V-Quasi-Bi-Slant Riemannian Maps

Abstract

In this work, we define a v-quasi-bi-slant Riemannian map (in brief, v-QBSR map) from almost Hermitian manifolds to Riemannian manifolds. This notion generalizes both a v-hemi slant Riemannian map and a v-semi slant Riemannian map. The geometry of leaves of distributions that are associated with the definition of such maps is studied. The conditions for v-QBSR maps to be integrable and totally geodesic are also obtained in the paper. Finally, we provide the examples of v-QBSR maps.

1. Introduction

Let $\pi$ be a $C^{\infty }$-map from a Riemannian manifold $(N_1,g_1)$ to another Riemannian manifold $(N_2,g_2)$. The notion of a Riemannian map between two Riemannian manifolds was introduced by Fischer [1], and generalizes both the notion of an isometric immersion and a Riemannian submersion. The theory of Riemannian submersion is discussed in [2]. In [1], Fischer also defined the concept of a Riemannian map in the following way: Let $\pi :(N_1,g_1)\to (N_2,g_2)$ be a differentiable map between Riemannian manifolds such that $0<rank$ ${\pi }_{\ast }<min\{m,n\}$ $(dim{N}_1=m,dim{N}_2=n)$. If we denote the $kernel$ space of ${\pi }_{\ast }$ by $ker{\pi }_{\ast }$ and the orthogonal complementary space of $ker{\pi }_{\ast }$ by ${(ker{\pi }_{\ast })}^{\perp }$ in $T{N}_1$, then $T{N}_1$ has the following orthogonal decomposition:

$$T{N}_1={(ker{\pi }_{\ast })}^{\perp }\oplus ker{\pi }_{\ast }.$$

Here, if the $range$ of ${\pi }_{\ast }$ is denoted by $range{\pi }_{\ast }$ and for a point $q\in N_1$, the orthogonal complementary space of $range{\pi }_{\ast \pi \left(q\right)}$ by ${\left(range{\pi }_{\ast \pi \left(q\right)}\right)}^{\perp }$ in $T_{\pi \left(q\right)}{N}_2.$ Then, the tangent space $T_{\pi \left(q\right)}{N}_2$ has the following orthogonal decomposition:

$$T_{\pi \left(q\right)}{N}_2=\left(range{\pi }_{\ast \pi \left(q\right)}\right)\oplus {\left(range{\pi }_{\ast \pi \left(q\right)}\right)}^{\perp }.$$

A differentiable map $\pi :(N_1,g_1)\to (N_2,g_2)$ is named a Riemannian map at $q\in N_1$ if the horizontal restriction ${\pi }_{\ast q}^h:{(ker{\pi }_{\ast q})}^{\perp }\to \left(range{\pi }_{\ast \pi \left(q\right)}\right)$ is a linear isometry between the inner product space $({(ker{\pi }_{\ast q})}^{\perp },{\left(g_1\right)}_{\left(q\right)}{|}_{{(ker{\pi }_{\ast q})}^{\perp }})$ and $(range{\pi }_{\ast {\pi }_{\left(q\right)}},$ ${\left(g_2\right)}_{\left(\pi \right(q)}{\left)\right|}_{\left(range{\pi }_{\ast q}\right)}).$ We see that a peculiar property of a Riemannian map is that it satisfies the generalized eikonal equation $2e\left(\pi \right)=\left|\right|{\pi }_{\ast }{\left|\right|}^2=rank\pi$. This equation works as a bridge between physical and geometrical optics [1]. By using Cauchy’s method of characteristics, the eikonal equation of geometrical optics was solved with the help of Cauchy’s method of characteristic.

The Riemannian maps have several important applications in mathematics as well as in physics, especially within the Yang–Mills theory [3], Kaluza–Klein theory [4], supergravity and superstring theories [5], redundant robotic chains [6] etc. General relativity, which relates the gravitational force to the curvature of space-time, provides a respectable theory of gravity on a larger scale. Supergravity theories permit extra dimensions in space-time, beyond the familiar three dimensions of space and one of time. Supergravity models in higher dimensions reduce to the familiar four-dimensional space-time if it is postulated that the extra dimensions are compacted or curled up in such a way that they are not noticeable. The advantage of the extra dimensions is that they allow supergravity theories to incorporate the electromagnetic, weak, and strong forces as well as gravity.

Moreover, Şahin introduced many types of Riemannian maps ([7,8,9,10,11,12]; see also [13,14,15,16,17,18]). One may consult the references [19,20,21] for further studies.

In [22], Park introduced v-semi-slant submersions from almost Hermitian manifolds, and in [23], Sepet and Bozok proposed v-semi-slant submersions from almost product Riemannian manifolds onto Riemannian manifolds. In this paper, we are interested in studying the above idea in the setting of v-QBSR maps. The article is organized as follows: Some basic information about Riemannian maps is given in Section 2. In Section 3, using v-QBSR maps, we obtain some properties, results and decomposition theorems. In Section 4, we present the examples of the v-QBSR maps.

2. Preliminaries

Let $N_1$ be a $C^{\infty }$-manifold of even dimension [24]. Let J be a $(1,1)$ tensor field defined on $N_1$ such that $J^2=-I,$ where I denotes the identity operator; then, J is called an almost complex structure on $N_1$, which is an almost complex manifold. We know that an almost complex manifold is orientable. The Nijenhuis tensor N of type $(1,2)$ of an almost complex structure J is defined as:

$$N(Z_1,Z_2)=[J{Z}_1,J{Z}_2]-[Z_1,Z_2]-J[J{Z}_1,Z_2]-J[Z_1,J{Z}_2]\mathrm{for}\mathrm{any}{Z}_1,Z_2\in \Gamma \left(T{N}_1\right).$$

If the Nijenhuis tensor N of an almost complex structure J on the manifold $N_1$ vanishes then the manifold $N_1$ becomes a complex manifold.

Let us consider a Riemannian metric $g_1$ on the manifold $N_1$ as:

$$g_1(J{V}_1,J{V}_2)=g_1(V_1,V_2),\mathrm{for}\mathrm{any}{V}_1,V_2\in \Gamma \left(T{N}_1\right),$$

(1)

then $g_1$ is called a Hermitian metric on the manifold $N_1$ and $N_1$ with the metric $g_1$ called an almost Hermitian manifold. The Levi–Civita connection ∇ on the manifold $N_1$ can be extended to the whole tensor algebra on the manifold $N_1.$ The tensor fields $\left({\nabla }_{V_1}J\right)$ are defined as:

$$\left({\nabla }_{V_1}J\right)V_2={\nabla }_{V_1}J{V}_2-J{\nabla }_{V_1}{V}_2$$

for any $V_1,V_2\in \Gamma \left(T{N}_1\right)$.

An almost Hermitian manifold $(N_1,g_1,J)$ is called a Kähler manifold if:

$$\left({\nabla }_{V_1}J\right)V_2=0$$

(2)

for any $V_1,V_2\in \Gamma \left(T{N}_1\right)$.

Now, we mention the following definition for further use:

Definition 1.

Ref. [22]: Let $\pi :(N_1,g_1,J)\to (N_2,g_2)$ be a Riemannian map. Then we say that π is a v-semi-slant Riemannian map if there is a distribution ${\mathcal{D}}_1\subset {(ker{\pi }_{\ast })}^{\perp }$ such that:

$${(ker{\pi }_{\ast })}^{\perp }={\mathcal{D}}_2\oplus {\mathcal{D}}_1,J\left({\mathcal{D}}_1\right)={\mathcal{D}}_1,$$

and the angle $\theta =\theta \left(Z_1\right)$ between $J{Z}_1$ and the space ${\left({\mathcal{D}}_2\right)}_p$ is constant for non-zero $Z_1\in {\left({\mathcal{D}}_2\right)}_p$ and $p\in N_1,$ where ${\mathcal{D}}_2$ is the orthogonal complement of ${\mathcal{D}}_1$ in ${(ker{\pi }_{\ast })}^{\perp }.$ We call the angle θ a v-semi-slant angle.

Let $\mathcal{T}$ and $\mathcal{A}$ be O’Neill’s tensors defined as [25]:

$${\mathcal{A}}_{E_1}{E}_2=\mathcal{H}{\nabla }_{\mathcal{H}{E}_1}\mathcal{V}{E}_2+\mathcal{V}{\nabla }_{\mathcal{H}{E}_1}\mathcal{H}{E}_2,$$

(3)

$${\mathcal{T}}_{E_1}{E}_2=\mathcal{H}{\nabla }_{\mathcal{V}{E}_1}\mathcal{V}{E}_2+\mathcal{V}{\nabla }_{\mathcal{V}{E}_1}\mathcal{H}{E}_2$$

(4)

For any vector fields $E_1,E_2$ on $N_1.$ It is well known that ${\mathcal{T}}_{E_1}$ and ${\mathcal{A}}_{E_1}$ are operators on the tangent bundle of $N_1,$ which are skew-symmetric and reversing the vertical and horizontal distributions.

Using Equations (3) and (4), we obtain:

$${\nabla }_{W_1}{W}_2={\mathcal{T}}_{W_1}{W}_2+\mathcal{V}{\nabla }_{W_1}{W}_2,$$

(5)

$${\nabla }_{W_1}{V}_1={\mathcal{T}}_{W_1}{V}_1+\mathcal{H}{\nabla }_{W_1}{V}_1,$$

(6)

$${\nabla }_{V_1}{W}_1={\mathcal{A}}_{V_1}{W}_1+\mathcal{V}{\nabla }_{V_1}{W}_1,$$

(7)

$${\nabla }_{V_1}{V}_2=\mathcal{H}{\nabla }_{V_1}{V}_2+{\mathcal{A}}_{V_1}{V}_2$$

(8)

For any $W_1,W_2\in \Gamma (ker{\pi }_{\ast })$ and $V_1,V_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp },$ where $\mathcal{H}{\nabla }_{W_1}{V}_1={\mathcal{A}}_{V_1}{W}_1,$ if $V_1$ is basic.

Let $\pi :(N_1,g_1)\to (N_2,g_2)$ be a $C^{\infty }$-map; then the second fundamental form [26] of $\pi$ is given by:

$$(\nabla {\pi }_{\ast })(Z_1,Z_2)={\nabla }_{Z_1}^{\pi }{\pi }_{\ast }\left(Z_2\right)-{\pi }_{\ast }\left({\nabla }_{Z_1}^{N_1}{Z}_2\right)$$

(9)

For any $Z_1,Z_2\in \Gamma \left(T{N}_1\right),$ where ${\nabla }^{\pi }$ is the pullback connection. Conveniently, the Riemannian connections of the metrics $g_1$ and $g_2$ [27] are denoted by ∇.

Furthermore, a $C^{\infty }$-map $\pi$ between $(N_1,g_1)$ and $(N_2,g_2)$ is called totally geodesic [27] if:

$$(\nabla {\pi }_{\ast })(Z_1,Z_2)=0,\mathrm{for}\mathrm{any}{Z}_1,Z_2\in \Gamma \left(T{N}_1\right).$$

(10)

3. V-QBSR Maps

In this section, v-QBSR maps from ($N_1,g_1,J$) to ($N_2,g_2$) are defined and studied. We now present the notion of a v-QBSR map as follows:

Definition 2.

A Riemannian map

$$\pi :(N_1,g_1,J)\to (N_2,g_2),$$

is called a v-QBSR map if there exist orthogonal distributions $D,D_1$ and $D_2$ such that:

(i)

${(ker{\pi }_{\ast })}^{\perp }=D\oplus D_1\oplus D_2;$

(ii)

D is invariant, that is, $J\left(D\right)=D$;

(iii)

$J\left(D_1\right)\perp D_2$ and $J\left(D_2\right)\perp D_1;$

(iv)

for any non-zero $Z_i\in {\left(D_i\right)}_{q_i}$, $q_i\in N_1,$ the angle ${\theta }_i$ between $J{Z}_i$ and ${\left(D_i\right)}_{q_i}$ is constant and not dependent on points $q_i$ and $Z_i$ in ${\left(D_i\right)}_{q_i},$ where $i=1,2.$ The angles ${\theta }_1$ and ${\theta }_2$ are called v-slant angles of the Riemannian map.

Let $\pi$ be a v-QBSR map from $(N_1,g_{1,}J)$ to $(N_2,g_2).$ Then we have:

$$T{N}_1=ker{\pi }_{\ast }\oplus {(ker{\pi }_{\ast })}^{\perp }.$$

(11)

Furthermore, we put

$$V_1=P{V}_1+Q{V}_1+R{V}_1,\mathrm{for}\mathrm{any}{V}_1\in \Gamma {(ker{\pi }_{\ast })}^{\perp },$$

(12)

where $P,Q$ and R are projection morphisms of ${(ker{\pi }_{\ast })}^{\perp }$ onto $D,D_1$ and $D_2,$ respectively. For $V_2\in {(\Gamma ker{\pi }_{\ast })}^{\perp },$ we set

$$J{V}_2=B{V}_2+C{V}_2,$$

(13)

where $B{V}_2\in (\Gamma ker{\pi }_{\ast })$ and $C{V}_2\in {(\Gamma ker{\pi }_{\ast })}^{\perp }.$

Using Equations (12) and (13), we obtain:

$$\begin{array}{ccc}\hfill J{V}_1& =& J\left(P{V}_1\right)+J\left(Q{V}_1\right)+J\left(R{V}_1\right),\hfill \\ & =& B\left(P{V}_1\right)+C\left(P{V}_1\right)+B\left(Q{V}_1\right)+C\left(Q{V}_1\right)+B\left(R{V}_1\right)+C\left(R{V}_1\right).\hfill \end{array}$$

Since D is an invariant, we have $BP{V}_1=0.$ Thus, the above equation reduces to

$$J{V}_1=C\left(P{V}_1\right)+B\left(Q{V}_1\right)+C\left(Q{V}_1\right)+B\left(R{V}_1\right)+C\left(R{V}_1\right).$$

(14)

Additionally, we can express

$$J{(ker{\pi }_{\ast })}^{\perp }=D\oplus (B{D}_1\oplus B{D}_2)\oplus (C{D}_1\oplus C{D}_2).$$

(15)

If $Z_1\in \Gamma \left(D_1\right)$ and $Z_2\in \Gamma \left(D_2\right);$ then:

$$g_1(Z_1,Z_2)=0.$$

From Definition $2\left(iii\right),$ we have:

$$g_1(J{Z}_1,Z_2)=0\mathrm{and}{g}_1(Z_1,J{Z}_2)=0.$$

Now, we consider:

$$\begin{array}{ccc}\hfill g_1(C{Z}_1,Z_2)& =& g_1(J{Z}_1-B{Z}_1,Z_2),\hfill \\ & =& g_1(J{Z}_1,Z_2),\hfill \\ & =& 0.\hfill \end{array}$$

In a similarly way, we obtain:

$$g_1(Z_1,C{Z}_2)=0.$$

Let $V_1\in \Gamma \left(D\right)$ and $V_2\in \Gamma \left(D_1\right).$ Then, we have:

$$\begin{array}{ccc}\hfill g_1(C{V}_1,V_2)& =& g_1(J{V}_1-B{V}_1,V_2),\hfill \\ & =& g_1(J{V}_1,V_2),\hfill \\ & =& -g_1(V_1,J{V}_2),\hfill \\ & =& 0,\hfill \end{array}$$

As D is an invariant, i.e., $J{V}_1\in \Gamma \left(D\right).$

Similarly, for $V_1\in \Gamma \left(D\right)$ and $V_3\in \Gamma \left(D_2\right),$ we obtain:

$$g_1(C{V}_1,V_3)=0.$$

From the above equations, we have:

$$g_1(C{Z}_1,C{Z}_2)=0,$$

and

$$g_1(B{Z}_1,B{Z}_2)=0,$$

for any $Z_1\in \Gamma \left(D_1\right)$ and $Z_2\in \Gamma \left(D_2\right).$

Thus, we can write

$$C{D}_1\cap C{D}_2=\left\{0\right\},B{D}_1\cap B{D}_2=\left\{0\right\}.$$

If ${\theta }_2=\frac{\pi }{2},$ then $CR=0$ and $D_2$ is anti-invariant, i.e., $J\left(D_2\right)\subseteq (ker{\pi }_{\ast }).$ In this case, we denote $D_2$ by $D^{\perp }.$ We also have

$$J{(ker{\pi }_{\ast })}^{\perp }=D\oplus C{D}_1\oplus J{D}^{\perp }.$$

(16)

Since $B{D}_1\subseteq (ker{\pi }_{\ast }),$ $B{D}_2\subseteq (ker{\pi }_{\ast })$, we can write

$$(ker{\pi }_{\ast })=B{D}_1\oplus B{D}_2\oplus \mathcal{V},$$

where $\mathcal{V}$ denotes the orthogonal complement of $(B{D}_1\oplus B{D}_2)$ in $(ker{\pi }_{\ast }).$

Additionally, for any $V_1\in \Gamma (ker{\pi }_{\ast }),$ we have

$$J{V}_1=\varphi V_1+\omega V_1,$$

(17)

where $\varphi V_1\in \Gamma (ker{\pi }_{\ast })$ and $\omega V_1\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Lemma 1.

For $\pi ,$ we have:

$${\varphi }^2{V}_1+B\omega V_1=-V_1,\omega \varphi V_1+C\omega V_1=0,$$

$$\omega B{V}_2+C^2{V}_2=-V_2,\varphi B{V}_2+BC{V}_2=0$$

for any $V_1\in \Gamma (ker{\pi }_{\ast })$ and $V_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

With the help of Equations (13) and (17) along with the condition $J^2=-I,$ we obtain Lemma 1. □

Lemma 2.

For $\pi ,$ we obtain:

$\left(i\right)$$C^2{V}_i=-\left({cos}^2{\theta }_i\right)V_i;$

$\left(ii\right)$$g_1(C{V}_i,C{U}_i)={cos}^2{\theta }_i{g}_1(V_i,U_i);$

$\left(iii\right)$$g_1(B{V}_i,B{U}_i)={sin}^2{\theta }_i{g}_1(V_i,U_i),$ for $V_i,U_i\in \Gamma \left(D_i\right),$ where $i=1,2.$

Proof. 

The proof of Lemma 2 is the same as the one for v-semi-slant submersion (see Proposition (3.5) and Remark (3.6) of [22]). Hence we omit it. □

Lemma 3.

For $\pi ,$ we have:

$$\mathcal{V}{\nabla }_{Z_1}\varphi Z_2+{\mathcal{T}}_{Z_1}\omega Z_2=\varphi \mathcal{V}{\nabla }_{Z_1}{Z}_2+B{\mathcal{T}}_{Z_1}{Z}_2,$$

(18)

$${\mathcal{T}}_{Z_1}\varphi Z_2+\mathcal{H}{\nabla }_{Z_1}\omega Z_2=\omega \mathcal{V}{\nabla }_{Z_1}{Z}_2+C{\mathcal{T}}_{Z_1}{Z}_2,$$

(19)

$$\mathcal{V}{\nabla }_{W_1}B{W}_2+{\mathcal{A}}_{W_1}C{W}_2=\varphi {\mathcal{A}}_{W_1}{W}_2+B\mathcal{H}{\nabla }_{W_1}{W}_2,$$

(20)

$${\mathcal{A}}_{W_1}B{W}_2+\mathcal{H}{\nabla }_{W_1}C{W}_2=\omega {\mathcal{A}}_{W_1}{W}_2+C\mathcal{H}{\nabla }_{W_1}{W}_2,$$

(21)

$$\mathcal{V}{\nabla }_{Z_1}B{W}_1+{\mathcal{T}}_{Z_1}C{W}_1=\varphi {\mathcal{T}}_{Z_1}{W}_1+B\mathcal{H}{\nabla }_{Z_1}{W}_1,$$

(22)

$${\mathcal{T}}_{Z_1}B{W}_1+\mathcal{H}{\nabla }_{Z_1}C{W}_1=\omega {\mathcal{T}}_{Z_1}{W}_1+C\mathcal{H}{\nabla }_{Z_1}{W}_1,$$

(23)

$$\mathcal{V}{\nabla }_{W_1}\varphi Z_1+\mathcal{A}{\nabla }_{W_1}\omega Z_1=B{\mathcal{A}}_{W_1}{Z}_1+\varphi \mathcal{V}{\nabla }_{W_1}{Z}_1,$$

(24)

$${\mathcal{A}}_{W_1}\varphi Z_1+\mathcal{H}{\nabla }_{W_1}\omega Z_1=C{\mathcal{A}}_{W_1}{Z}_1+\omega \mathcal{V}{\nabla }_{W_1}{Z}_1$$

(25)

for any $Z_1,Z_2\in \Gamma (ker{\pi }_{\ast })$ and $W_1,W_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

By Equations (5)–(8), (13) and (17), we obtain Equations (18)–(25). □

Next, we define

$$\left({\nabla }_{Z_1}\varphi \right)Z_2=\mathcal{V}{\nabla }_{Z_1}\varphi Z_2-\varphi \mathcal{V}{\nabla }_{Z_1}{Z}_2,$$

(26)

$$\left({\nabla }_{Z_1}\omega \right)Z_2=\mathcal{H}{\nabla }_{Z_1}\omega Z_2-\omega \mathcal{V}{\nabla }_{Z_1}{Z}_2,$$

(27)

$$\left({\nabla }_{W_1}C\right)W_2=\mathcal{H}{\nabla }_{W_1}C{W}_2-C\mathcal{H}{\nabla }_{W_1}{W}_2,$$

(28)

$$\left({\nabla }_{W_1}B\right)W_2=\mathcal{V}{\nabla }_{W_1}B{W}_2-B\mathcal{H}{\nabla }_{W_1}{W}_2$$

(29)

for any $Z_1,Z_2\in \Gamma (ker{\pi }_{\ast })$ and $W_1,W_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Lemma 4.

For $\pi ,$ we obtain:

$$\left({\nabla }_{Z_1}\varphi \right)Z_2=B{\mathcal{T}}_{Z_1}{Z}_2-{\mathcal{T}}_{Z_1}\omega Z_2,$$

$$\left({\nabla }_{Z_1}\omega \right)Z_2=C{\mathcal{T}}_{Z_1}{Z}_2-{\mathcal{T}}_{Z_1}\varphi Z_2,$$

$$\left({\nabla }_{W_1}C\right)W_2=\omega {\mathcal{A}}_{W_1}{W}_2-{\mathcal{A}}_{W_1}B{W}_2,$$

$$\left({\nabla }_{W_1}B\right)W_2=\varphi {\mathcal{A}}_{W_1}{W}_2-{\mathcal{A}}_{W_1}C{W}_2$$

for any $Z_1,Z_2\in \Gamma (ker{\pi }_{\ast })$ and $W_1,W_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

On the account of Equations (18)–(21) and (26)–(29), we obtain the required results of Lemma $4.$ □

Consequently, if $\varphi$ and $\omega$ are parallel tensor with respect to ∇ defined on $N_1$, we obtain:

$$B{\mathcal{T}}_{Z_1}{Z}_2={\mathcal{T}}_{Z_1}\omega Z_2,C{\mathcal{T}}_{Z_1}{Z}_2={\mathcal{T}}_{Z_1}\varphi Z_2$$

for any $Z_1,Z_2\in \Gamma \left(T{N}_1\right).$

Theorem 1.

Let π be a v-QBSR map. Then, D is integrable if and only if

$$g_1(\mathcal{H}{\nabla }_{V_2}J{V}_1-\mathcal{H}{\nabla }_{V_1}J{V}_2,CQ{Y}_1+CR{Y}_1)=g_1({\mathcal{A}}_{V_1}J{V}_2-{\mathcal{A}}_{V_2}J{V}_1,BQ{Y}_1+BR{Y}_1)$$

for $V_1,V_2\in \Gamma \left(D\right)$ and $Y_1\in \Gamma (D_1\oplus D_2)$.

Proof. 

For $V_1,V_2\in \Gamma \left(D\right)$ and $Y_1\in \Gamma (D_1\oplus D_2),$ using Equations (2), (8), (12) and (13), we have

$$\begin{array}{ccc}& & g_1([V_1,V_2],Y_1)\hfill \\ & =& g_1({\nabla }_{V_1}J{V}_2,J{Y}_1)-g_1({\nabla }_{V_2}J{V}_1,J{Y}_1),\hfill \\ & =& g_1({\nabla }_{V_1}J{V}_2,JQ{Y}_1+JR{Y}_1)-g_1({\nabla }_{V_2}J{V}_1,JQ{Y}_1+JR{Y}_1),\hfill \\ & =& g_1(\mathcal{H}{\nabla }_{V_1}J{V}_2-\mathcal{H}{\nabla }_{V_2}J{V}_1,CQ{Y}_1+CR{Y}_1)+g_1({\mathcal{A}}_{V_1}J{V}_2-{\mathcal{A}}_{V_2}J{V}_1,B{Y}_1),\hfill \end{array}$$

which completes the proof. □

Theorem 2.

Let π be a v-QBSR map. Then, $D_1$ is integrable if and only if

$$\begin{array}{ccc}& & g_1({\mathcal{A}}_{X_1}B{X}_2-{\mathcal{A}}_{X_2}B{X}_1,JP{Z}_1+CR{Z}_1)\hfill \\ & =& g_1({\mathcal{A}}_{X_1}BC{X}_2-{\mathcal{A}}_{X_2}BC{X}_1,Z_1)-g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2-\mathcal{V}{\nabla }_{X_2}B{X}_1,BR{Z}_1)\hfill \end{array}$$

for $X_1,X_2\in \Gamma \left(D_1\right)$ and $Z_1\in \Gamma (D\oplus D_2).$

Proof. 

For $X_1,X_2\in \Gamma \left(D_1\right)$ and $Z_1\in \Gamma (D\oplus D_2)$, we have

$$g_1([X_1,X_2],Z_1)=g_1({\nabla }_{X_1}{X}_2,Z_1)-g_1({\nabla }_{X_1}{X}_2,Z_1).$$

Using Equations (2), (7), (12) and (13) and Lemma $2,$ we have:

$$\begin{array}{ccc}& & g_1([X_1,X_2],Z_1)\hfill \\ & =& g_1({\nabla }_{X_1}J{X}_2,J{Z}_1)-g_1({\nabla }_{X_2}J{X}_1,J{Z}_1),\hfill \\ & =& g_1({\nabla }_{X_1}B{X}_2,JP{Z}_1+BR{Z}_1+CR{Z}_1)-g_1({\nabla }_{X_2}B{X}_1,JP{Z}_1+BR{Z}_1+CR{Z}_1)+\hfill \\ & & g_1({\nabla }_{X_1}C{X}_2,J{Z}_1)-g_1({\nabla }_{X_2}C{X}_1,J{Z}_1),\hfill \\ & =& {cos}^2{\theta }_1{g}_1([X_1,X_2],Z_1)+g_1({\mathcal{A}}_{X_1}B{X}_2-{\mathcal{A}}_{X_2}B{X}_1,JP{Z}_1+CR{Z}_1)-\hfill \\ & & g_1({\mathcal{A}}_{X_1}BC{X}_2-{\mathcal{A}}_{X_2}BC{X}_1,Z_1)+g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2-\mathcal{V}{\nabla }_{X_2}B{X}_1,BR{Z}_1).\hfill \end{array}$$

Now, we have:

$$\begin{array}{ccc}& & {sin}^2{\theta }_1{g}_1([X_1,X_2],Z_1)\hfill \\ & =& g_1({\mathcal{A}}_{X_1}B{X}_2-{\mathcal{A}}_{X_2}B{X}_1,JP{Z}_1+CR{Z}_1)-g_1({\mathcal{A}}_{X_1}BC{X}_2-{\mathcal{A}}_{X_2}BC{X}_1,Z_1)\hfill \\ & & +g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2-\mathcal{V}{\nabla }_{X_2}B{X}_1,BR{Z}_1),\hfill \end{array}$$

from which the proof follows. □

Theorem 3.

Let π be a v-QBSR map. Then, $D_2$ is integrable if and only if

$$\begin{array}{ccc}& & g_1({\mathcal{A}}_{Y_1}B{Y}_2-{\mathcal{A}}_{Y_2}B{Y}_1,JP{V}_1+CR{V}_1)\hfill \\ & =& g_1({\mathcal{A}}_{Y_1}BC{Y}_2-{\mathcal{A}}_{Y_2}BC{Y}_1,V_1)-g_1(\mathcal{V}{\nabla }_{Y_1}B{Y}_2-\mathcal{V}{\nabla }_{Y_2}B{Y}_1,BR{V}_1),\hfill \end{array}$$

for $Y_1,Y_2\in \Gamma \left(D_2\right)$ and $V_1\in \Gamma (D\oplus D_1).$

Proof. 

By considering the similar approach as in the proof of Theorem $2,$ we obtain the above result. □

Theorem 4.

Let π be a v-QBSR map. The distribution, ${(ker{\pi }_{\ast })}^{\perp }$ becomes a totally geodesic foliation on $N_1$ if and only if

$$\begin{array}{ccc}& & g_1({\mathcal{A}}_{Z_1}P{Z}_2+{cos}^2{\theta }_1{\mathcal{A}}_{Z_1}Q{Z}_2+{cos}^2{\theta }_2{\mathcal{A}}_{Z_1}R{Z}_2,V_1)\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{Z_1}BCP{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCQ{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCR{Z}_2,V_1)\hfill \\ & & +g_1({\mathcal{A}}_{Z_1}B{Z}_2,\omega V_1)+g_1(\mathcal{V}{\nabla }_{Z_1}B{Z}_2,\varphi V_1),\hfill \end{array}$$

for $Z_1,Z_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }$ and $V_1\in \Gamma (ker{\pi }_{\ast }).$

Proof. 

For $Z_1,Z_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }$ and $V_1\in \Gamma (ker{\pi }_{\ast }),$ we have

$$g_1({\nabla }_{Z_1}{Z}_2,V_1)=g_1({\nabla }_{Z_1}J{Z}_2,J{V}_1).$$

Using Equations (2), (7), (8), (12), (13) and (17) and Lemma $2,$ we have:

$$\begin{array}{ccc}\hfill g_1({\nabla }_{Z_1}{Z}_2,V_1)& =& g_1({\nabla }_{Z_1}JP{Z}_2,J{V}_1)+g_1({\nabla }_{Z_1}JQ{Z}_2,J{V}_1)+g_1({\nabla }_{Z_1}JR{Z}_2,J{V}_1),\hfill \\ & =& g_1({\mathcal{A}}_{Z_1}P{Z}_2+{cos}^2{\theta }_1{\mathcal{A}}_{Z_1}Q{Z}_2+{cos}^2{\theta }_2{\mathcal{A}}_{Z_1}R{Z}_2,V_1)-\hfill \\ & & g_1(\mathcal{V}{\nabla }_{Z_1}BCP{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCQ{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCR{Z}_2,V_1)+\hfill \\ & & g_1({\nabla }_{Z_1}BP{Z}_2+{\nabla }_{Z_1}BQ{Z}_2+{\nabla }_{Z_1}BR{Z}_2,\varphi V_1)+\hfill \\ & & g_1({\nabla }_{Z_1}BP{Z}_2+{\nabla }_{Z_1}BQ{Z}_2+{\nabla }_{Z_1}BR{Z}_2,\omega V_1).\hfill \end{array}$$

Now, since $BP{Z}_2+BQ{Z}_2+BR{Z}_2=B{Z}_2$ and $BP{Z}_2=0,$ we obtain:

$$\begin{array}{ccc}\hfill g_1({\nabla }_{Z_1}{Z}_2,V_1)& =& g_1({\mathcal{A}}_{Z_1}P{Z}_2+{cos}^2{\theta }_1{\mathcal{A}}_{Z_1}Q{Z}_2+{cos}^2{\theta }_2{\mathcal{A}}_{Z_1}R{Z}_2,V_1)-\hfill \\ & & g_1(\mathcal{V}{\nabla }_{Z_1}BCP{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCQ{Z}_2+\mathcal{V}{\nabla }_{Z_1}BCR{Z}_2,V_1)+\hfill \\ & & +g_1({\mathcal{A}}_{Z_1}B{Z}_2,\omega V_1)+g_1(\mathcal{V}{\nabla }_{Z_1}B{Z}_2,\varphi V_1).\hfill \end{array}$$

□

Theorem 5.

Let π be a v-QBSR map. The distribution $(ker{\pi }_{\ast })$ becomes a totally geodesic foliation on $N_1$ if and only if

$$\begin{array}{ccc}& & g_1({\mathcal{T}}_{X_1}{X}_2,P{Z}_1+{cos}^2{\theta }_{1}Q{Z}_1+{cos}^2{\theta }_{2}R{Z}_1)\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{X_1}{X}_2,BCP{Z}_1+BCQ{Z}_1+BCR{Z}_1)+\hfill \\ & & g_1({\mathcal{T}}_{X_1}\omega X_2,B{Z}_1)+g_1(\mathcal{V}{\nabla }_{X_1}\varphi X_2,B{Z}_1),\hfill \end{array}$$

for $X_1,X_2\in \Gamma (ker{\pi }_{\ast })$ and $Z_1\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

For $X_1,X_2\in \Gamma (ker{\pi }_{\ast })$; $Z_1\in \Gamma {(ker{\pi }_{\ast })}^{\perp },$ with the help of Equations (2) and (12), we have

$$\begin{array}{ccc}& & g_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ & =& g_1({\nabla }_{X_1}{X}_2,P{Z}_1+Q{Z}_1+R{Z}_1).\hfill \end{array}$$

Now, using Equations (5), (6), (13) and (17) and Lemma $2,$ we have:

$$\begin{array}{ccc}& & g_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ & =& g_1({\nabla }_{X_1}J{X}_2,JP{Z}_1)+g_1({\nabla }_{X_1}J{X}_2,JQ{Z}_1)+g_1({\nabla }_{X_1}J{X}_2,JR{Z}_1),\hfill \\ & =& g_1({\mathcal{T}}_{X_1}{X}_2,P{Z}_1+{cos}^2{\theta }_{1}Q{Z}_1+{cos}^2{\theta }_{2}R{Z}_1)-\hfill \\ & & g_1(\mathcal{V}{\nabla }_{X_1}{X}_2,BCP{Z}_1+BCQ{Z}_1+BCR{Z}_1)+\hfill \\ & & g_1({\nabla }_{X_1}\varphi X_2+{\nabla }_{X_1}\omega X_2,BP{Z}_1+BQ{Z}_1+BR{Z}_1).\hfill \end{array}$$

Now, since $BP{Z}_1+BQ{Z}_1+BR{Z}_1=B{Z}_1$ and $BP{Z}_1=0,$ we have:

$$\begin{array}{ccc}& & g_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ & =& g_1({\mathcal{T}}_{X_1}{X}_2,P{Z}_1+{cos}^2{\theta }_{1}Q{Z}_1+{cos}^2{\theta }_{2}R{Z}_1)-\hfill \\ & & g_1(\mathcal{V}{\nabla }_{X_1}{X}_2,BCP{Z}_1+BCQ{Z}_1+BCR{Z}_1)+\hfill \\ & & g_1({\mathcal{T}}_{X_1}\omega X_2+\mathcal{V}{\nabla }_{X_1}\varphi X_2,B{Z}_1).\hfill \end{array}$$

□

Theorem 6.

Let π be a v-QBSR map. Then, D defines a totally geodesic foliation on $N_1$ if and only if

$$\begin{array}{ccc}& & g_1(A_{Y_1}P{Y}_2,BCQ{U}_1+BCR{U}_1)\hfill \\ & =& g_1({\mathcal{A}}_{Y_1}JP{Y}_2,BQ{U}_1+BR{U}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}P{Y}_2,{cos}^2{\theta }_{1}Q{U}_1+{cos}^2{\theta }_{2}R{U}_1),\hfill \end{array}$$

and

$$g_1({\mathcal{A}}_{Y_1}JP{Y}_2,\varphi W_1)=-g_1(\mathcal{H}{\nabla }_{Y_1}JP{Y}_2,\omega W_1),$$

for $Y_1,Y_2\in \Gamma \left(D\right),U_1\in \Gamma (D_1\oplus D_2)$ and $W_1\in \Gamma (ker{\pi }_{\ast }).$

Proof. 

For $Y_1,Y_2\in \Gamma \left(D\right),U_1\in \Gamma (D_1\oplus D_2)$ and $W_1\in \Gamma (ker{\pi }_{\ast }),$ using Equations (2), (8), (12), (13) and (17) and Lemma $2,$ we have:

$$\begin{array}{ccc}& & g_1({\nabla }_{Y_1}{Y}_2,U_1)\hfill \\ & =& g_1({\nabla }_{Y_1}J{Y}_2,J{U}_1),\hfill \\ & =& g_1({\nabla }_{Y_1}JP{Y}_2,JQ{U}_1+JR{U}_1),\hfill \\ & =& g_1({\mathcal{A}}_{Y_1}JP{Y}_2,BQ{U}_1+BR{U}_1)+g_1(\mathcal{H}{\nabla }_{Y_1}P{Y}_2,{cos}^2{\theta }_{1}Q{U}_1+{cos}^2{\theta }_{2}R{U}_1)-\hfill \\ & & g_1(A_{Y_1}P{Y}_2,BCQ{U}_1+BCR{U}_1).\hfill \end{array}$$

Now, again using Equations (2), (8), (12) and (17) we have:

$$\begin{array}{ccc}\hfill g_1({\nabla }_{Y_1}{Y}_2,W_1)& =& g_1({\nabla }_{Y_1}J{Y}_2,J{W}_1),\hfill \\ & =& g_1({\nabla }_{Y_1}JP{Y}_2,\varphi W_1+\omega W_1),\hfill \\ & =& g_1({\mathcal{A}}_{Y_1}JP{Y}_2,\varphi W_1)+g_1(\mathcal{H}{\nabla }_{Y_1}JP{Y}_2,\omega W_1),\hfill \end{array}$$

which completes the proof. □

Theorem 7.

Let π be a v-QBSR map. Then, $D_1$ defines a totally geodesic foliation on $N_1$ if and only if

$$g_1({\mathcal{A}}_{X_1}BC{X}_2,Z_1)=g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2,BR{Z}_1)+g_1({\mathcal{A}}_{X_1}B{X}_2,JP{Z}_1+CR{Z}_1),$$

$$g_1(\mathcal{V}{\nabla }_{X_1}BC{X}_2,Z_2)=g_1({\mathcal{A}}_{X_1}BQ{X}_2,\omega Z_2)+g_1(\mathcal{V}{\nabla }_{X_1}BQ{X}_2,\varphi Z_2),$$

for $X_1,X_2\in \Gamma \left(D_1\right),Z_1\in \Gamma (D\oplus D_2)$ and $Z_2\in \Gamma (ker{\pi }_{\ast }).$

Proof. 

For $X_1,X_2\in \Gamma \left(D_1\right),Z_1\in \Gamma (D\oplus D_2)$ and $Z_2\in \Gamma (ker{\pi }_{\ast }),$ by using Equations (2), (7), (8), (12), (13) and (17) and Lemma $2,$ we have:

$$\begin{array}{ccc}& & g_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ & =& g_1({\nabla }_{X_1}B{X}_2,J{Z}_1)+g_1({\nabla }_{X_1}C{X}_2,J{Z}_1),\hfill \\ & =& {cos}^2{\theta }_1{g}_1({\nabla }_{X_1}{X}_2,Z_1)-g_1({\mathcal{A}}_{X_1}BC{X}_2,Z_1)\hfill \\ & & +g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2,BR{Z}_1)+g_1({\mathcal{A}}_{X_1}B{X}_2,JP{Z}_1+CR{Z}_1).\hfill \end{array}$$

Now, we have:

$$\begin{array}{ccc}& & {sin}^2{\theta }_1{g}_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ & =& -g_1({\mathcal{A}}_{X_1}BC{X}_2,Z_1)+g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2,BR{Z}_1)+\hfill \\ & & g_1({\mathcal{A}}_{X_1}B{X}_2,JP{Z}_1+CR{Z}_1).\hfill \end{array}$$

From Equations (2), (7), (8), (13) and (17) and Lemma $2,$ we have:

$$\begin{array}{ccc}\hfill g_1({\nabla }_{X_1}{X}_2,Z_2)& =& g_1({\nabla }_{X_1}B{X}_2,J{Z}_2)+g_1({\nabla }_{X_1}C{X}_2,J{Z}_2),\hfill \\ & =& {cos}^2{\theta }_1{g}_1({\nabla }_{X_1}{X}_2,Z_2)-g_1(\mathcal{V}{\nabla }_{X_1}BC{X}_2,Z_2)\hfill \\ & & +g_1({\mathcal{A}}_{X_1}B{X}_2,\omega Z_2)+g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2,\varphi Z_2).\hfill \end{array}$$

Therefore,

$$\begin{array}{ccc}& & {sin}^2{\theta }_1{g}_1({\nabla }_{X_1}{X}_2,Z_2)\hfill \\ & =& -g_1(\mathcal{V}{\nabla }_{X_1}BC{X}_2,Z_2)+g_1({\mathcal{A}}_{X_1}B{X}_2,\omega Z_2)+g_1(\mathcal{V}{\nabla }_{X_1}B{X}_2,\varphi Z_2).\hfill \end{array}$$

□

Theorem 8.

Let π be a v-QBSR map. Then, $D_2$ defines a totally geodesic foliation on $N_1$ if and only if

$$g_1({\mathcal{A}}_{Y_1}BC{Y}_2,V_1)=g_1(\mathcal{V}{\nabla }_{Y_1}B{Y}_2,JP{V}_1+BR{V}_1)+g_1({\mathcal{A}}_{Y_1}B{Y}_2,CR{V}_1),$$

$$g_1(\mathcal{V}{\nabla }_{Y_1}BC{Y}_2,V_2)=g_1({\mathcal{A}}_{Y_1}B{Y}_2,\omega V_2)+g_1(\mathcal{V}{\nabla }_{Y_1}B{Y}_2,\varphi V_2)$$

for $Y_1,Y_2\in \Gamma \left(D_2\right),V_1\in \Gamma (D\oplus D_1)$ and $V_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

By considering a similar approach as in the proof of Theorem $7,$ we obtain the above result. □

Theorem 9.

Let π be a v-QBSR map. Then, π is a totally geodesic map if and only if

$$\begin{array}{ccc}& & g_1({\mathcal{T}}_{V_1}{V}_2,P{X}_1+{cos}^2{\theta }_{1}Q{X}_1+{cos}^2{\theta }_{2}R{X}_1))\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{V_1}{V}_2,BCP{X}_1+BCQ{X}_1+BCR{X}_1)-g_1(\mathcal{V}{\nabla }_{V_1}\varphi V_2,B{X}_1)-g_1({\mathcal{T}}_{V_1}\omega V_2,B{X}_1),\hfill \end{array}$$

and

$$\begin{array}{ccc}& & g_1({\mathcal{A}}_{X_1}{V}_1,P{X}_2+{cos}^2{\theta }_{1}Q{X}_2+{cos}^2{\theta }_{2}R{X}_2)\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{X_1}{V}_1,BCP{X}_2+BCQ{X}_2+BCR{X}_2)-g_1(\mathcal{V}{\nabla }_{X_1}\varphi V_1,B{X}_2)-g_1({\mathcal{A}}_{X_1}\omega V_1,B{X}_2)\hfill \end{array}$$

for $V_1,V_2\in \Gamma (ker{\pi }_{\ast })$ and $X_1,X_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

Proof. 

Since π is a Riemannian map, we have

$$(\nabla {\pi }_{\ast })(X_1,X_2)=0,$$

for $X_1,X_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp }.$

For $V_1,V_2\in \Gamma (ker{\pi }_{\ast })$ and $X_1,X_2\in \Gamma {(ker{\pi }_{\ast })}^{\perp },$ using Equations (2), (9), (12), (13) and (17), we have:

$$\begin{array}{ccc}& & g_2((\nabla {\pi }_{\ast })(V_1,V_2),{\pi }_{\ast }\left(X_1\right))\hfill \\ & =& -g_1({\nabla }_{V_1}{V}_2,X_1)\hfill \\ & =& -g_1({\nabla }_{V_1}J{V}_2,J{X}_1)\hfill \\ & =& -g_1({\nabla }_{V_1}J{V}_2,JP{X}_1)-g_1({\nabla }_{V_1}J{V}_2,JQ{X}_1)-g_1({\nabla }_{V_1}J{V}_2,JR{X}_1),\hfill \\ & =& -g_1({\nabla }_{V_1}J{V}_2,CP{X}_1+CQ{X}_1+CR{X}_1)-g_1({\nabla }_{V_1}J{V}_2,BP{X}_1+BQ{X}_1+BR{X}_1),\hfill \\ & =& -g_1({\nabla }_{V_1}{V}_2,P{X}_1+{cos}^2{\theta }_{1}Q{X}_1+{cos}^2{\theta }_{2}R{X}_1)+g_1({\nabla }_{V_1}{V}_2,BCP{X}_1+BCQ{X}_1+BCR{X}_1)\hfill \\ & & -g_1({\nabla }_{V_1}\varphi V_2,B{X}_1)-g_1({\nabla }_{V_1}\omega V_2,B{X}_1)\hfill \\ & =& -g_1({\mathcal{T}}_{V_1}{V}_2,P{X}_1+{cos}^2{\theta }_{1}Q{X}_1+{cos}^2{\theta }_{2}R{X}_1)+g_1(\mathcal{V}{\nabla }_{V_1}{V}_2,BCP{X}_1+BCQ{X}_1+BCR{X}_1)\hfill \\ & & -g_1(\mathcal{V}{\nabla }_{V_1}\varphi V_2,B{X}_1)-g_1({\mathcal{T}}_{V_1}\omega V_2,B{X}_1).\hfill \end{array}$$

Furthermore, using Equations (2), (9), (7), (8), (12), (13) and (17), we obtain:

$$\begin{array}{ccc}& & g_2((\nabla {\pi }_{\ast })(X_1,V_1),{\pi }_{\ast }\left(X_2\right))\hfill \\ & =& -g_1({\nabla }_{X_1}{V}_1,X_2),\hfill \\ & =& -g_1({\nabla }_{X_1}J{V}_1,J{X}_2),\hfill \\ & =& -g_1({\nabla }_{X_1}J{V}_1,JP{X}_2+JQ{X}_2+JR{X}_2),\hfill \\ & =& -g_1({\nabla }_{X_1}J{V}_1,BP{X}_2+BQ{X}_2+BR{X}_2)-g_1({\nabla }_{X_1}J{V}_1,CP{X}_2+CQ{X}_2+CR{X}_2),\hfill \\ & =& -g_1(\mathcal{V}{\nabla }_{X_1}\varphi V_1,B{X}_2)-g_1({\mathcal{A}}_{X_1}\omega V_1,B{X}_2)+g_1(\mathcal{V}{\nabla }_{X_1}{V}_1,BCP{X}_2+BCQ{X}_2+BCR{X}_2)-\hfill \\ & & g_1({\mathcal{A}}_{X_1}{V}_1,P{X}_2+{cos}^2{\theta }_{1}Q{X}_2+{cos}^2{\theta }_{2}R{X}_2).\hfill \end{array}$$

□

4. Example

Let $R^{2t}$ be the Euclidean space. Let $(y_1,y_2,\dots \dots ,y_{2t-1},y_{2t})$ be the coordinates of $R^{2t}.$ Define an almost complex structure J on $R^{2t}$ as follows:

$$\begin{array}{ccc}& & J(a_1\frac{\partial }{\partial y_1}+a_2\frac{\partial }{\partial y_2}+\dots \dots \dots \dots +a_{2t-1}\frac{\partial }{\partial y_{2t-1}}+a_{2t}\frac{\partial }{\partial y_{2t}})\hfill \\ & =& -a_2\frac{\partial }{\partial y_1}+a_1\frac{\partial }{\partial y_2}+\dots \dots \dots \dots -a_{2t}\frac{\partial }{y_{2t-1}}+a_{2t-1}\frac{\partial }{\partial y_{2t}},\hfill \end{array}$$

where $a_1,a_2,\dots \dots \dots \dots ,a_{2t}$ are $C^{\infty }$-functions on $R^{2t}.$ Throughout this section, we will use this notation.

Example 1.

Define a map $\pi :R^{12}\to R^8$ by:

$$\pi (y_1,y_2,\dots \dots \dots ,y_{12})=(y_{1}sin{\theta }_1+y_{3}cos{\theta }_1,y_4,202,303,y_{5}sin{\theta }_2-y_{7}cos{\theta }_2,y_8,y_9,y_{10}),$$

which is a v-QBSR map such that

$$ker{\pi }_{\ast }=⟨cos{\theta }_1\frac{\partial }{\partial y_1}-sin{\theta }_1\frac{\partial }{\partial y_3},\frac{\partial }{\partial y_2},cos{\theta }_2\frac{\partial }{\partial y_5}+sin{\theta }_2\frac{\partial }{\partial y_7},\frac{\partial }{\partial y_6},\frac{\partial }{\partial y_{11}},\frac{\partial }{\partial y_{12}}⟩$$

$${(ker{\pi }_{\ast })}^{\perp }=⟨sin{\theta }_1\frac{\partial }{\partial y_1}+cos{\theta }_1\frac{\partial }{\partial y_3},\frac{\partial }{\partial y_4},sin{\theta }_2\frac{\partial }{\partial y_5}+cos{\theta }_2\frac{\partial }{\partial y_7},\frac{\partial }{\partial y_8},\frac{\partial }{\partial y_9},\frac{\partial }{\partial y_{10}}⟩,$$

$${(ker{\pi }_{\ast })}^{\perp }=D\oplus D_1\oplus D_2,$$

where:

$$D=⟨\frac{\partial }{\partial y_9},\frac{\partial }{\partial y_{10}}⟩,D_1=⟨sin{\theta }_1\frac{\partial }{\partial y_1}+cos{\theta }_1\frac{\partial }{\partial y_3},\frac{\partial }{\partial y_4}⟩,D_2=⟨sin{\theta }_2\frac{\partial }{\partial y_5}+cos{\theta }_2\frac{\partial }{\partial y_7},\frac{\partial }{\partial y_8}⟩$$

with v-quasi-bi-slant angles ${\theta }_1$ and ${\theta }_2.$

Example 2.

Define a map $\pi :R^{12}\to R^8$ by:

$$\pi (y_1,y_2,\dots \dots \dots ,y_{12})=(y_1,y_2,\frac{y_3-y_5}{\sqrt{2}},y_6,2021,\frac{\sqrt{3}{y}_7+y_9}{2},y_8,2022),$$

which is a v-quasi-bi-slant map such that

$$(ker{\pi }_{\ast })=⟨\frac{1}{\sqrt{2}}(\frac{\partial }{\partial y_3}+\frac{\partial }{\partial y_5}),\frac{\partial }{\partial y_4},\frac{1}{2}(\frac{\partial }{\partial y_7}-\sqrt{3}\frac{\partial }{\partial y_9}),\frac{\partial }{\partial y_{10}},\frac{\partial }{\partial y_{11}},\frac{\partial }{\partial y_{12}}⟩,$$

$${(ker{\pi }_{\ast })}^{\perp }=⟨\frac{\partial }{\partial y_1},\frac{\partial }{\partial y_2},\frac{1}{\sqrt{2}}(\frac{\partial }{\partial y_3}-\frac{\partial }{\partial y_5}),\frac{\partial }{\partial y_6},\frac{1}{2}(\sqrt{3}\frac{\partial }{\partial y_7}+\frac{\partial }{\partial y_9}),\frac{\partial }{\partial y_8}⟩,$$

$${(ker{\pi }_{\ast })}^{\perp }=D\oplus D_1\oplus D_2,$$

where:

$$D=<\frac{\partial }{\partial y_1},\frac{\partial }{\partial y_2}>,D_1=<\frac{1}{\sqrt{2}}(\frac{\partial }{\partial y_3}-\frac{\partial }{\partial y_5}),\frac{\partial }{\partial y_6}>,D_2=<\frac{1}{2}(\sqrt{3}\frac{\partial }{\partial y_7}+\frac{\partial }{\partial y_9}),\frac{\partial }{\partial y_8}>,$$

with v-quasi-bi-slant angles ${\theta }_1=\frac{\pi }{4}$ and ${\theta }_2=\frac{\pi }{6}.$