A Study of Clairaut Semi-Invariant Riemannian Maps from Cosymplectic Manifolds

Abstract

In the present note, we characterize Clairaut semi-invariant Riemannian maps from cosymplectic manifolds to Riemannian manifolds. Moreover, we provide a nontrivial example of such a Riemannian map.

1. Introduction

The theory of Riemannian maps between Riemannian manifolds is widely used to compare the geometric structures between two Riemannian manifolds, initiated by Fischer [1]. Let $({\mathcal{M}}_1,g_1)$ and $({\mathcal{M}}_2,g_2)$ be two Riemannian manifolds of dimensions m and n, respectively. Let a Riemannian map $\mathsf{\Pi }:({\mathcal{M}}_1,g_1)\to ({\mathcal{M}}_2,g_2)$ be a differentiable map between $({\mathcal{M}}_1,g_1)$ and $({\mathcal{M}}_2,g_2)$ such that $0<rank{\mathsf{\Pi }}_{*}<min\{m,n\}$, where ${\mathsf{\Pi }}_{*}$ represents a differential map of $\mathsf{\Pi }$. If we denote the $kernel$ space of ${\mathsf{\Pi }}_{*}$ by $ker{\mathsf{\Pi }}_{*}$ and the orthogonal complementary space of $ker{\mathsf{\Pi }}_{*}$ by ${(ker{\mathsf{\Pi }}_{*})}^{\perp }$ in $T{\mathcal{M}}_1$, then the $T{\mathcal{M}}_1$ has the following orthogonal decomposition:

$$T{\mathcal{M}}_1=ker{\mathsf{\Pi }}_{*}\oplus {(ker{\mathsf{\Pi }}_{*})}^{\perp }.$$

(1)

We denote the range of ${\mathsf{\Pi }}_{*}$ by $range{\mathsf{\Pi }}_{*}$ and for a point $q\in {\mathcal{M}}_1$ the orthogonal complementary space of $range{\mathsf{\Pi }}_{*\mathsf{\Pi }\left(q\right)}$ by ${\left(range{\mathsf{\Pi }}_{*\mathsf{\Pi }\left(q\right)}\right)}^{\perp }$ in $T_{\mathsf{\Pi }\left(q\right)}{\mathcal{M}}_2.$ The tangent space $T_{\mathsf{\Pi }\left(q\right)}{\mathcal{M}}_2$ has the following orthogonal decomposition:

$$T_{\mathsf{\Pi }\left(q\right)}{\mathcal{M}}_2=\left(range{\mathsf{\Pi }}_{*\mathsf{\Pi }\left(q\right)}\right)\oplus {\left(range{\mathsf{\Pi }}_{*\mathsf{\Pi }\left(q\right)}\right)}^{\perp }.$$

A differentiable map $\mathsf{\Pi }:({\mathcal{M}}_1,g_1)\to ({\mathcal{M}}_2,g_2)$ is called a Riemannian map at $q\in {\mathcal{M}}_1$ if the horizontal restriction ${\mathsf{\Pi }}_{*q}^h:{(ker{\mathsf{\Pi }}_{*q})}^{\perp }\to \left(range{\mathsf{\Pi }}_{*\mathsf{\Pi }\left(q\right)}\right)$ is a linear isometric between the inner product spaces $({(ker{\mathsf{\Pi }}_{*q})}^{\perp },{\left(g_1\right)}_{\left(q\right)}{|}_{{(ker{\mathsf{\Pi }}_{*q})}^{\perp }})$ and $(range{\mathsf{\Pi }}_{*{\mathsf{\Pi }}_{\left(q\right)}},$${\left(g_2\right)}_{\left(\mathsf{\Pi }\right(q)}{\left)\right|}_{\left(range{\mathsf{\Pi }}_{*q}\right)}).$

Further, the notion of the Riemannian map has been studied from different perspectives, such as invariant and anti-invariant Riemannian maps [2], semi-invariant Riemannian maps [3], slant Riemannian maps [4,5,6], semi-slant Riemannian maps [7,8,9], hemi-slant Riemannian maps [10], quasi-hemi-slant Riemannian maps [11] etc.

On the other side, in the theory of the geodesics upon a surface of revolution, the prestigious Clairaut’s theorem states that for any geodesic $c(c:I_1\subset R\to {\mathcal{M}}_1$ on ${\mathcal{M}}_1$) on the revolution surface ${\mathcal{M}}_1$ the product $rsin\theta$ is constant along c, where $\theta \left(s\right)$ is the angle between $c\left(s\right)$ and the meridian curve through $c\left(s\right),$ $s\in I_1$. This means that it is independent of s. In 1972, Bishop [12] studied Riemannian submersions which are a generalization of Clairaut’s theorem. According to him, a submersion $\mathsf{\Pi }:{\mathcal{M}}_1\to {\mathcal{M}}_2$ is said to be a Clairaut submersion if there is a function $r:{\mathcal{M}}_1\to R^{+}$ such that for every geodesic making an angle $\theta$ with the horizontal subspaces, $rsin\theta$ is constant. This notion has also been studied in Lorentzian spaces, time-like and space-like spaces, by the authors [13,14,15]. Later, in [16], it was shown that such submersions have their applications in static spacetimes.

Moreover, Clairaut submersions were further generalized in [17]. We recommend the papers [18,19,20,21,22,23,24,25,26,27,28,29,30,31,32] and the references therein for more details about the further related studies.

In this paper, we are interested in studying the above idea in contact manifolds. Throughout the manuscript, we denote semi-invariant Riemannian maps by SIR maps and Clairaut semi-invariant Riemannian maps by CSIR maps. The article is organized as follows: In Section 2, we gather some basic facts that are needed for this paper. In Section 3, we define a CSIR map from an almost contact metric manifold to a Riemannian manifold and study its geometry. In Section 4, we give a nontrivial example of the CSIR map from cosymplectic manifolds to Riemannian manifolds.

2. Preliminaries

An odd-dimensional smooth manifold ${\mathcal{M}}_1$ is said to have an almost contact structure [33] if there exist on ${\mathcal{M}}_1$ a tensor field $\varphi$ of type $(1,1)$, a vector field $\xi$, and 1-form $\eta$ such that

$${\varphi }^2{V}_1=-V_1+\eta \left(V_1\right)\xi ,\eta \circ \varphi =0,\varphi \xi =0,$$

(2)

$$\eta \left(\xi \right)=1.$$

(3)

If there exists a Riemannian metric $g_1$ on an almost contact manifold ${\mathcal{M}}_1$ satisfying:

$$\begin{array}{ccc}\hfill g_1(\varphi V_1,\varphi V_2)& =& g_1(V_1,V_2)-\eta \left(V_1\right)\eta \left(V_2\right),\hfill \\ \hfill g_1(V_1,\varphi V_2)& =& -g_1(\varphi V_1,V_2),\hfill \end{array}$$

(4)

$$g_1(V_1,\xi )=\eta \left(V_1\right),$$

(5)

where $V_1,V_2$ are any vector fields on ${\mathcal{M}}_1$, then ${\mathcal{M}}_1$ is called an almost contact metric manifold [34] with an almost contact structure $(\varphi ,\xi ,\eta ,g_1)$ and is represented by $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$.

An almost contact structure $(\varphi ,\xi ,\eta )$ is said to be normal if the almost complex structure J on the product manifold ${\mathcal{M}}_1\times R$ is given by

$$J(V_1,\mathcal{F}\frac{d}{dt})=(\varphi V_1-\mathcal{F}\xi ,\eta \left(V_1\right)\frac{d}{dt}),$$

(6)

where $J^2=-I$ and $\mathcal{F}$ is a differentiable function on ${\mathcal{M}}_1\times R$ that has no torsion, i.e., J is integrable. The condition for normality in terms of $\varphi$, $\xi$, and $\eta$ is given by $[\varphi ,\varphi ]+2d\eta \otimes \xi =0$ on ${\mathcal{M}}_1$, where $[\varphi ,\varphi ]$ is the Nijenhuis tensor of $\varphi$. Further, the fundamental 2-form $\mathsf{\Phi }$ is defined by $\mathsf{\Phi }(V_1,V_2)=g_1(V_1,\varphi V_2)$.

A manifold ${\mathcal{M}}_1$ with the structure $(\varphi ,\xi ,\eta ,g_1)$ is said to be cosymplectic [33] if

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

(7)

for any vector fields $V_1,V_2$ on ${\mathcal{M}}_1$, where ∇ stands for the Riemannian connection of the metric $g_1$ on ${\mathcal{M}}_1$. For a cosymplectic manifold, we have

$${\nabla }_{V_1}\xi =0$$

(8)

for any vector field $V_1$ on ${\mathcal{M}}_1.$

O’Neill’s tensors [35] $\mathcal{T}$ and $\mathcal{A}$ are given by

$${\mathcal{A}}_{F_1}{F}_2=\mathcal{H}{\nabla }_{\mathcal{H}{F}_1}\mathcal{V}{F}_2+\mathcal{V}{\nabla }_{\mathcal{H}{F}_1}\mathcal{H}{F}_2,$$

(9)

$${\mathcal{T}}_{F_1}{F}_2=\mathcal{H}{\nabla }_{\mathcal{V}{F}_1}\mathcal{V}{F}_2+\mathcal{V}{\nabla }_{\mathcal{V}{F}_1}\mathcal{H}{F}_2,$$

(10)

for any $F_1,F_2$ on ${\mathcal{M}}_1.$ It is easy to see that ${\mathcal{T}}_{F_1}$ and ${\mathcal{A}}_{F_1}$ are skew-symmetric operators on the tangent bundle of ${\mathcal{M}}_1$ reversing the vertical and the horizontal distributions. In addition, for any vertical vector fields $X_1$ and $X_2$, the tensor field $\mathcal{T}$ has the symmetry property, i.e.,

$${\mathcal{T}}_{X_1}{X}_2={\mathcal{T}}_{X_2}{X}_1,$$

(11)

while for horizontal vector fields $Z_1,Z_2,$ the tensor field $\mathcal{A}$ has alternation property, i.e.,

$${\mathcal{A}}_{Z_1}{Z}_2=-{\mathcal{A}}_{Z_2}{Z}_1.$$

(12)

From Equations (9) and (10) we have

$${\nabla }_{U_1}{U}_2={\mathcal{T}}_{U_1}{U}_2+\mathcal{V}{\nabla }_{U_1}{U}_2,$$

(13)

$${\nabla }_{U_1}{W}_1={\mathcal{T}}_{U_1}{W}_1+\mathcal{H}{\nabla }_{U_1}{W}_1,$$

(14)

$${\nabla }_{W_1}{U}_1={\mathcal{A}}_{W_1}{U}_1+\mathcal{V}{\nabla }_{W_1}{U}_1,$$

(15)

$${\nabla }_{W_1}{W}_2=\mathcal{H}{\nabla }_{W_1}{W}_2+{\mathcal{A}}_{W_1}{W}_2$$

(16)

for all $U_1,U_2\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*})$ and $W_1,W_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp },$ where $\mathcal{H}{\nabla }_{U_1}{W}_1={\mathcal{A}}_{W_1}{U}_1$ and $W_1$ is basic. It can be easily seen that $\mathcal{T}$ acts on the fibers as the second fundamental form, while $\mathcal{A}$ acts on the horizontal distribution and measures the obstruction to the integrability of the distribution.

It is noticed that for $p\in {\mathcal{M}}_1,$$Z_1\in {\mathcal{V}}_p$ and $X_1\in {\mathcal{H}}_p$ the linear operators

$${\mathcal{A}}_{X_1},{\mathcal{T}}_{Z_1}:T_p{\mathcal{M}}_1\to T_p{\mathcal{M}}_1$$

are skew-symmetric, i.e.,

$$g_1({\mathcal{A}}_{X_1}{F}_1,F_2)=-g_1(F_1,{\mathcal{A}}_{X_1}{F}_2)\mathrm{and}{g}_1({\mathcal{T}}_{Z_1}{F}_1,F_2)=-g_1(F_1,{\mathcal{T}}_{Z_1}{F}_2)$$

(17)

for each $F_1,F_2\in$$T_P{\mathcal{M}}_1.$ Since ${\mathcal{T}}_{Z_1}$ is skew-symmetric, we observe that $\mathsf{\Pi }$ has totally geodesic fibres if and only if $\mathcal{T}\equiv 0$.

The map $\mathsf{\Pi }$ between two Riemannian manifolds is totally geodesic if

$$(\nabla {\mathsf{\Pi }}_{*})(V_1,V_2)=0\forall V_1,V_2\in \mathsf{\Gamma }\left(T{\mathcal{M}}_1\right).$$

A totally umbilical map is a Riemannian map with totally umbilical fibers [36] if

$${\mathcal{T}}_{Y_1}{Y}_2=g_1(Y_1,Y_2)H$$

(18)

for all $Y_1,Y_2\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*}),$ where H denotes the mean curvature vector field of fibers.

The map ${\mathsf{\Pi }}_{*}$ can be observed as a section of the bundle $Hom(T{\mathcal{M}}_1,{\mathsf{\Pi }}^{-1}T{\mathcal{M}}_2)$⟶${\mathcal{M}}_1,$ where ${\mathsf{\Pi }}^{-1}T{\mathcal{M}}_2$ is the bundle which has fibers ${\left({\mathsf{\Pi }}^{-1}T{\mathcal{M}}_2\right)}_x=T_{\mathsf{\Pi }\left(x\right)}{\mathcal{M}}_2$ and has a connection ∇ induced from the Riemannian connection ${\nabla }^{{\mathcal{M}}_1}$ and the pullback connection ${\nabla }^{\mathsf{\Pi }}$, then the second fundamental form of $\mathsf{\Pi }$ is given by

$$(\nabla {\mathsf{\Pi }}_{*})(W_1,W_2)={\nabla }_{W_1}^{\mathsf{\Pi }}{\mathsf{\Pi }}_{*}\left(W_2\right)-{\mathsf{\Pi }}_{*}\left({\nabla }_{W_1}^{{\mathcal{M}}_1}{W}_2\right)$$

(19)

for the vector fields $W_1,W_2\in \mathsf{\Gamma }\left(T{\mathcal{M}}_1\right)$. We know that the second fundamental form is symmetric.

Now, we have the following lemma [2]:

Lemma 1.

Let $\mathsf{\Pi }:({\mathcal{M}}_1,g_1)\to ({\mathcal{M}}_2,g_2)$ be a map between Riemannian manifolds. Then

$$g_2((\nabla {\mathsf{\Pi }}_{*})(Y_1,Y_2),{\mathsf{\Pi }}_{*}\left(Y_3\right))=0\forall Y_1,Y_2,Y_3\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$$

(20)

As a result of above Lemma, we obtain

$$(\nabla {\mathsf{\Pi }}_{*})(Z_1,Z_2)\in \left(\mathsf{\Gamma }{\left(range{\mathsf{\Pi }}_{*}\right)}^{\perp }\right)\forall Z_1,Z_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$$

(21)

3. CSIR Map from Cosymplectic Manifolds

Let S be a revolution surface in $R^3$ with rotation axis L. For any $q\in$S, we denote the distance from q to L by $r\left(q\right)$. Given a geodesic $\alpha :I\subset R\to S$ on S, let $\theta \left(t\right)$ be the angle between $\alpha \left(t\right)$ and the meridian curve through $\alpha \left(t\right)$, $t\in I$. A well-known Clairaut’s theorem says that for any geodesic $\alpha$ on S, the product $rsin\theta \left(t\right)$ is constant along $\alpha$, i.e., it is independent of t.

Recently, Sahin [30] initiated the study of Clairaut Riemannian maps. He defined a map $\mathsf{\Pi }:({\mathcal{M}}_1,g_1)\to ({\mathcal{M}}_2,g_2)$ called a Clairaut Riemannian map if there exists a positive function r on ${\mathcal{M}}_1,$ such that for any geodesic $\alpha$ on ${\mathcal{M}}_1,$ the function $(r\circ \alpha )sin\theta$ is constant, where for any $t,$$\theta \left(t\right)$ is the angle between $\stackrel{.}{\alpha }\left(t\right)$ and the horizontal space at $\alpha \left(t\right).$ Moreover, he obtained the following necessary and sufficient condition for a Riemannian map to be a Clairaut Riemannian map:

Theorem 1

([30]). Let $\mathsf{\Pi }:({\mathcal{M}}_1,g_1)\to ({\mathcal{M}}_2,g_2)$ be a Riemannian map with connected fibers. Then, Π is a Clairaut Riemannian map with $r=e^f$ if each fiber is totally umbilical and has the mean curvature vector field $H=-\nabla f$, where $\nabla f$ is the gradient of the function f with respect to $g_1.$

Definition 1

([3]). Let Π be a Riemannian map from an almost contact metric manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2).$ Then, we say that Π is an SIR map if there is a distribution ${\mathfrak{D}}_1\subseteq ker{\mathsf{\Pi }}_{*}$ such that

$$ker{\mathsf{\Pi }}_{*}={\mathfrak{D}}_1\oplus {\mathfrak{D}}_2,\varphi \left({\mathfrak{D}}_1\right)={\mathfrak{D}}_1,\varphi \left({\mathfrak{D}}_2\right)\subseteq {(ker{\mathsf{\Pi }}_{*})}^{\perp },$$

where ${\mathfrak{D}}_1$ and ${\mathfrak{D}}_2$ are mutually orthogonal distributions in ($ker{\mathsf{\Pi }}_{*}).$

Let $\mu$ denote the complementary orthogonal subbundle to $\varphi \left({\mathfrak{D}}_2\right)$ in ${(ker{\mathsf{\Pi }}_{*})}^{\perp }.$ Then, we have

$${(ker{\mathsf{\Pi }}_{*})}^{\perp }=\varphi \left({\mathfrak{D}}_2\right)\oplus \mu .$$

Obviously, $\mu$ is an invariant subbundle of ${(ker{\mathsf{\Pi }}_{*})}^{\perp }$ with respect to the contact structure $\varphi$. We say that an SIR map $\mathsf{\Pi }:{\mathcal{M}}_1\to {\mathcal{M}}_2$ admits a vertical Reeb vector field $\xi$ if it is tangent to $(ker{\mathsf{\Pi }}_{*})$ and it admits a horizontal Reeb vector field $\xi$ if it is normal to $(ker{\mathsf{\Pi }}_{*})$. It is easy to see that $\mu$ contains the Reeb vector field in case the Riemannian map admits horizontal Reeb vector field.

Now, we define the notion of the CSIR map in contact manifolds as follows:

Definition 2.

An SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ is called a CSIR map if it satisfies the condition of a Clairaut Riemannian map.

For any vector field $Z_1\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*})$, we input

$$Z_1=P{Z}_1+Q{Z}_1,$$

(22)

where P and Q are projection morphisms [36] of $ker{\mathsf{\Pi }}_{*}$ onto ${\mathfrak{D}}_1$ and ${\mathfrak{D}}_2,$ respectively.

For any $V_1\in (ker{\mathsf{\Pi }}_{*}),$ we obtain

$$\varphi V_1=\psi V_1+\omega V_1,$$

(23)

where $\psi V_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_1\right)$ and $\omega V_1\in \mathsf{\Gamma }\left(\varphi {\mathfrak{D}}_2\right).$ In addition, for $V_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp },$ we have

$$\varphi V_2=B{V}_2+C{V}_2,$$

(24)

where $B{V}_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $C{V}_2\in \mathsf{\Gamma }\left(\mu \right).$

Definition 3

([14]). Let Π be an SIR map from an almost contact metric manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$. If $\mu =\left\{0\right\}$ or $\mu =<\xi >,$ i.e., ${(ker{\mathsf{\Pi }}_{*})}^{\perp }=\varphi \left({\mathfrak{D}}_2\right)$ or ${(ker{\mathsf{\Pi }}_{*})}^{\perp }=\varphi \left({\mathfrak{D}}_2\right)\oplus <\xi >$, respectively. Then we call ϕ a Lagrangian Riemannian map. In this case, for any horizontal vector field $V_1,$ we have

$$B{V}_1=\varphi V_1\mathrm{and}C{V}_1=0.$$

(25)

Lemma 2.

Let Π be an SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ admitting vertical or horizontal Reeb vector field. Then, we obtain

$$\mathcal{V}{\nabla }_{Y_1}\psi Y_2+{\mathcal{T}}_{Y_1}\omega Y_2=B{\mathcal{T}}_{Y_1}{Y}_2+\psi \mathcal{V}{\nabla }_{Y_1}{Y}_2,$$

(26)

$${\mathcal{T}}_{Y_1}\psi Y_2+\mathcal{H}{\nabla }_{Y_1}\omega Y_2=C{\mathcal{T}}_{Y_1}{Y}_2+\omega \mathcal{V}{\nabla }_{Y_1}{Y}_2,$$

(27)

$$\mathcal{V}{\nabla }_{V_1}B{V}_2+{\mathcal{A}}_{V_1}C{V}_2=B\mathcal{H}{\nabla }_{V_1}{V}_2+\psi {\mathcal{A}}_{V_1}{V}_2,$$

(28)

$${\mathcal{A}}_{V_1}B{V}_2+\mathcal{H}{\nabla }_{V_1}C{V}_2=C\mathcal{H}{\nabla }_{V_1}{V}_2+\omega {\mathcal{A}}_{V_1}{V}_2,$$

(29)

$$\mathcal{V}{\nabla }_{Y_1}B{V}_1+{\mathcal{T}}_{Y_1}C{V}_1=\psi {\mathcal{T}}_{Y_1}{V}_1+B\mathcal{H}{\nabla }_{Y_1}{V}_1,$$

(30)

$${\mathcal{T}}_{Y_1}B{V}_1+\mathcal{H}{\nabla }_{Y_1}C{V}_1=\omega {\mathcal{T}}_{Y_1}{V}_1+C\mathcal{H}{\nabla }_{Y_1}{V}_1,$$

(31)

$$\mathcal{V}{\nabla }_{V_1}\psi Y_1+{\mathcal{A}}_{V_1}\omega Y_1=B{\mathcal{A}}_{V_1}{Y}_1+\psi \mathcal{V}{\nabla }_{V_1}{Y}_1,$$

(32)

$${\mathcal{A}}_{V_1}\psi Y_1+\mathcal{H}{\nabla }_{V_1}\omega Y_1=C{\mathcal{A}}_{V_1}{Y}_1+\omega \mathcal{V}{\nabla }_{V_1}{Y}_1,$$

(33)

where $Y_1,Y_2\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*})$ and $V_1,V_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$

Proof. 

Using Equations (7), (13)–(16), (23) and (24), we obtain Lemma 2. □

Corollary 1.

Let Π be a Lagrangian Riemannian map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ admitting vertical or horizontal Reeb vector field. Then, we obtain

$$\mathcal{V}{\nabla }_{X_1}\psi X_2+{\mathcal{T}}_{X_1}\omega X_2=B{\mathcal{T}}_{X_1}{X}_2+\psi \mathcal{V}{\nabla }_{X_1}{X}_2,{\mathcal{T}}_{X_1}\psi X_2+\mathcal{H}{\nabla }_{X_1}\omega X_2=\omega \mathcal{V}{\nabla }_{X_1}{X}_2,$$

$$\mathcal{V}{\nabla }_{Z_1}B{Z}_2=B\mathcal{H}{\nabla }_{Z_1}{Z}_2+\psi {\mathcal{A}}_{Z_1}{Z}_2,{\mathcal{A}}_{Z_1}B{Z}_2=\omega {\mathcal{A}}_{Z_1}{Z}_2,$$

$$\mathcal{V}{\nabla }_{X_1}B{Z}_1=\psi {\mathcal{T}}_{X_1}{Z}_1+B\mathcal{H}{\nabla }_{X_1}{Z}_1,{\mathcal{T}}_{X_1}B{Z}_1=\omega {\mathcal{T}}_{X_1}{Z}_1,$$

$$\mathcal{V}{\nabla }_{Z_1}\psi X_1+{\mathcal{A}}_{Z_1}\omega X_1=B{\mathcal{A}}_{Z_1}{X}_1+\psi \mathcal{V}{\nabla }_{Z_1}{X}_1,{\mathcal{A}}_{Z_1}\psi X_1+\mathcal{H}{\nabla }_{Z_1}\omega X_1=\omega \mathcal{V}{\nabla }_{Z_1}{X}_1,$$

where $X_1,X_2\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*})$ and $Z_1,Z_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$

Lemma 3.

Let Π be an SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ admitting vertical or horizontal Reeb vector field. Then, we have

$${\mathcal{T}}_{U_1}\xi =0,$$

(34)

$${\mathcal{A}}_{U_2}\xi =0$$

(35)

for $U_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }$ and $U_2\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$

Proof. 

Using Equations (8), (14) and (16) we obtain Lemma 3. □

Lemma 4.

Let Π be an SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$. If $\alpha :I_2\subset R\to {\mathcal{M}}_1$ is a regular curve and $Y_1\left(t\right)$ and $Y_2\left(t\right)$ are the vertical and horizontal components of the tangent vector field $\stackrel{.}{\alpha }=E$ of $\alpha \left(t\right),$ respectively, then α is a geodesic if and only if along α the following relations hold:

$$\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{Y}_2+\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}\psi Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})C{Y}_2+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})\omega Y_1=0,$$

(36)

$$\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{Y}_2+\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}\omega Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})B{Y}_2+({\mathcal{A}}_{Y_2}+{\mathcal{T}}_{Y_1})\psi Y_1=0.$$

(37)

Proof. 

Let $\alpha :I_2\to {\mathcal{M}}_1$ be a regular curve on ${\mathcal{M}}_1.$ Since $Y_1\left(t\right)$ and $Y_2\left(t\right)$ are the vertical and horizontal parts of the tangent vector field $\stackrel{.}{\alpha }\left(t\right),$ i.e., $\stackrel{.}{\alpha }\left(t\right)=Y_1\left(t\right)+Y_2\left(t\right)$, from Equations (2), (7), (13)–(16), (23) and (24) we obtain

$$\begin{array}{ccc}\hfill \varphi {\nabla }_{\stackrel{.}{\alpha }}\stackrel{.}{\alpha }& =& {\nabla }_{\stackrel{.}{\alpha }}\varphi \stackrel{.}{\alpha }\hfill \\ & =& {\nabla }_{Y_1}\psi Y_1+{\nabla }_{Y_1}\omega Y_1+{\nabla }_{Y_2}\psi Y_1+{\nabla }_{Y_2}\omega Y_1\hfill \\ & & +{\nabla }_{Y_1}B{Y}_2+{\nabla }_{Y_1}C{Y}_2+{\nabla }_{Y_2}B{Y}_2+{\nabla }_{Y_2}C{Y}_2\hfill \\ & =& \mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{Y}_2+\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}\psi Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})C{Y}_2+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})\omega Y_1\hfill \\ & & +\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{Y}_2+\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}\omega Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})B{Y}_2+({\mathcal{A}}_{Y_2}+{\mathcal{T}}_{Y_1})\psi Y_1.\hfill \end{array}$$

Taking the vertical and horizontal components in the above equation, we have

$$\mathcal{V}\varphi {\nabla }_{\stackrel{.}{\alpha }}\stackrel{.}{\alpha }=\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{Y}_2+\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}\psi Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})C{Y}_2+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})\omega Y_1,$$

$$\mathcal{H}\varphi {\nabla }_{\stackrel{.}{\alpha }}\stackrel{.}{\alpha }=\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{Y}_2+\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}\omega Y_1+({\mathcal{T}}_{Y_1}+{\mathcal{A}}_{Y_2})B{Y}_2+({\mathcal{A}}_{Y_2}+{\mathcal{T}}_{Y_1})\psi Y_1.$$

Thus, $\alpha$ is a geodesic on ${\mathcal{M}}_1$ if and only if $\mathcal{V}\varphi {\nabla }_{\stackrel{.}{\alpha }}\stackrel{.}{\alpha }=0$ and $\mathcal{H}\varphi {\nabla }_{\stackrel{.}{\alpha }}\stackrel{.}{\alpha }=0$; this completes the proof. □

Theorem 2.

Let Π be an SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2).$ Then, Π is a CSIR map with $r=e^f$ if and only if

$$\begin{array}{ccc}& & g_1(\nabla f,V_2)\left|\right|V_1{\left|\right|}^2\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{V}_2,\psi V_1)+g_1({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})C{V}_2,\psi V_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{V}_2,\omega V_1)+g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})B{V}_2,\omega V_1),\hfill \end{array}$$

where $\alpha :I_2\to {\mathcal{M}}_1$ is a geodesic on ${\mathcal{M}}_1$; $V_1\left(t\right)$ and $V_2\left(t\right)$ are vertical and horizontal components of $\stackrel{.}{\alpha }\left(t\right),$ respectively.

Proof. 

Let $\alpha :I_2\to {\mathcal{M}}_1$ be a geodesic on ${\mathcal{M}}_1$ with $V_1\left(t\right)=\mathcal{V}\stackrel{.}{\alpha }\left(t\right)$ and $V_2\left(t\right)=\mathcal{H}\stackrel{.}{\alpha }\left(t\right)$. We denote the angle in $[0,\pi ]$ between $\stackrel{.}{\alpha }\left(t\right)$ and $V_2\left(t\right)$ by $\theta \left(t\right)$. Assuming $\upsilon =\left|\right|\stackrel{.}{\alpha }\left(t\right){\left|\right|}^2$, then we obtain

$$g_1(V_1\left(t\right),V_1\left(t\right))=\upsilon {sin}^2\theta \left(t\right),$$

(38)

$$g_1(V_2\left(t\right),V_2\left(t\right))=\upsilon {cos}^2\theta \left(t\right).$$

(39)

Now, differentiating (38), we obtain

$$\begin{array}{ccc}\hfill \frac{d}{dt}{g}_1(V_1\left(t\right),V_1\left(t\right))& =& 2\upsilon sin\theta \left(t\right)cos\theta \left(t\right)\frac{d\theta }{dt},\hfill \\ \hfill g_1({\nabla }_{\stackrel{.}{\alpha }}{V}_1\left(t\right),V_1\left(t\right))& =& \upsilon sin\theta \left(t\right)cos\theta \left(t\right)\frac{d\theta }{dt}.\hfill \end{array}$$

Using Equations (4) and (7) in the above equation, we obtain

$$g_1({\nabla }_{\stackrel{.}{\alpha }}\varphi V_1\left(t\right),\varphi V_1\left(t\right))=\upsilon sin\theta \left(t\right)cos\theta \left(t\right)\frac{d\theta }{dt}.$$

(40)

Thus, we obtain

$$\begin{array}{ccc}& & g_1({\nabla }_{\stackrel{.}{\alpha }}\varphi V_1,\varphi V_1)\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}\psi V_1,\psi V_1)+g_1(\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}\omega V_1,\omega V_1)\hfill \\ & & +g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})\psi V_1,\omega V_1)+g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})\omega V_1,\psi V_1).\hfill \end{array}$$

(41)

Using Equations (36) and (37) in (41), we have

$$\begin{array}{ccc}& & g_1({\nabla }_{\stackrel{.}{\alpha }}\varphi V_1,\varphi V_1)\hfill \\ & =& -g_1(\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{V}_2,\psi V_1)-g_1({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})C{V}_2,\psi V_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{V}_2,\omega V_1)-g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})B{V}_2,\omega V_1).\hfill \end{array}$$

(42)

From Equations (40) and (42), we have

$$\begin{array}{ccc}& & \upsilon sin\theta \left(t\right)cos\theta \left(t\right)\frac{d\theta }{dt}\hfill \\ & =& -g_1(\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{V}_2,\psi V_1)-g_1({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})C{V}_2,\psi V_1)\hfill \\ & & -g_1(\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{V}_2,\omega V_1)-g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})B{V}_2,\omega V_1).\hfill \end{array}$$

(43)

Moreover, $\pi$ is a Clairaut semi-invariant Riemannian map with $r=e^f$ if and only if $\frac{d}{dt}(e^{f\circ \alpha }sin\theta )=0,$ i.e., $e^{f\circ \alpha }(cos\theta \frac{d\theta }{dt}+sin\theta \frac{df}{dt})=0,$ which, by multiplying with nonzero factor $\upsilon sin\theta$, gives

$$\begin{array}{ccc}\hfill -\upsilon cos\theta sin\theta \frac{d\theta }{dt}& =& \upsilon {sin}^2\theta \frac{df}{dt}\hfill \\ \hfill \upsilon cos\theta sin\theta \frac{d\theta }{dt}& =& -g_1(V_1,V_1)\frac{df}{dt}\hfill \\ \hfill \upsilon cos\theta sin\theta \frac{d\theta }{dt}& =& -g_1(\nabla f,\stackrel{.}{\alpha })\left|\right|V_1{\left|\right|}^2\hfill \\ \hfill \upsilon cos\theta sin\theta \frac{d\theta }{dt}& =& -g_1(\nabla f,V_2)\left|\right|V_1{\left|\right|}^2.\hfill \end{array}$$

(44)

Thus, from Equations (43) and (44) we have

$$\begin{array}{ccc}& & g_1(\nabla f,V_2)\left|\right|V_1{\left|\right|}^2\hfill \\ & =& g_1(\mathcal{V}{\nabla }_{\stackrel{.}{\alpha }}B{V}_2,\psi V_1)+g_1({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})C{V}_2,\psi V_1)\hfill \\ & & +g_1(\mathcal{H}{\nabla }_{\stackrel{.}{\alpha }}C{V}_2,\omega V_1)+g_1(({\mathcal{T}}_{V_1}+{\mathcal{A}}_{V_2})B{V}_2,\omega V_1).\hfill \end{array}$$

Hence, Theorem 2 is proved. □

Corollary 2.

Let Π be an SIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ admitting horizontal Reeb vector field. Then, we obtain

$$g_1(\nabla f,\xi )=0.$$

Theorem 3.

Let Π be a CSIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ with $r=e^f,$ then at least one of the following statement is true:

$\left(i\right)$f is constant on $\varphi \left({\mathfrak{D}}_2\right)$;

$\left(ii\right)$ The fibers are one-dimensional;

$\left(iii\right)$${\stackrel{\mathsf{\Pi }}{\nabla }}_{\varphi U_1}{\mathsf{\Pi }}_{*}\left(Z_1\right)=-Z_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi U_1\right),$ for all $U_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right),Z_1\in \mathsf{\Gamma }\left(\mu \right)$ and $\xi \ne Z_1.$

Proof. 

Let $\mathsf{\Pi }$ be a CSIR map from a cosymplectic manifold to a Riemannian manifold. Then, for $V_1,V_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right),$ using Equation (18) and Theorem 1 we obtain

$${\mathcal{T}}_{V_1}{V}_2=-g_1(V_1,V_2)gradf.$$

(45)

Taking the inner product of Equation (45) with $\varphi U_1,$ we have

$$g_1({\mathcal{T}}_{V_1}{V}_2,\varphi U_1)=-g_1(V_1,V_2)g_1(gradf,\varphi U_1)$$

(46)

for all $U_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right).$

From Equations (4), (7), (13) and (46) we obtain

$$g_1({\nabla }_{V_1}\varphi V_2,U_1)=g_1(V_1,V_2)g_1(gradf,\varphi U_1).$$

Since ∇ is a metric connection, by using Equations (14) and (45) in the above equation, we obtain

$$g_1(V_1,U_1)g_1(gradf,\varphi V_2)=g_1(V_1,V_2)g_1(gradf,\varphi U_1).$$

(47)

Taking $U_1=V_2$ and interchanging the role of $V_1$ and $V_2,$ we obtain

$$g_1(V_2,V_2)g_1(gradf,\varphi V_1)=g_1(V_1,V_2)g_1(gradf,\varphi V_2).$$

(48)

From Equations (47) and (48), we obtain

$$g_1(gradf,\varphi V_1)=\frac{{\left(g_1(V_1,V_2)\right)}^2}{\left|\right|V_1{\left|\right|}^2\left|\right|V_2{\left|\right|}^2}{g}_1(gradf,\varphi V_1).$$

(49)

If $gradf\in \mathsf{\Gamma }\left(\varphi \left({\mathfrak{D}}_2\right)\right),$ then Equation (49) and the condition of equality in the Schwarz inequality imply that either f is constant on $\varphi \left({\mathfrak{D}}_2\right)$ or the fibers are one-dimensional. This implies the proof of $\left(i\right)$ and $\left(ii\right)$.

Now, from Equations (13) and (45), we obtain

$$g_1({\nabla }_{V_1}{U}_1,Z_1)=-g_1(V_1,U_1)g_1(gradf,Z_1),$$

(50)

for all $Z_1\in \mathsf{\Gamma }\left(\mu \right)$ and $\xi \ne Z_1.$ Using Equations (4), (7) and (50) we have

$$g_1({\nabla }_{V_1}\varphi U_1,\varphi Z_1)=-g_1(V_1,U_1)g_1(gradf,Z_1),$$

which implies

$$g_1({\nabla }_{\varphi U_1}{V}_1,\varphi Z_1)=-g_1(V_1,U_1)g_1(gradf,Z_1).$$

(51)

Since ∇ is a metric connection, then by using Equations (47) and (51) we have

$$g_1(\mathcal{H}{\nabla }_{\varphi U_1}{Z}_1,\varphi V_1)=-g_1(\varphi V_1,\varphi U_1)g_1(gradf,Z_1).$$

In addition, for the Riemannian map $\mathsf{\Pi },$ we have

$$g_2({\mathsf{\Pi }}_{*}\left({\nabla }_{\varphi U_1}^{{\mathcal{M}}_1}{Z}_1\right),{\mathsf{\Pi }}_{*}\left(\varphi V_1\right))=-g_2({\mathsf{\Pi }}_{*}\left(\varphi V_1\right),{\mathsf{\Pi }}_{*}\left(\varphi U_1\right))g_1(gradf,Z_1).$$

(52)

Again, using Equations (19), (21) and (52) we obtain

$$g_2({\stackrel{\mathsf{\Pi }}{\nabla }}_{\varphi U_1}{\mathsf{\Pi }}_{*}\left(Z_1\right),{\mathsf{\Pi }}_{*}\left(\varphi V_1\right))=-g_2({\mathsf{\Pi }}_{*}\left(\varphi V_1\right),{\mathsf{\Pi }}_{*}\left(\varphi U_1\right))g_1(gradf,Z_1),$$

which implies.

$${\stackrel{\mathsf{\Pi }}{\nabla }}_{\varphi U_1}{\mathsf{\Pi }}_{*}\left(Z_1\right)=-Z_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi U_1\right).$$

(53)

If $gradf\in \mathsf{\Gamma }\left(\mu \right)\setminus \left\{\xi \right\},$ then (53) implies $\left(iii\right).$ This completes the proof. □

Corollary 3.

Let Π be a CSIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ with $r=e^f$ and $dim\left(D_2\right)>1.$ Then, the fibers of Π are totally geodesic if and only if ${\stackrel{\mathsf{\Pi }}{\nabla }}_{\varphi U_1}{\mathsf{\Pi }}_{*}\left(Z_1\right)=0$$\forall U_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $Z_1\in \mathsf{\Gamma }\left(\mu \right).$

Lemma 5.

Let Π be a CSIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ with $r=e^f$ and $dim\left(D_2\right)>1.$ Then, ${\stackrel{\mathsf{\Pi }}{\nabla }}_{Z_1}{\mathsf{\Pi }}_{*}\left(\varphi X_1\right)=Z_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi X_1\right)$$\forall X_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $Z_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\}.$

Proof. 

Let $\mathsf{\Pi }$ be a CSIR map from a cosymplectic manifold to a Riemannian manifold. From Theorem $1,$ fibers are totally umbilical with mean curvature vector field $H=-gradf,$ then we have

$$\begin{array}{ccc}\hfill -g_1({\nabla }_{X_1}{Z}_1,X_2)& =& g_1({\nabla }_{X_1}{X}_2,Z_1)\hfill \\ \hfill -g_1({\nabla }_{X_1}{Z}_1,X_2)& =& -g_1(X_1,X_2)g_1(gradf,Z_1)\hfill \end{array}$$

for all $X_1,X_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $Z_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\}.$

Using Equations (4) and (7) in the above equation, we obtain

$$g_1({\nabla }_{Z_1}\varphi X_1,\varphi X_2)=g_1(\varphi X_1,\varphi X_2)g_1(gradf,Z_1).$$

(54)

Since $\mathsf{\Pi }$ is an SIR map, by using Equation (54) we have

$$g_2({\mathsf{\Pi }}_{*}\left({\nabla }_{Z_1}^{\mathsf{\Pi }}\varphi X_1\right),{\mathsf{\Pi }}_{*}\left(\varphi X_2\right))=g_2({\mathsf{\Pi }}_{*}\left(\varphi X_1\right),{\mathsf{\Pi }}_{*}\left(\varphi X_2\right))g_1(gradf,Z_1).$$

(55)

From (19) and (55) we obtain

$$g_2({\stackrel{\mathsf{\Pi }}{\nabla }}_{Z_1}{\mathsf{\Pi }}_{*}\left(\varphi X_1\right),{\mathsf{\Pi }}_{*}\left(\varphi X_2\right))=g_2({\mathsf{\Pi }}_{*}\left(\varphi X_1\right),{\mathsf{\Pi }}_{*}\left(\varphi X_2\right))g_1(gradf,Z_1),$$

(56)

which implies ${\stackrel{\mathsf{\Pi }}{\nabla }}_{Z_1}{\mathsf{\Pi }}_{*}\left(\varphi X_1\right)=Z_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi X_1\right)$ $\forall X_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $Z_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\}.$ □

Corollary 4.

Let Π be a CSIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ with $r=e^f$ and $dim\left(D_2\right)>1.$ Then, ${\stackrel{\mathsf{\Pi }}{\nabla }}_{Z_1}{\mathsf{\Pi }}_{*}\left(\varphi X_1\right)=0$$\forall X_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ and $Z_1=\xi \in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }.$

Theorem 4.

Let Π be a CSIR map with $r=e^f$ from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2).$ If $\mathcal{T}$ is not identically zero, then the invariant distribution ${\mathfrak{D}}_1$ does not define a totally geodesic foliation on ${\mathcal{M}}_1.$

Proof. 

For $V_1,V_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_1\right)$ and $Z_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$, using Equations (4), (7), (13) and (18) we obtain

$$\begin{array}{ccc}\hfill g_1({\nabla }_{V_1}{V}_2,Z_1)& =& g_1({\nabla }_{V_1}\varphi V_2,\varphi Z_1)\hfill \\ & =& g_1({\mathcal{T}}_{V_1}\varphi V_2,\varphi Z_1)\hfill \\ & =& -g_1(V_1,\varphi V_2)g_1(gradf,\varphi Z_1).\hfill \end{array}$$

Thus, the assertion can be seen from the above equation and the fact that $gradf\in \varphi \left({\mathfrak{D}}_2\right).$ □

Theorem 5.

Let Π be a CSIR map with $r=e^f$ from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2).$ Then, the fibers of Π are totally geodesic, or anti-invariant distribution ${\mathfrak{D}}_2$ is one-dimensional.

Proof. 

If the fibers of $\mathsf{\Pi }$ are totally geodesic, it is obvious. For the second one, since $\mathsf{\Pi }$ is a Clairaut proper semi-invariant Riemannian map, then either $dim\left({\mathfrak{D}}_2\right)=1$ or $dim\left({\mathcal{D}}_2\right)>1.$ If $dim\left({\mathcal{D}}_2\right)>1,$ then we can choose $V_1,V_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_2\right)$ such that $\{V_1,V_2\}$ is orthonormal. From Equations (14), (23) and (24) we obtain

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{V_1}\varphi V_2+\mathcal{H}{\nabla }_{V_1}\varphi V_2& =& {\nabla }_{V_1}\varphi V_2\hfill \\ \hfill {\mathcal{T}}_{V_1}\varphi V_2+\mathcal{H}{\nabla }_{V_1}\varphi V_2& =& B{\mathcal{T}}_{V_1}{V}_2+C{\mathcal{T}}_{V_1}{V}_2+\psi \mathcal{V}{\nabla }_{V_1}{V}_2+\omega \mathcal{V}{\nabla }_{V_1}{V}_2.\hfill \end{array}$$

Taking the inner product of the above equation with $V_1,$ we obtain

$$g_1({\mathcal{T}}_{V_1}\varphi V_2,V_1)=g_1(B{\mathcal{T}}_{V_1}{V}_2,V_1)+g_1(\psi \mathcal{V}{\nabla }_{V_1}{V}_2,V_1).$$

(57)

From Equation (7) we have

$$g_1({\mathcal{T}}_{V_1}{V}_1,\varphi V_2)=-g_1({\mathcal{T}}_{V_1}\varphi V_2,V_1)=g_1({\mathcal{T}}_{V_1}{V}_2,\varphi V_1).$$

(58)

Now, using Equations (18) and (58) we obtain

$$g_1({\mathcal{T}}_{V_1}{V}_1,\varphi V_2)=-g_1(gradf,\varphi V_2).$$

(59)

From Equations (18), (58) and (59) we obtain

$$-g_1(gradf,\varphi V_2)=g_1({\mathcal{T}}_{V_1}{V}_1,\varphi V_2)=-g_1({\mathcal{T}}_{V_1}\varphi V_2,V_1)=g_1({\mathcal{T}}_{V_1}{V}_2,\varphi V_1),$$

(60)

from which we obtain

$$\begin{array}{ccc}\hfill g_1(gradf,\varphi V_2)& =& -g_1({\mathcal{T}}_{V_1}{V}_2,\varphi V_1)\hfill \\ \hfill g_1(gradf,\varphi V_2)& =& g_1(V_1,V_2)g_1(gradf,\varphi V_1)\hfill \\ \hfill g_1(gradf,\varphi V_2)& =& 0.\hfill \end{array}$$

Thus, we obtain

$$gradf\perp \varphi \left({\mathfrak{D}}_2\right).$$

Therefore, the dimension of ${\mathfrak{D}}_2$ must be one. □

Theorem 6.

Let Π be a CSIR map from a cosymplectic manifold $({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)$ to a Riemannian manifold $({\mathcal{M}}_2,g_2)$ with $r=e^f$ and $dim\left(D_2\right)>1.$ Then, we obtain

$$\sum _{\kappa =1}^{\omega }{g}_1({\mathcal{A}}_{X_1}{x}_{\kappa },{\mathcal{A}}_{X_1}{x}_{\kappa })=\sum _{\kappa =1}^{\omega }{g}_2({\nabla }_{X_1}^{\mathsf{\Pi }}{\mathsf{\Pi }}_{*}\left(\varphi x_{\kappa }\right),{\nabla }_{X_1}^{\mathsf{\Pi }}{\mathsf{\Pi }}_{*}\left(\varphi x_{\kappa }\right)),$$

(61)

$$\sum _{i=1}^{\beta +\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(F_i,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,F_i))=\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})({\vartheta }_l,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,{\vartheta }_l)),$$

(62)

$$\sum _{j=1}^{\beta }{g}_1({\mathcal{A}}_{X_1}{w}_j,{\mathcal{A}}_{X_1}{w}_j)={\left(X_1\left(f\right)\right)}^2\sum _{j=1}^{\beta }{g}_1(w_j,w_j),$$

(63)

$\forall X_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\},$ where $\{x_1,x_2,....,x_{\omega }\},\{w_1,w_2,....,w_{\beta }\},\{F_1,F_2,....,F_{\beta +\mathcal{s}}\}$ and

$\{{\vartheta }_1,{\vartheta }_2,....{\vartheta }_{\mathcal{s}}\}$ are orthonormal frames of ${\mathfrak{D}}_1,{\mathfrak{D}}_2,\varphi {\left({\mathfrak{D}}_2\right)}^{\perp }\oplus \mu$ and μ, respectively.

Proof. 

Let $\mathsf{\Pi }:$$({\mathcal{M}}_1,\varphi ,\xi ,\eta ,g_1)\to ({\mathcal{M}}_2,g_2)$ be a CSIR map, then for all $X_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\},$ we have

$$\sum _{\kappa =1}^{\omega }{g}_1({\mathcal{A}}_{X_1}{x}_{\kappa },{\mathcal{A}}_{X_1}{x}_{\kappa })=\sum _{\kappa =1}^{\omega }{g}_1(\mathcal{H}{\nabla }_{X_1}\varphi x_{\kappa },\mathcal{H}{\nabla }_{X_1}\varphi x_{\kappa }).$$

(64)

Since $\mathsf{\Pi }$ is a Riemannian map, in view of Equation (19), Equation (64) transforms to

$$\begin{array}{ccc}\hfill \sum _{\kappa =1}^{\omega }{g}_1({\mathcal{A}}_{X_1}{x}_{\kappa },{\mathcal{A}}_{X_1}{x}_{\kappa })& =& \sum _{\kappa =1}^{\omega }{g}_2({\mathsf{\Pi }}_{*}\left({\nabla }_{X_1}^{{\mathcal{M}}_1}\varphi x_{\kappa }\right),{\mathsf{\Pi }}_{*}\left({\nabla }_{X_1}^{{\mathcal{M}}_1}\varphi x_{\kappa }\right))\hfill \\ & =& \sum _{\kappa =1}^{\omega }{g}_2({\nabla }_{X_1}^{\mathsf{\Pi }}{\mathsf{\Pi }}_{*}\left(\varphi x_{\kappa }\right),{\nabla }_{X_1}^{\mathsf{\Pi }}{\mathsf{\Pi }}_{*}\left(\varphi x_{\kappa }\right)).\hfill \end{array}$$

Now, for all $X_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\},$ we obtain

$$\begin{array}{ccc}& & \sum _{i=1}^{\beta +\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(F_i,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,F_i))\hfill \\ & =& \sum _{j=1}^{\beta }\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j+{\vartheta }_l,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j+{\vartheta }_l)).\hfill \end{array}$$

Since $\varphi w_j\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }$ and $(\nabla {\mathsf{\Pi }}_{*})$ is linear, from the above equation, we have

$$\begin{array}{ccc}& & \sum _{i=1}^{\beta +\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(F_i,X_1),(\nabla {\mathsf{\Pi }}_{*})(F_i,X_1))\hfill \\ & =& \sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))\hfill \\ & & +\sum _{j=1}^{\beta }\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})({\vartheta }_l,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))\hfill \\ & & +\sum _{j=1}^{\beta }\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,{\vartheta }_l))\hfill \\ & & +\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})({\vartheta }_l,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,{\vartheta }_l)).\hfill \end{array}$$

(65)

Thus, (61) holds.

On the other side, using (19) in the first term of the right-hand side of (65), we have

$$\begin{array}{ccc}& & \sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))\hfill \\ & =& \sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),\stackrel{\mathsf{\Pi }}{{\nabla }_{X_1}}{\mathsf{\Pi }}_{*}\left(\varphi w_j\right)-{\mathsf{\Pi }}_{*}\left({\nabla }_{X_1}^{{\mathcal{M}}_1}\varphi w_j\right)),\hfill \end{array}$$

which, by using Equations (4), (7) and (65), turns into

$$\begin{array}{ccc}& & \sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))\hfill \\ & =& \sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),\stackrel{\mathsf{\Pi }}{{\nabla }_{X_1}}{\mathsf{\Pi }}_{*}\left(\varphi w_j\right)).\hfill \end{array}$$

(66)

Now, by using Lemma 4 in (66) we obtain

$$\sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))=\sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),X_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi w_j\right)).$$

This implies that

$$\sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))=\sum _{j=1}^{\beta }{X}_1\left(f\right)g_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),{\mathsf{\Pi }}_{*}\left(\varphi w_j\right)).$$

By using Equation (20) in the above equation, it follows that

$$\sum _{j=1}^{\beta }{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))=0.$$

(67)

Similarly, we find

$$\sum _{j=1}^{\beta }\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})({\vartheta }_l,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))=0,$$

(68)

$$\sum _{j=1}^{\beta }\sum _{l=1}^{\mathcal{s}}{g}_2((\nabla {\mathsf{\Pi }}_{*})(\varphi w_j,X_1),(\nabla {\mathsf{\Pi }}_{*})(X_1,{\vartheta }_l))=0.$$

(69)

Thus, by using Equations (67)–(69) in Equation (65) we obtain (62).

Further, for $X_1\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }\setminus \left\{\xi \right\},$ we obtain

$$\begin{array}{ccc}\hfill \sum _{j=1}^{\beta }{g}_1({\mathcal{A}}_{X_1}{w}_j,{\mathcal{A}}_{X_1}{w}_j)& =& \sum _{j=1}^{\beta }{g}_1(\mathcal{H}{\nabla }_{X_1}{w}_j,\mathcal{H}{\nabla }_{X_1}{w}_j)\hfill \\ & =& \sum _{j=1}^{\beta }{g}_1(\mathcal{H}{\nabla }_{X_1}\varphi w_j,\mathcal{H}{\nabla }_{X_1}\varphi w_j).\hfill \end{array}$$

Since $\mathsf{\Pi }$ is a Riemannian map, in view of Equation (19), the above equation becomes

$$\begin{array}{ccc}& & \sum _{j=1}^{\beta }{g}_1({\mathcal{A}}_{X_1}{w}_j,{\mathcal{A}}_{X_1}{w}_j)\hfill \\ & =& \sum _{j=1}^{\beta }\{g_2((\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j),(\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j))\hfill \\ & & -2g_2((\nabla {\mathsf{\Pi }}_{*})(X_1,\varphi w_j),{\stackrel{\mathsf{\Pi }}{\nabla }}_{X_1}{\mathsf{\Pi }}_{*}\left(\varphi w_j\right))\hfill \\ & & +g_2({\stackrel{\mathsf{\Pi }}{\nabla }}_{X_1}{\mathsf{\Pi }}_{*}\left(\varphi w_j\right),\stackrel{\mathsf{\Pi }}{{\nabla }_{X_1}}{\mathsf{\Pi }}_{*}\left(\varphi w_j\right))\},\hfill \end{array}$$

(70)

which, by using Lemma 4 and Equations (21) and (67) in (70) we obtain

$$\begin{array}{ccc}\hfill \sum _{j=1}^{\beta }{g}_1({\mathcal{A}}_{X_1}{w}_j,{\mathcal{A}}_{X_1}{w}_j)& =& \sum _{j=1}^{\beta }{g}_2(X_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi w_j\right),X_1\left(f\right){\mathsf{\Pi }}_{*}\left(\varphi w_j\right))\hfill \\ & =& {\left(X_1\left(f\right)\right)}^2\sum _{j=1}^{\beta }{g}_2({\mathsf{\Pi }}_{*}\left(\varphi w_j\right),{\mathsf{\Pi }}_{*}\left(\varphi w_j\right)).\hfill \end{array}$$

(71)

Since $\varphi w_j\in \mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }$ and $\mathsf{\Pi }$ is a Riemannian map, from (71) we obtain

$$\sum _{j=1}^{\beta }{g}_1({\mathcal{A}}_{X_1}{w}_j,{\mathcal{A}}_{X_1}{w}_j)={\left(X_1\left(f\right)\right)}^2\sum _{j=1}^{\beta }{g}_1(w_j,w_j).$$

(72)

Thus, from Equations (4) and (72) we obtain (63). □

4. Example

Let ${\mathcal{M}}_1$ be a differentiable manifold given by ${\mathcal{M}}_1=\{(x_1,x_2,x_{3,}{x}_4,x_5,x_6,x_7)\in R^7:x_7>0\}$. We define the Riemannian metric $g_1$ on ${\mathcal{M}}_1$ by $g_1=e^{2x_7}d{x}_1^2+e^{2x_7}d{x}_2^2+e^{2x_7}d{x}_3^2+e^{2x_7}d{x}_4^2+e^{2x_7}d{x}_5^2+e^{2x_7}d{x}_6^2+d{x}_7^2$, and the cosymplectic structure $(\varphi ,\xi ,\eta ,g_1)$ on ${\mathcal{M}}_1$ is defined as

$$\begin{array}{c}\hfill \varphi (x_1\frac{\partial }{\partial x_1}+x_2\frac{\partial }{\partial x_2}+x_3\frac{\partial }{\partial x_3}+x_4\frac{\partial }{\partial x_4}+x_5\frac{\partial }{\partial x_5}+x_6\frac{\partial }{\partial x_6}+x_7\frac{\partial }{\partial x_7})\\ \hfill =(x_4\frac{\partial }{\partial x_1}+x_5\frac{\partial }{\partial x_2}+x_6\frac{\partial }{\partial x_3}-x_1\frac{\partial }{\partial x_4}-x_2\frac{\partial }{\partial x_5}-x_3\frac{\partial }{\partial x_6}),\end{array}$$

$\xi =\frac{\partial }{\partial x_7}$, $\eta =d{x}_7$, and $g_1$ was earlier defined.

Let ${\mathcal{M}}_2=\{(v_1,v_2,v_3,v_4)\in R^4\}$ be a Riemannian manifold with Riemannian metric $g_2$ on ${\mathcal{M}}_2$ given by $g_2=e^{2x_7}d{v}_1^2+e^{2x_7}d{v}_2^2+e^{2x_7}d{v}_3^2+d{v}_4^2.$ Define a map $\mathsf{\Pi }:R^7\to R^4$ by

$$\mathsf{\Pi }(x_1,x_2,x_3,x_4,x_5,x_6,x_7)=(\frac{x_2-x_5}{\sqrt{2}},101,x_6,x_7).$$

Then, we have

$$ker{\mathsf{\Pi }}_{*}={\mathfrak{D}}_1\oplus {\mathfrak{D}}_2,$$

where

$${\mathfrak{D}}_1=<V_1=e_1,V_2=e_4>,{\mathfrak{D}}_2=<V_3=e_2+e_5,V_4=e_3>,$$

and

$${(ker{\mathsf{\Pi }}_{*})}^{\perp }=<H_1=e_2-e_5,H_2=e_6,H_3=e_7>,$$

where $\{e_1=e^{-x_7}\frac{\partial }{\partial x_1},e_2=e^{-x_7}\frac{\partial }{\partial x_2},e_3=e^{-x_7}\frac{\partial }{\partial x_3},e_4=e^{-x_7}\frac{\partial }{\partial x_4},e_5=e^{-x_7}\frac{\partial }{\partial x_5},e_6=e^{-x_7}\frac{\partial }{\partial x_6},e_7=\frac{\partial }{\partial x_7}\},$$\{e_1^{*}=\frac{\partial }{\partial v_1},e_2^{*}=\frac{\partial }{\partial v_2},e_3^{*}=\frac{\partial }{\partial v_3},e_4^{*}=\frac{\partial }{\partial v_4}\}$ are bases on $T_p{\mathcal{M}}_1$ and $T_{\mathsf{\Pi }\left(p\right)}{\mathcal{M}}_2$, respectively, for all $p\in {\mathcal{M}}_1.$ By direct computations, we can see that ${\mathsf{\Pi }}_{*}\left(H_1\right)=\sqrt{2}{e}^{-x_7}{e}_1^{*},{\mathsf{\Pi }}_{*}\left(H_2\right)=e^{-x_7}{e}_2^{*},{\mathsf{\Pi }}_{*}\left(H_3\right)=e_3^{*}$ and $g_1(H_i,H_j)=g_2({\mathsf{\Pi }}_{*}{H}_i,{\mathsf{\Pi }}_{*}{H}_j)$ for all $H_i,H_j\in$$\mathsf{\Gamma }{(ker{\mathsf{\Pi }}_{*})}^{\perp }$, $i,j=1,2,3$. Thus, $\mathsf{\Pi }$ is a Riemannian map with ${\left(range{\mathsf{\Pi }}_{*}\right)}^{\perp }=<e_4^{*}>$. Moreover, it is easy to see that $\varphi V_3=H_1$ and $\varphi V_4=H_2.$ Therefore, $\mathsf{\Pi }$ is an SIR map.

Now, we will find the smooth function f on ${\mathcal{M}}_1$ satisfying $T_{V}V=g_1(V,V)\nabla f$$\forall V\in \mathsf{\Gamma }(ker{\mathsf{\Pi }}_{*}).$ The covariant derivative for the vector fields $E=E_i\frac{\partial }{\partial x_i},F=F_j\frac{\partial }{\partial x_j}$ on ${\mathcal{M}}_1$ is defined as

$${\nabla }_{E}F=E_i{F}_j{\nabla }_{\frac{\partial }{\partial x_i}}\frac{\partial }{\partial x_j}+E_i\frac{\partial F_j}{\partial x_i}\frac{\partial }{\partial x_j},$$

(73)

where the covariant derivative of basis vector fields $\frac{\partial }{\partial x_j}$ and $\frac{\partial }{\partial x_i}$ is defined by

$${\nabla }_{\frac{\partial }{\partial x_i}}\frac{\partial }{\partial x_j}={\mathsf{\Gamma }}_{ij}^k\frac{\partial }{\partial x_k},$$

(74)

and Christoffel symbols are defined by

$${\mathsf{\Gamma }}_{ij}^k=\frac{1}{2}{g}^{kl}(\frac{\partial g_{1jl}}{\partial x_i}+\frac{\partial g_{1il}}{\partial x_j}-\frac{\partial g_{1ij}}{\partial x_l}).$$

(75)

Thus, we obtain

$$\begin{array}{ccc}\hfill g_{1ij}& =& \left[\begin{array}{ccccccc}{e}^{2x_7}& 0& 0& 0& 0& 0& 0\\ 0& e^{2x_7}& 0& 0& 0& 0& 0\\ 0& 0& e^{2x_7}& 0& 0& 0& 0\\ 0& 0& 0& e^{2x_7}& 0& 0& 0\\ 0& 0& 0& 0& e^{2x_7}& 0& 0\\ 0& 0& 0& 0& 0& e^{2x_7}& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right],\hfill \\ \hfill g_1^{ij}& =& \left[\begin{array}{ccccccc}{e}^{-2x_7}& 0& 0& 0& 0& 0& 0\\ 0& e^{-2x_7}& 0& 0& 0& 0& 0\\ 0& 0& e^{-2x_7}& 0& 0& 0& 0\\ 0& 0& 0& e^{-2x_7}& 0& 0& 0\\ 0& 0& 0& 0& e^{-2x_7}& 0& 0\\ 0& 0& 0& 0& 0& e^{-2x_7}& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right].\hfill \end{array}$$

(76)

By using Equations (75) and (76) we find

$$\begin{array}{ccc}\hfill {\mathsf{\Gamma }}_{11}^1& =& {\mathsf{\Gamma }}_{11}^2={\mathsf{\Gamma }}_{11}^3={\mathsf{\Gamma }}_{11}^4={\mathsf{\Gamma }}_{11}^5={\mathsf{\Gamma }}_{11}^6=0,{\mathsf{\Gamma }}_{11}^7=-e^{2x_7},\hfill \\ \hfill {\mathsf{\Gamma }}_{22}^1& =& {\mathsf{\Gamma }}_{22}^2={\mathsf{\Gamma }}_{22}^3={\mathsf{\Gamma }}_{22}^4={\mathsf{\Gamma }}_{22}^5={\mathsf{\Gamma }}_{22}^6=0,{\mathsf{\Gamma }}_{22}^7=-e^{2x_7},\hfill \\ \hfill {\mathsf{\Gamma }}_{33}^1& =& {\mathsf{\Gamma }}_{33}^2={\mathsf{\Gamma }}_{33}^3={\mathsf{\Gamma }}_{33}^4={\mathsf{\Gamma }}_{33}^5={\mathsf{\Gamma }}_{33}^6=0,{\mathsf{\Gamma }}_{33}^7=-e^{2x_7},\hfill \\ \hfill {\mathsf{\Gamma }}_{44}^1& =& {\mathsf{\Gamma }}_{44}^2={\mathsf{\Gamma }}_{44}^3={\mathsf{\Gamma }}_{44}^4={\mathsf{\Gamma }}_{44}^5={\mathsf{\Gamma }}_{44}^6=0,{\mathsf{\Gamma }}_{44}^7=-e^{2x_7},\hfill \\ \hfill {\mathsf{\Gamma }}_{55}^1& =& {\mathsf{\Gamma }}_{55}^2={\mathsf{\Gamma }}_{55}^3={\mathsf{\Gamma }}_{55}^4={\mathsf{\Gamma }}_{55}^5={\mathsf{\Gamma }}_{55}^6=0,{\mathsf{\Gamma }}_{55}^7=-e^{2x_7},\hfill \\ \hfill {\mathsf{\Gamma }}_{12}^1& =& {\mathsf{\Gamma }}_{12}^2={\mathsf{\Gamma }}_{12}^3={\mathsf{\Gamma }}_{12}^4={\mathsf{\Gamma }}_{12}^5={\mathsf{\Gamma }}_{12}^6={\mathsf{\Gamma }}_{12}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{21}^1& =& {\mathsf{\Gamma }}_{21}^2={\mathsf{\Gamma }}_{21}^3={\mathsf{\Gamma }}_{21}^4={\mathsf{\Gamma }}_{21}^5={\mathsf{\Gamma }}_{21}^6={\mathsf{\Gamma }}_{21}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{13}^1& =& {\mathsf{\Gamma }}_{13}^2={\mathsf{\Gamma }}_{13}^3={\mathsf{\Gamma }}_{13}^4={\mathsf{\Gamma }}_{13}^5={\mathsf{\Gamma }}_{13}^6={\mathsf{\Gamma }}_{13}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{31}^1& =& {\mathsf{\Gamma }}_{31}^2={\mathsf{\Gamma }}_{31}^3={\mathsf{\Gamma }}_{31}^4={\mathsf{\Gamma }}_{31}^5={\mathsf{\Gamma }}_{31}^6={\mathsf{\Gamma }}_{31}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{14}^1& =& {\mathsf{\Gamma }}_{14}^2={\mathsf{\Gamma }}_{14}^3={\mathsf{\Gamma }}_{14}^4={\mathsf{\Gamma }}_{14}^5={\mathsf{\Gamma }}_{14}^6={\mathsf{\Gamma }}_{14}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{41}^1& =& {\mathsf{\Gamma }}_{41}^2={\mathsf{\Gamma }}_{41}^3={\mathsf{\Gamma }}_{41}^4={\mathsf{\Gamma }}_{41}^5={\mathsf{\Gamma }}_{41}^6={\mathsf{\Gamma }}_{41}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{15}^1& =& {\mathsf{\Gamma }}_{15}^2={\mathsf{\Gamma }}_{15}^3={\mathsf{\Gamma }}_{15}^4={\mathsf{\Gamma }}_{15}^5={\mathsf{\Gamma }}_{15}^6={\mathsf{\Gamma }}_{15}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{51}^1& =& {\mathsf{\Gamma }}_{51}^2={\mathsf{\Gamma }}_{51}^3={\mathsf{\Gamma }}_{51}^4={\mathsf{\Gamma }}_{51}^5={\mathsf{\Gamma }}_{51}^6={\mathsf{\Gamma }}_{51}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{23}^1& =& {\mathsf{\Gamma }}_{23}^2={\mathsf{\Gamma }}_{23}^3={\mathsf{\Gamma }}_{23}^4={\mathsf{\Gamma }}_{23}^5={\mathsf{\Gamma }}_{23}^6={\mathsf{\Gamma }}_{23}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{32}^1& =& {\mathsf{\Gamma }}_{32}^2={\mathsf{\Gamma }}_{32}^3={\mathsf{\Gamma }}_{32}^4={\mathsf{\Gamma }}_{32}^5={\mathsf{\Gamma }}_{32}^6={\mathsf{\Gamma }}_{32}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{24}^1& =& {\mathsf{\Gamma }}_{24}^2={\mathsf{\Gamma }}_{24}^3={\mathsf{\Gamma }}_{24}^4={\mathsf{\Gamma }}_{24}^5={\mathsf{\Gamma }}_{24}^6={\mathsf{\Gamma }}_{24}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{42}^1& =& {\mathsf{\Gamma }}_{42}^2={\mathsf{\Gamma }}_{42}^3={\mathsf{\Gamma }}_{42}^4={\mathsf{\Gamma }}_{42}^5={\mathsf{\Gamma }}_{42}^6={\mathsf{\Gamma }}_{42}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{25}^1& =& {\mathsf{\Gamma }}_{25}^2={\mathsf{\Gamma }}_{25}^3={\mathsf{\Gamma }}_{25}^4={\mathsf{\Gamma }}_{25}^5={\mathsf{\Gamma }}_{25}^6={\mathsf{\Gamma }}_{25}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{52}^1& =& {\mathsf{\Gamma }}_{52}^2={\mathsf{\Gamma }}_{52}^3={\mathsf{\Gamma }}_{52}^4={\mathsf{\Gamma }}_{52}^5={\mathsf{\Gamma }}_{52}^6={\mathsf{\Gamma }}_{52}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{34}^1& =& {\mathsf{\Gamma }}_{34}^2={\mathsf{\Gamma }}_{34}^3={\mathsf{\Gamma }}_{34}^4={\mathsf{\Gamma }}_{34}^5={\mathsf{\Gamma }}_{34}^6={\mathsf{\Gamma }}_{34}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{43}^1& =& {\mathsf{\Gamma }}_{43}^2={\mathsf{\Gamma }}_{43}^3={\mathsf{\Gamma }}_{43}^4={\mathsf{\Gamma }}_{43}^5={\mathsf{\Gamma }}_{43}^6={\mathsf{\Gamma }}_{43}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{35}^1& =& {\mathsf{\Gamma }}_{35}^2={\mathsf{\Gamma }}_{35}^3={\mathsf{\Gamma }}_{35}^4={\mathsf{\Gamma }}_{35}^5={\mathsf{\Gamma }}_{35}^6={\mathsf{\Gamma }}_{35}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{53}^1& =& {\mathsf{\Gamma }}_{53}^2={\mathsf{\Gamma }}_{53}^3={\mathsf{\Gamma }}_{53}^4={\mathsf{\Gamma }}_{53}^5={\mathsf{\Gamma }}_{53}^6={\mathsf{\Gamma }}_{53}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{45}^1& =& {\mathsf{\Gamma }}_{45}^2={\mathsf{\Gamma }}_{45}^3={\mathsf{\Gamma }}_{45}^4={\mathsf{\Gamma }}_{45}^5={\mathsf{\Gamma }}_{45}^6={\mathsf{\Gamma }}_{45}^7=0,\hfill \\ \hfill {\mathsf{\Gamma }}_{54}^1& =& {\mathsf{\Gamma }}_{54}^2={\mathsf{\Gamma }}_{54}^3={\mathsf{\Gamma }}_{54}^4={\mathsf{\Gamma }}_{54}^5={\mathsf{\Gamma }}_{54}^6={\mathsf{\Gamma }}_{54}^7=0.\hfill \end{array}$$

(77)

Using Equations (73), (74) and (77) we calculate

$$\begin{array}{ccc}\hfill {\nabla }_{e_1}{e}_1& =& {\nabla }_{e_2}{e}_2={\nabla }_{e_3}{e}_3={\nabla }_{e_4}{e}_4=-\frac{\partial }{\partial x_7},\hfill \\ \hfill {\nabla }_{e_1}{e}_2& =& {\nabla }_{e_1}{e}_3={\nabla }_{e_1}{e}_4={\nabla }_{e_2}{e}_1={\nabla }_{e_2}{e}_3={\nabla }_{e_2}{e}_4=0,\hfill \\ \hfill {\nabla }_{e_3}{e}_1& =& {\nabla }_{e_3}{e}_2={\nabla }_{e_3}{e}_4={\nabla }_{e_4}{e}_1={\nabla }_{e_4}{e}_2={\nabla }_{e_4}{e}_3=0.\hfill \end{array}$$

(78)

Thus, we find

$$\begin{array}{ccc}\hfill {\nabla }_{V_1}{V}_1& =& {\nabla }_{e_1}{e}_1=-\frac{\partial }{\partial x_7},{\nabla }_{V_2}{V}_2={\nabla }_{e_4}{e}_4=-\frac{\partial }{\partial x_7},\hfill \\ \hfill {\nabla }_{V_3}{V}_3& =& {\nabla }_{e_2+e_5}{e}_2+e_5=-2\frac{\partial }{\partial x_7},{\nabla }_{V_4}{V}_4={\nabla }_{e_3}{e}_3=-\frac{\partial }{\partial x_7}.\hfill \end{array}$$

(79)

Now, from

$${\mathcal{T}}_{V}V={\mathcal{T}}_{{\lambda }_1{V}_1+{\lambda }_2{V}_2+{\lambda }_3{V}_3+{\lambda }_4{V}_4}{\lambda }_1{V}_1+{\lambda }_2{V}_2+{\lambda }_3{V}_3+{\lambda }_4{V}_4,{\lambda }_1,{\lambda }_2,{\lambda }_3,{\lambda }_4\in R,$$

we lead to

$$\begin{array}{ccc}\hfill {\mathcal{T}}_{V}V& =& {\lambda }_1^2{\mathcal{T}}_{V_1}{V}_1+{\lambda }_2^2{\mathcal{T}}_{V_2}{V}_2+{\lambda }_3^2{\mathcal{T}}_{V_3}{V}_3+{\lambda }_4^2{\mathcal{T}}_{V_4}{V}_4\hfill \\ & & +2{\lambda }_1{\lambda }_2{\mathcal{T}}_{V_1}{V}_2+2{\lambda }_1{\lambda }_3{\mathcal{T}}_{V_1}{V}_3+2{\lambda }_1{\lambda }_4{\mathcal{T}}_{V_1}{V}_4+2{\lambda }_2{\lambda }_3{\mathcal{T}}_{V_2}{V}_3\hfill \\ & & +2{\lambda }_2{\lambda }_4{\mathcal{T}}_{V_2}{V}_4+2{\lambda }_3{\lambda }_4{\mathcal{T}}_{V_3}{V}_4.\hfill \end{array}$$

(80)

From Equations (13) and (79), we obtain

$$\begin{array}{c}\hfill {\mathcal{T}}_{V_1}{V}_1=-\frac{\partial }{\partial x_7},{\mathcal{T}}_{V_2}{V}_2=-\frac{\partial }{\partial x_7},{\mathcal{T}}_{V_3}{V}_3=-2\frac{\partial }{\partial x_7},{\mathcal{T}}_{V_4}{V}_4=-\frac{\partial }{\partial x_7},\\ \hfill {\mathcal{T}}_{V_1}{V}_2=0,{\mathcal{T}}_{V_1}{V}_3=0,{\mathcal{T}}_{V_1}{V}_4=0,{\mathcal{T}}_{V_2}{V}_3=0,{\mathcal{T}}_{V_2}{V}_4=0,{\mathcal{T}}_{V_3}{V}_4=0.\end{array}$$

(81)

Thus, by using Equations (80) and (81) we obtain

$${\mathcal{T}}_{V}V=-({\lambda }_1^2+{\lambda }_2^2+2{\lambda }_3^2+{\lambda }_1^4)\frac{\partial }{\partial x_7}.$$

(82)

Since $V={\lambda }_1{V}_1+{\lambda }_2{V}_2+{\lambda }_3{V}_3+{\lambda }_4{V}_4,$$g_1({\lambda }_1{V}_1+{\lambda }_2{V}_2+{\lambda }_3{V}_3+{\lambda }_4{V}_4,{\lambda }_1{V}_1+{\lambda }_2{V}_2+{\lambda }_3{V}_3+{\lambda }_4{V}_4)={\lambda }_1^2+{\lambda }_2^2+2{\lambda }_3^2+{\lambda }_1^4.$ For any smooth function f on $R^7$, the gradient of f with respect to the metric $g_1$ is given by $\nabla f={\sum }_{i,j=1}^7{g}_1^{ij}\frac{\partial f}{\partial x_i}\frac{\partial }{\partial x_j}.$ Hence, $\nabla f=$$e^{-2x_7}\frac{\partial f}{\partial x_1}\frac{\partial }{\partial x_1}+e^{-2x_7}\frac{\partial f}{\partial x_2}\frac{\partial }{\partial x_2}+e^{-2x_7}\frac{\partial f}{\partial x_3}\frac{\partial }{\partial x_3}+e^{-2x_7}\frac{\partial f}{\partial x_4}\frac{\partial }{\partial x_4}+e^{-2x_7}\frac{\partial f}{\partial x_5}\frac{\partial }{\partial x_5}+e^{-2x_7}\frac{\partial f}{\partial x_6}\frac{\partial }{\partial x_6}+\frac{\partial f}{\partial x_7}\frac{\partial }{\partial x_7}.$ Hence, $\nabla f=\frac{\partial }{\partial x_7}$ for the function $f=x_7.$ Then, it is easy to see that ${\mathcal{T}}_{V}V=-g_1(V,V)\nabla f$; thus, by Theorem 1, $\mathsf{\Pi }$ is a CSIR map from cosymplectic manifold onto Riemannian manifold.

5. Conclusions

In the last few years, Riemannian maps have been extensively studied between different kinds of the manifolds. Recently, a special type of Riemannian map, namely, the "Clairaut Riemannian map" was introduced and studied by Sahin [30]; moreover, he, in [37], gave an open problem to find characterizations for Clairaut Riemannian maps. As a continuation of this study, we tried to study Clairaut semi-invariat Riemannian maps in contact geometry. Here, we investigated the various most fundamental geometric properties on the fibers and distributions of these maps. In the future, we plan to focus on studying Clairaut’s semi-slant Riemannian maps, Clairaut’s hemi-slant Riemannian maps, and Clairaut’s bi-slant Riemannian maps between different kinds of the manifolds.