\perp$ be a mixed totally geodesic warped product pseudo-slant submanifold of a nearly Kenmotsu manifold. Then, we have

$$\begin{array}{c}\hfill {\left|\right|\mathbf{B}\left|\right|}^2\ge n_2{cot}^2\varphi \left(\left|\right|\overrightarrow{\nabla }ln{f}_2{\left|\right|}^2-1\right),\end{array}$$

where $dim{\mathbf{M}}_{\perp }=n_2.$

3.2. Some Applications Related to Mathematical Physics

In this section, we investigate the Dirichlet energy, which satisfies the following for a compact submanifold $\mathbf{M}$ and differentiable function $\theta :\mathbf{M}⟶\mathbb{R}$:

$$\begin{array}{c}\hfill E\left(\theta \right)=\frac{1}{2}{\int }_{\mathbf{M}}{\parallel \nabla \theta \parallel }^{2}dV,\end{array}$$

(68)

where $dV$ is a volume element. Considering this, we give the following Theorem by combining (40) and (68), where ${\mathbf{M}}_T$ is compact without boundary.

Theorem 15.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, such that ${\mathbf{M}}_T$ is compact without boundary. Then, we have

$$\begin{array}{cc}\hfill 4n_{2}E(ln{f}_1)+& 2n_3\left(\frac{2k}{9}{cot}^2\varphi +2{csc}^2\varphi \right)E(ln{f}_2)\hfill \\ \hfill & \le {\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}\times \left\{n_3\right\}}\{{\parallel \mathbf{B}\parallel }^2+n_3{\beta }^2\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\hfill \\ \hfill & +2n_2\left({\beta }^2-{\alpha }^2\right)-n_3{\alpha }^2\}dV,\hfill \end{array}$$

where $E(ln{f}_1)$ and $E(ln{f}_2)$ are the Dirichlet energies of the warping functions $f_1$ and $f_2$, respectively. The equality cases are the same as in Theorem 1

Similarly, if ${\mathbf{M}}_{\varphi }$ is a compact base without boundary, then Theorems 11 and (68) give the following:

Theorem 16.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$. such that ${\mathbf{M}}_{\varphi }$ is compact without boundary. Then, we have

$$\begin{array}{cc}\hfill 2n_1{csc}^2\varphi (1+& {cos}^2\varphi )E(ln{f}_1)+n_2{cot}^2\varphi E(ln{f}_2)\hfill \\ \hfill & \le {\int }_{{\mathbf{M}}_{\varphi }\times \left\{n_1\right\}\times \left\{n_2\right\}}\left\{{\parallel \mathbf{B}\parallel }^2+{\beta }^2\left(n_2{cot}^2\varphi +n_1{csc}^2\varphi \left(1+{cos}^2\varphi \right)\right)\right\}dV.\hfill \end{array}$$

As immediate applications of Theorem 15, we give corollaries in the following.

Corollary 1.

Assume that $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }$ is a CR-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, such that ${\mathbf{M}}_T$ is compact without boundary. Then, we have

$$\begin{array}{c}\hfill E(ln{f}_1)\le \frac{1}{4n_2}{\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}}\left({\parallel \mathbf{B}\parallel }^2+2n_2\left({\beta }^2-{\alpha }^2\right)\right)dV.\end{array}$$

Corollary 2.

Assume that $\mathbf{M}={\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\varphi }$ is a warped product semi-slant submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, such that ${\mathbf{M}}_T$ is compact without boundary. Then, we have

$$\begin{array}{c}\hfill E(ln{f}_2)\le \left(\frac{9{sin}^2\varphi }{8k\left(10-{sin}^2\varphi \right)}\right){\int }_{{\mathbf{M}}_T\times \left\{n_3\right\}}\left\{{\parallel \mathbf{B}\parallel }^2+2k{\beta }^2\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\theta \right)\right\}dV.\end{array}$$

On the other hand, substituting $n_1=0$ and $n_2=0$ into Theorem 16, we can derive the following corollaries.

Corollary 3.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_{\perp }$ be a warped product pseudo-slant submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, such that ${\mathbf{M}}_{\varphi }$ is compact without boundary. Then, we have

$$\begin{array}{c}\hfill E(ln{f}_2)\le \frac{1}{n_2{cot}^2\varphi }{\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}}\left\{{\parallel \mathbf{B}\parallel }^2+{\beta }^2{n}_2{cot}^2\varphi \right\}dV.\end{array}$$

Corollary 4.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_2}{\mathbf{M}}_T$ be a warped product semi-slant submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$, such that ${\mathbf{M}}_{\varphi }$ is compact without boundary. Then, we have

$$\begin{array}{cc}\hfill 2n_1{csc}^2\varphi (1+& {cos}^2\varphi )E(ln{f}_1)\le {\int }_{{\mathbf{M}}_T\times \left\{n_1\right\}}\left\{{\parallel \mathbf{B}\parallel }^2+{\beta }^2{n}_1{csc}^2\varphi \left(1+{cos}^2\varphi \right)\right\}dV.\hfill \end{array}$$

For the Laplacian, many applications in mathematics as well as in physics can be found. This is possible due to the eigenvalue problem of $\Delta$. The corresponding Laplace eigenvalue equation is defined as follows: A real number $\lambda$ is called an eigenvalue if there exists a non-vanishing function $\theta$ which satisfies the following equation:

$$\begin{array}{c}\hfill \Delta \theta =\lambda \theta ,\mathrm{on}\mathbf{M},\end{array}$$

with appropriate boundary conditions. Considering a Riemannian manifold ${\mathbf{M}}^n$ with no boundary, the first non-zero eigenvalue of $\Delta$, defined as ${\lambda }_1$, includes variational properties (cf. [34]):

$$\begin{array}{c}\hfill {\lambda }_1=inf\left\{{\displaystyle \frac{{\int }_{\mathbf{M}}{\left|\right|\nabla \theta \left|\right|}^{2}dV}{{\int }_{\mathbf{M}}{\parallel \theta \parallel }^{2}dV}}|\theta \in W^{1,2}\left({\mathbf{M}}^n\right)\backslash \left\{0\right\},{\int }_{\mathbf{M}}\theta dV=0\right\}.\end{array}$$

Inspired by the above characterization, using the first non-zero eigenvalue of the Laplace operator and the maximum principle for the first non-zero eigenvalue ${\lambda }_1$, we deduce the following:

Theorem 17.

Let $\mathbf{M}={\mathbf{M}}_T{\times }_{f_1}{\mathbf{M}}_{\perp }{\times }_{f_2}{\mathbf{M}}_{\varphi }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}$ with $\zeta \in {\mathbf{M}}_T$ and such that ${\mathbf{M}}_T$ is a compact base without boundary. If ${\lambda }_1$ and ${\mu }_1$ are the first eigenvalues of eigenfunctions $f_1$ and $f_2$, respectively, then we have following inequality:

$$\begin{array}{cc}\hfill {\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}\times \left\{n_3\right\}}{\parallel \mathbf{B}\parallel }^{2}dV\ge & 2n_2{\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}\times \left\{n_3\right\}}\left\{{\lambda }_1{(ln{f}_1)}^2+{\alpha }^2-{\beta }^2\right\}dV\hfill \\ \hfill & +n_3{\int }_{{\mathbf{M}}_T\times \left\{n_2\right\}\times \left\{n_3\right\}}\left\{\left(\frac{2}{9}{cot}^2\varphi +2{csc}^2\varphi \right)\left\{{\mu }_1{(ln{f}_2)}^2-{\beta }^2\right\}+{\alpha }^2\right\}dV.\hfill \end{array}$$

(69)

Proof. 

Assuming that $\theta$ is a non-constant warping function, by use of the minimum principle for the first eigenvalue ${\lambda }_1$, one can obtain (p. 186, [34]):

$$\begin{array}{c}\hfill {\lambda }_1{\int }_{{\mathbf{M}}^n}{\left(\theta \right)}^{2}dV\le {\int }_{{\mathbf{M}}^n}{\parallel \nabla \theta \parallel }^{2}dV.\end{array}$$

(70)

The equality holds if and only if $\Delta \theta ={\lambda }_1\theta$. Taking the integral in (40) and using (70), we get the required result (69). □

Similarly, using Theorem 11, we have

Theorem 18.

Let $\mathbf{M}={\mathbf{M}}_{\varphi }{\times }_{f_1}{\mathbf{M}}_T{\times }_{f_2}{\mathbf{M}}_{\perp }$ be a bi-warped product submanifold of a nearly trans-Sasakian manifold $\tilde{\mathbf{M}}.$ If ${\mathbf{D}}^{\perp }-{\mathbf{D}}^{\varphi }$ is mixed totally geodesic and ζ is tangent to the compact base ${\mathbf{M}}_{\varphi },$ then we get

$$\begin{array}{cc}\hfill n_1{csc}^2\varphi (1+& {cos}^2\varphi ){\lambda }_1{\int }_{{\mathbf{M}}_{\varphi }\times \left\{n_1\right\}\times \left\{n_2\right\}}{(ln{f}_1)}^{2}dV+n_2{cot}^2\varphi {\mu }_1{\int }_{{\mathbf{M}}_{\varphi }\times \left\{n_1\right\}\times \left\{n_2\right\}}{(ln{f}_2)}^{2}dV\hfill \\ \hfill & \le {\int }_{{\mathbf{M}}_{\varphi }\times \left\{n_1\right\}\times \left\{n_2\right\}}\left\{{\parallel \mathbf{B}\parallel }^2+{\beta }^2\left(n_2{cot}^2\varphi +n_1{csc}^2\varphi \left(1+{cos}^2\varphi \right)\right)\right\}dV.\hfill \end{array}$$

4. Conclusions

It is noted that the nearly trans-Sasakian structures generalize some remarkable geometric structures on manifolds, like nearly cosymplectic, nearly Sasakian, nearly Kenmotsu, nearly $\alpha$-Sasakian, and nearly $\beta$-Kenmotsu. The main target of this paper is to discuss the geometry of bi-warped product submanifolds of some special types in nearly trans-Sasakian manifolds. In particular, we derived some basic inequalities, which turn out to be generalization of various known results obtained by several mathematicians in the last decade. The eigenvalues inequalities are established and the Dirichlet energy inequalities are derived that give the new motivation of such studies.