Semi-invariant Conformal submersions with horizontal Reeb vector field

Uday Chand De, Shashikant Pandey and Punam Gupta

MSC Classification. 53C15, 53C40, 53C50.
Keywords. Conformal submersion, semi-invariant Riemannian submersion, semi-
invariant conformal $\zeta^{\perp}$ Riemannian submersion, almost contact
metric manifolds, totally geodesic.

Abstract. The present paper deals with the characterization of a new
submersion named semi-invariant conformal $\zeta^{\perp}$-Riemannian
submersion from almost contact metric manifolds onto Riemannian manifolds
which is the generalization of some known submersions on Riemannian manifolds.
We give important and adequate conditions for such submersions to be totally
geodesic and harmonic. Also, few examples are examined for such submersions
endowed with horizontal Reeb vector field.

##  1 Introduction

In Riemannian geometry, to develop Riemannian manifolds with non-negative
sectional curvature is an iconic problem. For this, O’Neill [25] and Gray [13]
initiate to study Riemannian submersions connecting Riemannian manifolds.
Riemannian submersions have several applications in mathematical physics,
specially in the Yang–Mills theory [9, 38], Kaluza-Klein theory [10, 15],
supergravity and superstring theories [16, 24]. Submersions were studied by
many authors between differential manifolds like almost Hermitian manifolds,
Kähler manifold, almost contact manifolds, nearly $K$-cosymplectic manifold,
Sasakian manifold etc. The authors studied semi-Riemannian submersion and
Lorentzian submersion [12], anti-invariant submersion [21, 23, 29, 35], anti-
invariant $\zeta^{\perp}$-submersion [22], semi-invariant submersion [31],
[19], semi-invariant $\zeta^{\perp}$-submersion [4] and many other type of
submersions studied by several authors [11, 20, 30, 37, 18, 27, 28], hemi-
slant submersion [33], anti-holomorphic semi-invariant submersion [34] etc.
between differential manifolds with different structures.

In comparison of conformal submersions, Riemannian submersions are specific.
The conformal maps don’t preserve distance between points but they preserve
angle between vector fields which allows us to transfer certain properties of
manifolds to another manifolds by deforming such properties. Ornea [26]
initiated the theory of conformal submersions between Riemannian manifolds.
Later, it was studied by many authors [1, 2, 3, 5, 14].

In the present manuscript, we establish the definition of semi-invariant
conformal $\zeta^{\perp}$-Riemannian submersion (in short, sic
$\zeta^{\perp}$-Rs) from almost contact metric (in short, acm) manifolds onto
Riemannian manifolds, which is the generalization of conformal anti-invariant
$\zeta^{\perp}$-submersions, semi-invariant conformal submersions and many
others. In section 2, we give preliminaries about Sasakian manifolds,
Riemannian submersions and conformal submersions. In section 3, we define sic
$\zeta^{\perp}$-Rs from acm manifolds onto Riemannian manifolds with examples.
We give the necessary and sufficient (briefly, ns) conditions for
distributions to be integrable. After that, we establish the ns condition for
sic $\zeta^{\perp}$-Rs to be homothetic map. Then we investigate the geometry
of leaves of horizontal and vertical distributions and obtain ns condition for
the distribution to be absolutely geodesic foliation on Sasakian manifolds. At
last, we obtain a condition when the fibers of sic $\zeta^{\perp}$-Rs from
Sasakian manifolds onto Riemannian manifolds to be locally product manifold.
In section 4, we provide ns conditions for a sic $\zeta^{\perp}$-Rs to be
harmonic. We also investigate the ns conditions for such submersions to be
completely geodesic.

##  2 Preliminaries

In this segment, we recall some definitions which will be required to study
throughout the paper.
Let $(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ be an odd dimensional
acm manifold [8] consisting of a $(1,1)$-tensor field $\varphi_{\ast}$, a
vector field $\zeta$, a $1$-form $\eta_{\ast}$ and a Riemannian metric
$g_{m}$. Then

$\varphi_{\ast}^{2}P=-P+\eta_{\ast}(P)\zeta,$ (2.1) $\varphi_{\ast}\,\zeta=0,\
\eta_{\ast}\circ\varphi_{\ast}=0,\ \eta_{\ast}(\zeta)=1,\
g_{m}(P,\zeta)=\eta_{\ast}(P),$ (2.2)

and

$g_{m}(\varphi_{\ast}P,\varphi_{\ast}Q)=g_{m}(P,Q)-\eta_{\ast}(P)\eta_{\ast}(Q),\quad
g_{m}(\varphi_{\ast}P,Q)=-g_{m}(P,\varphi_{\ast}Q),$ (2.3)

for any vector fields $P,Q\in\Gamma(TM^{\ast})$.

An acm manifold $(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ is called
_Sasakian_[32]__ if

$(\nabla_{P}\varphi_{\ast})Q=g_{m}(P,Q)\zeta-\eta_{\ast}(Q)P$ (2.4)

and

$\nabla_{P}\zeta=-\varphi_{\ast}P$ (2.5)

for any vector fields $P,Q\in\Gamma(TM^{\ast})$, where $\nabla$ is the Levi-
Civita connection.

Now, we give some useful definitions of submersions to extend our definition
of semi-invariant conformal Riemannian submersions endowed with the horizontal
Reeb vector field.

###### Definition 2.1

[25] Let $(M^{\ast},g_{m})$ and $(N^{\ast},g_{n})$ be Riemannian manifolds,
where $\dim(M^{\ast})=m$, $\dim(N^{\ast})=n$ and $m>n$. A Riemannian
submersion $\Psi:M^{\ast}\rightarrow N^{\ast}$ is a map of $M^{\ast}$ onto
$N^{\ast}$ satisfying the following conditions:

(i)

$\Psi$ has maximal rank.

(ii)

The metric of horizontal vectors is preserved by the differential
$\Psi_{\ast}$.

For each $z\in N^{\ast}$, $\Psi^{-1}(z)$ is an $(m-n)$-dimensional submanifold
of $M^{\ast}$. The submanifolds $\Psi^{-1}(z)$, $z\in N^{\ast}$, are called
_fibers_. A vector field on $M^{\ast}$ is called vertical and horizontal if it
is always tangent to fibers and orthogonal to fibers, respectively. A vector
field $P$ on $M$ is called basic if $P$ is horizontal and $\Psi$-related to a
vector field $P^{\prime}$ on $N$, that is,
$\Psi_{\ast}P_{x}=P_{\Psi_{\ast}(x)}^{\prime}$ for all $x\in M.$ The
projection morphisms on the distributions $\ker\Psi_{\ast}$ and
$(\ker\Psi_{\ast})^{\perp}$ are denoted by ${\mathcal{V}_{1}}$ and
${\mathcal{H}_{1}}$, respectively. The sections of ${\mathcal{V}_{1}}$ and
${\mathcal{H}_{1}}$ are called the vertical vector fields and horizontal
vector fields, respectively. So

${\mathcal{V}_{1}}_{p}=T_{p}\left(\Psi^{-1}(q)\right),\qquad{\mathcal{H}_{1}}_{p}=T_{p}\left(\Psi^{-1}(q)\right)^{\perp}.$

###### Definition 2.2

[26] Let $(M^{\ast},g_{m})$ and $(N^{\ast},g_{n})$ be Riemannian manifolds,
where $\dim(M^{\ast})=m$, $\dim(N^{\ast})=n$ and $m>n$. A (horizontally)
conformal submersion $\Psi:M^{\ast}\rightarrow N^{\ast}$ is a map of
$M^{\ast}$ onto $N^{\ast}$ satisfying the following axioms:

(i)

$\Psi$ has maximal rank.

(ii)

The angle between the horizontal vectors is preserved by the differential
$\Psi_{\ast}$,

$g_{n}(\Psi_{\ast}U,\Psi_{\ast}V)=\Lambda_{\ast}(p)g_{m}(U,V),\quad
U,V\in{\mathcal{H}_{1}}_{p_{\ast}}$

where $\Lambda_{\ast}(p)$ is a non-zero number and $p\in M^{\ast}$. The number
$\Lambda_{\ast}(p)$ is called the square dilation of $\Psi$ at $p$, it is
necessarily non-negative. Its square root
$\lambda_{\ast}(p)=\sqrt{\Lambda_{\ast}(p)}$ is called the dilation of $\Psi$
at $p$.

Clearly, Riemannian submersion is a horizontally conformal submersion with
$\lambda_{\ast}=1$.

###### Definition 2.3

[6] Let $\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ be a conformal
submersion. A vector field $E_{1}$ on $M^{\ast}$ is called projectiable if
there exist a vector field $\widehat{E_{1}}$ on $N^{\ast}$ such that
$\Psi_{\ast}({E_{1}}_{p})=\widehat{E_{1}}_{\Psi_{\ast}(p)}$ for any $p\in
M^{\ast}.$ In this case, $E_{1}$ and $\widehat{E_{1}}$ are called
$\Psi$-related. A horizontal vector field $V$ on $M^{\ast}$ is called basic,
if it is projectiable.

It is to be noted that if $\widehat{W}$ is a vector field on $N^{\ast},$ then
there exists a unique basic vector field $W$ on $M^{\ast}$ which is called the
horizontal lift of $\widehat{W}.$

###### Definition 2.4

[6] A horizontally conformal submersion
$\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ is called horizontally
homothetic if the slope of its dilation $\lambda_{\ast}$ is vertical, that is,

${\mathcal{H}_{1}}(grad\lambda_{\ast})=0$

at $p\in M^{\ast}$, where $\mathcal{H}{{}_{1}}$ is the projection on the
horizontal space $\left(\ker\Psi_{\ast}\right)^{\perp}$.

The second fundamental tensors of all fibers $\Psi^{-1}(q),\ q\in N^{\ast}$
give rise to tensor field $T$ and $A$ on $M^{\ast}$ defined by O’Neill [25]
for arbitrary vector field $E$ and $F$, which is

$T_{E}F={\mathcal{H}_{1}}\nabla_{{\mathcal{V}_{1}}E}^{M}{\mathcal{V}_{1}}F+{\mathcal{V}_{1}}\nabla_{{\mathcal{V}_{1}}E}^{M}{\mathcal{H}_{1}}F,$
(2.6)
$A_{E}F={\mathcal{H}_{1}}\nabla_{{\mathcal{H}_{1}}E}^{M}{\mathcal{V}_{1}}F+{\mathcal{V}_{1}}\nabla_{{\mathcal{H}_{1}}E}^{M}{\mathcal{H}_{1}}F,$
(2.7)

where ${\mathcal{V}_{1}}$ and ${\mathcal{H}_{1}}$ are the vertical and
horizontal projections, respectively.

From equations (2.6) and (2.7), we obtain the following equations

$\nabla_{V}W=T_{V}W+\widehat{\nabla}_{V}W,$ (2.8)
$\nabla_{V}P={\mathcal{H}_{1}}\nabla_{V}P+T_{V}P,$ (2.9)
$\nabla_{P}V=A_{P}V+{\mathcal{V}_{1}}\nabla_{P}V,$ (2.10)
$\nabla_{P}Q={\mathcal{H}_{1}}\nabla_{P}Q+A_{P}Q,$ (2.11)

for all $V,W\in\Gamma(\ker\Psi_{\ast})$ and
$P,Q\in\Gamma(\ker\Psi_{\ast})^{\perp},$ where
${\mathcal{V}_{1}}\nabla_{V}W=\widehat{\nabla}_{V}W.$ If $P$ is the basic
vector field, then $A_{P}Q={\mathcal{H}_{1}}\nabla_{Q}P.$

Clearly, for $p\in M,$ $U\in{\mathcal{V}_{1}}_{p}$ and
$P\in{\mathcal{H}_{1}}_{p}$, the linear operators

${T}_{U},{A}_{P}:T_{p}M^{\ast}\rightarrow T_{p}M^{\ast}$

are skew-symmetric, that is,

$g_{m}({A}_{P}E,F)=-g_{m}(E,{A}_{P}F){\rm
and}g_{m}({T}_{U}E,F)=-g_{m}(E,{T}_{U}F),$ (2.12)

for all $E,F\in$ $T_{p}M^{\ast}.$ It is clear, the restriction of ${T}$ to the
vertical distribution ${T}|_{\ker\Psi_{\ast}\times\ker\Psi_{\ast}}$ is
certainly the second fundamental form of the fibres of $\Psi$. From (2.12), it
is easy to say that $\Psi$ has totally geodesic fibres if and only if
${T}\equiv 0$.

Consider the smooth map $\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$
between Riemannian manifolds. Then the differential $\Psi_{\ast}$ of $\Psi$
can be noticed as a section of the bundle
$Hom(TM^{\ast},\Psi^{-1}TN^{\ast})\rightarrow M^{\ast}$, where
$\Psi^{-1}TN^{\ast}$ is the bundle which has fibres
$\left(\Psi^{-1}TN^{\ast}\right)_{x}=T_{f(x)}N^{\ast}$ , $x\in M^{\ast}$.
$Hom(TM^{\ast},\Psi^{-1}TN^{\ast})$ has a connection $\nabla$ induced from the
Riemannian connection $\nabla^{M^{\ast}}$ and the pullback connection. The
second fundamental form of $\Psi$ is

$(\nabla\Psi_{\ast})(E,F)=\nabla_{E}^{N^{\ast}}\Psi_{\ast}F-\Psi_{\ast}(\nabla_{E}^{M^{\ast}}F),{\rm
forall}E,F\in\Gamma(TM^{\ast}),$ (2.13)

where $\nabla^{N^{\ast}}$ is the pullback connection ([6, 7]). The map $\Psi$
is said to be totally geodesic [6] if $(\nabla\Psi_{\ast})(E,F)=0,$ for all
$E,F\in\Gamma(TM^{\ast})$.

###### Lemma 2.5

[36, Lemma 1.16, pp.129] Let
$\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ be a smooth map between
Riemannian manifolds $(M^{\ast},g_{m})$ and $(N^{\ast},g_{n})$. Then

$\Psi_{\ast}([P,Q])=\nabla_{P}^{N^{\ast}}\Psi_{\ast}Q-\nabla_{Q}^{N^{\ast}}\Psi_{\ast}P,$
(2.14)

for all $P,Q\in\Gamma(TM^{\ast})$.

From Lemma 2.5, we come to the conclusion

$[P,Q]\in\Gamma(ker\Psi_{\ast})$ (2.15)

for $P\in\Gamma(ker\Psi_{\ast})^{\perp}$ and $Q\in\Gamma(ker\Psi_{\ast})$.

A smooth map $\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ is said to be
harmonic [6] if and only if $\mathrm{trace}(\nabla\Psi_{\ast})=0$. The tension
field of $\Psi$ is the section $\tau(\Psi)$ of $\Gamma(\Psi^{-1}TN)$ and
defined by

$\tau(\Psi)={div}\Psi_{\ast}=\sum_{i=1}^{m}(\nabla\Psi_{\ast})(e_{i},e_{i}),$
(2.16)

where $\\{e_{1},e_{2},......,e_{m}\\}$ is the orthonormal basis on $M^{\ast}$.
Then it follows that ns condition for $\Psi$ to be harmonic is $\tau(\Psi)=0$
[6].

Now, we recall an important Lemma from [6], which will be needed in the study
of whole paper.

###### Lemma 2.6

Let $\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ be a horizontal
conformal submersion, $P,Q$ are horizontal vector fields and $V,W$ are
vertical vector fields. Then

$(a)$
$(\nabla\Psi_{\ast})(P,Q)=P(\ln\lambda_{\ast})\Psi_{\ast}(Q)+Q(\ln\lambda_{\ast})\Psi_{\ast}(P)-g_{m}(P,Q)\Psi_{\ast}({grad}\
ln\lambda_{\ast}),$

$(b)$ $(\nabla\Psi_{\ast})(V,W)=-\Psi_{\ast}({\mathcal{T}}_{V}W),$

$(c)$
$(\nabla\Psi_{\ast})(P,V)=-\Psi_{\ast}(\nabla_{P}^{M}V)=-\Psi_{\ast}(A_{P}V).$

##  3  Semi-invariant conformal $\zeta^{\perp}$-Riemannian submersions

In this segment, we deal with the definition and examples of sic
$\zeta^{\perp}$-Rs from acm manifolds onto Riemannian manifolds. We acquire
the integrability of distributions and also geometry of leaves of
$ker\Psi_{\ast}$ and $(ker\Psi_{\ast})^{\perp}$.

###### Definition 3.1

Let $\left(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m}\right)$ be an acm
manifold and $\left(N^{\ast},g_{n}\right)$ be a Riemannian manifold. A
horizontally conformal submersion
$\Psi:(M^{\ast},g_{m})\rightarrow(N^{\ast},g_{n})$ with slope
$\lambda_{\ast}$, is said to be a sic $\zeta^{\perp}$-Rs if $\zeta$ is normal
to $ker\Psi_{\ast}$ and there is a distribution $D^{\ast}_{1}\subseteq
ker\Psi_{{\ast}}$ such that

$ker\Psi_{\ast}=D^{\ast}_{1}\oplus D^{\ast}_{2},$ (3.1)
$\varphi_{\ast}(D^{\ast}_{1})=D^{\ast}_{1},\
\varphi_{\ast}(D^{\ast}_{2})\subseteq(ker\Psi_{\ast})^{\perp},$ (3.2)

where $D^{\ast}_{2}$ is orthogonal complementary to $D^{\ast}_{1}$ in
$ker\Psi_{{\ast}}$.

It is to be noted that the distribution $ker\Psi_{\ast}$ is integrable.

Further to prove the consistency of sic $\zeta^{\perp}$-Rs in acm manifolds,
we are putting some examples here.

###### Example 3.2

Every anti-invariant $\zeta^{\perp}$-Riemannian submersion from an acm
manifold onto a Riemannian manifold is a sic $\zeta^{\perp}$-Rs with
$\lambda_{\ast}=I$ and $D^{\ast}_{1}=\\{0\\}$, where $I$ denotes the identity
function.

###### Example 3.3

Every semi-invariant $\zeta^{\perp}$ submersion from an acm manifold onto a
Riemannian manifold is a sic $\zeta^{\perp}$-Rs with $\lambda_{\ast}=I.$

###### Example 3.4

[8] Let $(\mathbb{R}^{m},g,\varphi_{\ast},\zeta,\eta_{\ast})$, $(m=2n+1)$ be a
Sasakian manifold given by

$\eta_{\ast}=\frac{1}{2}\left(dw-\sum\limits_{i=1}^{n}v^{i}du^{i}\right),\quad\zeta=2\frac{\partial}{\partial
w},$
$g=\frac{1}{4}\eta_{\ast}\otimes\eta_{\ast}+\sum\limits_{i=1}^{n}(du^{i}\otimes
du^{i}+dv^{i}\otimes dv^{i}),$
$\varphi_{\ast}=\left(\begin{array}[]{ccc}0&\delta_{ij}&0\\\
-\delta_{ij}&0&0\\\ 0&v^{j}&0\end{array}\right),$

where $\\{(u^{i},v^{i},w)|i=1,\ldots,n\\}$ are the cartesian coordinates. The
vector fields $X_{i}=2\frac{\partial}{\partial v^{i}},\quad
X_{n+i}=2\left(\frac{\partial}{\partial u^{i}}+v^{i}\frac{\partial}{\partial
w}\right)$ and $\zeta$ form a $\varphi_{\ast}$-basis for the contact metric
structure.

Let
$\Psi:\left(\mathbb{R}^{7},g_{1}\right)\rightarrow\left(\mathbb{R}^{2},g_{2}\right)$
be a Riemannian submersion defined by

$\Psi(u_{1},u_{2},u_{3},v_{1},v_{2},v_{3},w)=(\sinh v_{3}\cos w,\cosh
v_{3}\sin w).$

Then it follows that

$ker\Psi_{\ast}=span\\{L_{1}=\partial u_{1},L_{2}=\partial
u_{2},L_{3}=\partial u_{3},L_{4}=\partial v_{1},L_{5}=\partial v_{2}\\}$

and

$(ker\Psi_{\ast})^{\bot}=span\left\\{W_{1}=\partial v_{3},W_{2}=\partial
w\right\\}.$

Hence, we have
$\varphi_{\ast}L_{1}=-L_{4},\varphi_{\ast}L_{2}=-L_{5},\varphi_{\ast}L_{3}=-W_{1}$,
$\varphi_{\ast}L_{4}=L_{1}$ and $\varphi_{\ast}L_{5}=L_{2}.$ Thus it follows
that $D_{1}^{\ast}=span\\{L_{1},L_{2},L_{4},L_{5}\\}$,
$D_{2}^{\ast}=span\\{L_{3}\\}$ and $\zeta\in(ker\Psi_{\ast})^{\bot}$. We can
easily compute that

$\displaystyle g_{1}\left(\Psi_{\ast}W_{1},\Psi_{\ast}W_{1}\right)$
$\displaystyle=$
$\displaystyle\left(\sinh^{2}v_{3}\sin^{2}w+\cosh^{2}v_{3}\cos^{2}w\right)g_{2}\left(W_{1},W_{1}\right),$
$\displaystyle g_{1}\left(\Psi_{\ast}W_{2},\Psi_{\ast}W_{2}\right)$
$\displaystyle=$
$\displaystyle\left(\sinh^{2}v_{3}\sin^{2}w+\cosh^{2}v_{3}\cos^{2}w\right)g_{2}\left(W_{2},W_{2}\right).$

Therefore, $\Psi$ is a sic $\zeta^{\perp}$-Rs with
$\lambda_{\ast}=\sqrt{\left(\sinh^{2}v_{3}\sin^{2}w+\cosh^{2}v_{3}\cos^{2}w\right)}.$

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Consider

$\left(\ker\Psi_{\ast}\right)^{\bot}=\varphi_{\ast}D^{\ast}_{2}\oplus\mu,$

where $\mu$ is the complementary distribution to $\varphi_{\ast}D^{\ast}_{2}$
in $(ker\Psi_{\ast})^{\bot}$ and $\varphi_{\ast}\mu\subset\mu$. Therefore
$\zeta\in\mu$.

For $V\in\Gamma(ker\Psi_{\ast})$, we write

$\varphi_{\ast}V=\phi V+\omega V,$ (3.3)

where $\phi V\in\Gamma(D^{\ast}_{1})$ and $\omega
V\in\Gamma(\varphi_{\ast}D^{\ast}_{2})$.

For $P\in\Gamma(ker\Psi_{\ast})^{\perp}$, we have

$\varphi_{\ast}P=\alpha P+\beta P,$ (3.4)

where $\alpha P\in\Gamma(D^{\ast}_{2})$ and $\beta P\in\Gamma(\mu)$.

Now, using (2.8), (2.9), (3.3) and (3.4), we get

$(\nabla_{V}^{M}\phi)W=\alpha T_{V}W-T_{V}\omega W,$ (3.5)
$(\nabla_{V}^{M}\omega)W=\beta T_{V}W-T_{V}\phi W$ (3.6)

for $V,W\in\Gamma(ker\Psi_{\ast})$, where

$(\nabla_{V}^{M}\phi)W=\hat{\nabla}_{V}\phi W-\phi\hat{\nabla}_{V}W$ (3.7)

and

$(\nabla_{V}^{M}\omega)W={\mathcal{H}_{1}}\nabla_{V}^{M}\omega
W-\omega\hat{\nabla}_{V}W.$ (3.8)

Now, we investigate the integrability of the distributions $D^{\ast}_{1}$ and
$D^{\ast}_{2}$.

###### Lemma 3.5

Let $\Psi$ be sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns condition for the distribution $D^{\ast}_{1}$
to be integrable is

$(\nabla\Psi_{\ast})(V,\varphi_{\ast}U)-(\nabla\Psi_{\ast})(U,\varphi_{\ast}V)\in\Gamma(\Psi_{\ast}(\mu))$

for $U,V\in\Gamma(D^{\ast}_{1})$.

Proof. Let $U,V\in\Gamma(D_{1}^{\ast})$ and $W\in\Gamma(D_{2}^{\ast})$. With
the help of (2.2), we get

$g_{m}(\nabla_{U}\varphi_{\ast}V,\varphi_{\ast}W)=g_{m}(\nabla_{U}V,W).$ (3.9)

So

$g_{m}([U,V],W)=g_{m}(\nabla_{U}V,W)-g_{m}(\nabla_{V}U,W).$

Using (2.8) and (3.9), we obtain

$\displaystyle g_{m}([U,V],W)$ $\displaystyle=$ $\displaystyle
g_{m}(\nabla_{U}\varphi_{\ast}V,\varphi_{\ast}W)-g_{m}(\nabla_{V}\varphi_{\ast}U,\varphi_{\ast}W)$
$\displaystyle=$ $\displaystyle
g_{m}(T_{U}\varphi_{\ast}V-T_{V}\varphi_{\ast}U,\varphi_{\ast}W).$

The distribution $D_{1}^{\ast}$ is integrable if and only if
$g_{m}([U,V],W)=g_{m}([U,V],Z)=0$ for $U,V\in\Gamma(D_{1}^{\ast})$,
$W\in\Gamma(D_{2}^{\ast})$ and $Z\in\Gamma((ker\Psi_{\ast})^{\perp})$. Since
$ker\Psi_{\ast}$ is integrable, so we immediately have $g_{m}([U,V],Z)=0$.
Thus the ns condition for the distribution $D_{1}^{\ast}$ to be integrable is
$g_{m}([U,V],W)=0$. Since $\Psi$ is a conformal submersion, in view of Lemma
2.6 $(b)$, we conclude that

$g_{m}([U,V],W)=\frac{1}{\lambda_{\ast}^{2}}g_{n}((\nabla\Psi_{\ast})(V,\varphi_{\ast}U)-(\nabla\Psi_{\ast})(U,\varphi_{\ast}V),\Psi_{\ast}\varphi_{\ast}W).$

###### Lemma 3.6

Let $\Psi$ be sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns condition for the integrability of
distribution $D^{\ast}_{2}$ is

$\hat{\nabla}_{W}\phi\xi+T_{W}\omega\xi-\hat{\nabla}_{\xi}\phi W-T_{\xi}\omega
W\in\Gamma(D^{\ast}_{2})$

for $W,\xi\in\Gamma(D^{\ast}_{2})$.

Proof. Let $U\in\Gamma(D^{\ast}_{1})$ and $W,\xi\in\Gamma(D^{\ast}_{2})$.
Using (2.2), (2.3), (2.6), (2.8), (2.9) and (3.9), we get

$\displaystyle g_{m}(\varphi_{\ast}[W,\xi],U)$ $\displaystyle=$ $\displaystyle
g_{m}(\nabla_{W}\varphi_{\ast}\xi,U)-g_{m}(\nabla_{\xi}\varphi_{\ast}W,U)$
$\displaystyle=$ $\displaystyle
g_{m}(T_{W}\phi\xi+\hat{\nabla}_{W}\phi\xi+T_{W}\omega\xi+{\mathcal{H}_{1}}\nabla_{W}\omega\xi$
$\displaystyle-T_{\xi}\phi W-\hat{\nabla}_{\xi}\phi W-T_{\xi}\omega
W-{\mathcal{H}_{1}}\nabla_{\xi}\omega W,U)$ $\displaystyle=$ $\displaystyle
g_{m}(\hat{\nabla}_{W}\phi\xi+T_{W}\omega\xi-\hat{\nabla}_{\xi}\phi
W-T_{\xi}\omega W,U).$

###### Theorem 3.7

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the distribution
$(ker\Psi_{\ast})^{\perp}$ to be integrable are

$A_{Q}\omega\alpha P-A_{P}\omega\alpha Q-\varphi_{\ast}A_{P}\beta
Q+\varphi_{\ast}A_{Q}\beta P\notin\Gamma(D^{\ast}_{1})$

and

$\displaystyle\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{V}\Psi_{\ast}CU-\nabla_{U}\Psi_{\ast}CV,\Psi_{\ast}\varphi_{\ast}Z)$
$\displaystyle=g_{m}(A_{V}BU-A_{U}BV-CV(ln\lambda_{\ast})U$
$\displaystyle+CU(ln\lambda_{\ast})V+2g_{m}(U,CV)grad\ ln\lambda_{\ast}$
$\displaystyle+\eta_{\ast}(U)V-\eta_{\ast}(V)U,\varphi_{\ast}Z)$

for $U,V\in\Gamma(ker\Psi_{\ast})^{\perp})$ and $Z\in\Gamma(D^{\ast}_{2})$.

Proof. The ns condition for the distribution $(ker\Psi_{\ast})^{\perp}$ to be
integrable on manifold $M^{\ast}$ is that

$g_{m}([P,Q],U)=0\ \ {\rm and}\ \ g_{m}([P,Q],W)=0,$

for $P,Q\in\Gamma((ker\Psi_{\ast})^{\perp})$, $U\in\Gamma(D^{\ast}_{1})$ and
$W\in\Gamma(D^{\ast}_{2})$.

Using (2.1), (2.2), (2.4) and (3.4), we infer

$\displaystyle g_{m}([P,Q],U)$
$\displaystyle=g_{m}(\varphi_{\ast}[P,Q],\varphi_{\ast}U)$
$\displaystyle=g_{m}(\varphi_{\ast}\nabla_{P}Q-\varphi_{\ast}\nabla_{Q}P,\varphi_{\ast}U)$
$\displaystyle=g_{m}(\nabla_{P}\alpha Q,\varphi_{\ast}U)+g_{m}(\nabla_{P}\beta
Q,\varphi_{\ast}U)$ $\displaystyle-g_{m}(\nabla_{Q}\alpha
P,\varphi_{\ast}U)-g_{m}(\nabla_{Q}\beta P,\varphi_{\ast}U)$
$\displaystyle=g_{m}(\varphi_{\ast}\nabla_{P}\alpha
Q,U)-g_{m}(\varphi_{\ast}\nabla_{P}\beta Q,U)$
$\displaystyle+g_{m}(\varphi_{\ast}\nabla_{Q}\alpha
P,U)+g_{m}(\varphi_{\ast}\nabla_{Q}\beta P,U).$

By simple calculations, we have

$\displaystyle g_{m}([P,Q],U)$
$\displaystyle=-g_{m}(\nabla_{P}\varphi_{\ast}\alpha
Q,U)+g_{m}(\nabla_{P}\beta Q,\varphi_{\ast}U)$
$\displaystyle+g_{m}(\nabla_{Q}\varphi_{\ast}\alpha P,U)-g_{m}(\nabla_{Q}\beta
P,\varphi_{\ast}U).$

By (3.3), we have $\varphi_{\ast}\alpha P=\phi\alpha P+\omega\alpha P$. Since
$\varphi_{\ast}\alpha P$, $\omega\alpha P\in\Gamma((ker\Psi_{\ast})^{\perp})$,
so $\phi\alpha P=0$, $\forall P\in\Gamma((ker\Psi_{\ast})^{\perp})$. Using
(2.4), (2.11) and (3.3), we get

$g_{m}([P,Q],U)=g_{m}(A_{Q}\omega\alpha P-A_{P}\omega\alpha
Q-\varphi_{\ast}A_{P}\beta Q+\varphi_{\ast}A_{Q}\beta P,U).$ (3.10)

With the help of (2.1), (2.2), (2.4) and (3.4), we deduce

$\displaystyle g_{m}([P,Q],W)$ $\displaystyle=g_{m}(\nabla_{P}\alpha
Q,\varphi_{\ast}W)+g_{m}(\nabla_{P}\beta Q,\varphi_{\ast}W)$ $\displaystyle-
g_{m}(\nabla_{Q}\alpha P,\varphi_{\ast}W)-g_{m}(\nabla_{Q}\beta
P,\varphi_{\ast}W)$
$\displaystyle+\eta_{\ast}(P)g_{m}(Q,\varphi_{\ast}W)-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W)$

Since $\Psi$ is sic $\zeta^{\perp}$-Rs, using (2.10), (2.11), (2.13) and Lemma
2.5, 2.6, we obtain

$\displaystyle g_{m}([P,Q],W)$
$\displaystyle=-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W)+\eta_{\ast}(P)g_{m}(Q,\varphi_{\ast}W)$
$\displaystyle-\frac{1}{\lambda_{\ast}^{2}}g_{n}((\nabla\Psi_{\ast})(P,\alpha
Q),\Psi_{\ast}\varphi_{\ast}W)+\frac{1}{\lambda_{\ast}^{2}}g_{n}((\nabla\Psi_{\ast})(Q,\alpha
P),\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}\\{-P(ln\lambda_{\ast})\Psi_{\ast}\beta
Q-\beta Q(ln\lambda_{\ast})\Psi_{\ast}P$ $\displaystyle+g_{m}(P,\beta
Q)\Psi_{\ast}(grad\ ln\lambda_{\ast})+\nabla_{P}^{\Psi}\Psi_{\ast}\beta
Q,\Psi_{\ast}\varphi_{\ast}W\\}$
$\displaystyle-\frac{1}{\lambda_{\ast}^{2}}g_{n}\\{-Q(ln\lambda_{\ast})\Psi_{\ast}\beta
P-\beta P(ln\lambda_{\ast})\Psi_{\ast}Q$ $\displaystyle+g_{m}(Q,\beta
P)\Psi_{\ast}(grad\ ln\lambda_{\ast})+\nabla_{Q}^{\Psi}\Psi_{\ast}\beta
P,\Psi_{\ast}\varphi_{\ast}W\\},$ $\displaystyle g_{m}([P,Q],W)$
$\displaystyle=-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W)+\eta_{\ast}(P)g_{m}(Q,\varphi_{\ast}W)-g_{m}(A_{P}\alpha
Q,\varphi_{\ast}W)$ $\displaystyle+g_{m}(A_{Q}\alpha
P,\varphi_{\ast}W)-\frac{1}{\lambda_{\ast}^{2}}\\{g_{m}(grad\
ln\lambda_{\ast},P)g_{n}(\Psi_{\ast}\beta Q,\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle+g_{m}(grad\ ln\lambda_{\ast},\beta
Q)g_{n}(\Psi_{\ast}P,\Psi_{\ast}\varphi_{\ast}W)$ $\displaystyle-g_{m}(P,\beta
Q)g_{n}(\Psi_{\ast}(grad\ ln\lambda_{\ast}),\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle-g_{m}(grad\ ln\lambda_{\ast},Q)g_{n}(\Psi_{\ast}\beta
P,\Psi_{\ast}\varphi_{\ast}W)$ $\displaystyle-g_{m}(grad\
ln\lambda_{\ast},\beta P)g_{n}(\Psi_{\ast}Q,\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle+g_{m}(Q,\beta P)g_{n}(\Psi_{\ast}(grad\
ln\lambda_{\ast}),\Psi_{\ast}\varphi_{\ast}W)$ $\displaystyle-
g_{n}(\nabla_{P}^{\Psi}\Psi_{\ast}\beta
Q,\Psi_{\ast}\varphi_{\ast}W)+g_{n}(\nabla_{Q}^{\Psi}\Psi_{\ast}\beta
P,\Psi_{\ast}\varphi_{\ast}W)\\}.$

Using definition 3.1, we have

$\displaystyle g_{m}([P,Q],W)$ $\displaystyle=$
$\displaystyle-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W)+\eta_{\ast}(P)g_{m}(Q,\varphi_{\ast}W)$
(3.11) $\displaystyle+g_{m}(A_{Q}\alpha P-A_{P}\alpha Q-\left(\beta
Q\right)(P\ln\lambda_{\ast})$ $\displaystyle+\left(\beta
P\right)(Q\ln\lambda_{\ast})+2g_{m}(P,\beta Q)grad\
ln\lambda_{\ast},\varphi_{\ast}W)$ $\displaystyle+g_{m}((\beta
Pln\lambda_{\ast})Q-(\beta Qln\lambda_{\ast})P,\varphi_{\ast}W)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{P}^{\Psi}\Psi_{\ast}\beta
Q-\nabla_{Q}^{\Psi}\Psi_{\ast}\beta P,\Psi_{\ast}\varphi_{\ast}W).$

From the above expression, we entail that

$\displaystyle\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{Q}^{\Psi}\Psi_{\ast}\beta
P-\nabla_{P}^{\Psi}\Psi_{\ast}\beta Q,\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle=g_{m}(A_{Q}\alpha P-A_{P}\alpha Q-\left(\beta
Q\right)(P\ln\lambda_{\ast})$ (3.12) $\displaystyle+\left(\beta
P\right)(Q\ln\lambda_{\ast})+2g_{m}(P,\beta Q)grad\ ln\lambda_{\ast}$
$\displaystyle+(\beta Pln\lambda_{\ast})Q-(\beta
Qln\lambda_{\ast})P+\eta_{\ast}(P)Q$
$\displaystyle-\eta_{\ast}(Q)P,\varphi_{\ast}W).$

So from (3.10) and (3.12), we obtain the results.
Now, we find out the required necessary and sufficient conditions for sic
$\zeta^{\perp}$-Rs to be homothetic map.

###### Theorem 3.8

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$ with integrable distribution $(ker\Psi_{\ast})^{\perp}$.
Then the ns condition for the $\Psi$ to be homothetic map is

$\displaystyle g_{n}(\nabla_{Q}^{\Psi}\Psi_{\ast}\beta
P-\nabla_{P}^{\Psi}\Psi_{\ast}\beta Q,\Psi_{\ast}\varphi_{\ast}W)$
$\displaystyle=\lambda_{\ast}^{2}g_{m}(A_{Q}\alpha P-A_{P}\alpha Q$ (3.13)
$\displaystyle+\eta_{\ast}(P)Q-\eta_{\ast}(Q)P,\varphi_{\ast}W)$

for $P,Q\in\Gamma((ker\Psi_{\ast})^{\perp})$ and $W\in\Gamma(D^{\ast}_{2})$.

Proof. Let $P,Q\in\Gamma((ker\Psi_{\ast})^{\perp})$ and
$W\in\Gamma(D^{\ast}_{2})$. Using (3.12), we infer

$\displaystyle g_{m}([P,Q],W)$
$\displaystyle=-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W)+\eta_{\ast}(P)g_{m}(Q,\varphi_{\ast}W)$
(3.14) $\displaystyle+g_{m}(A_{Q}\alpha P-A_{P}\alpha Q-\left(\beta
Q\right)(P\ln\lambda_{\ast})$ $\displaystyle+\left(\beta
P\right)(Q\ln\lambda_{\ast})+2g_{m}(P,\beta Q)grad\
ln\lambda_{\ast},\varphi_{\ast}W)$ $\displaystyle+g_{m}((\beta
Pln\lambda_{\ast})Q-(\beta Qln\lambda_{\ast})P,\varphi_{\ast}W)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{P}^{\Psi}\Psi_{\ast}\beta
Q-\nabla_{Q}^{\Psi}\Psi_{\ast}\beta P,\Psi_{\ast}\varphi_{\ast}W).$

If $\Psi$ is a parallel homothetic map, therefore we acquire (3.13).
Conversely, if (3.13) holds, then we gain

$\displaystyle 0$ $\displaystyle=g_{m}((\beta Pln\lambda_{\ast})Q-(\beta
Qln\lambda_{\ast})P,\varphi_{\ast}W)$ (3.15) $\displaystyle+g_{m}(-\left(\beta
Q\right)(P\ln\lambda_{\ast})+\left(\beta
P\right)(Q\ln\lambda_{\ast})+2g_{m}(P,\beta Q)grad\
ln\lambda_{\ast},\varphi_{\ast}W).$

Replacing $Q$ by $\varphi_{\ast}W$ for $W\in\Gamma(D^{\ast}_{2})$ in (3.15),
we entail that

$g_{m}({grad}\ln\lambda_{\ast},\beta
P)g_{m}(\varphi_{\ast}W,\varphi_{\ast}W)=0,$

which implies that $\lambda_{\ast}$ is a constant on $\Gamma(\mu)$ since
$||\varphi_{\ast}W||\neq 0$.
Now, again replace $Q$ by $\beta P$ for $P\in\Gamma(\mu)$ in (3.15), we have

$g_{m}(grad\ ln\lambda_{\ast},\varphi_{\ast}W)g_{m}(P,\beta^{2}P)=g_{m}(grad\
ln\lambda_{\ast},\varphi_{\ast}W)g_{m}(\beta P,\beta P)=0.$

Now, we see that the $||\beta P||\neq 0$, this concludes $\lambda_{\ast}$ is a
constant on $\Gamma(\varphi_{\ast}D^{\ast}_{2})$, which finalize the proof of
our theorem.

Now, we come to the following results for the geometry of leaves of horizontal
distributions.

###### Theorem 3.9

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the $(ker\Psi_{\ast}^{\perp})$
to be completely geodesic foliation on $M^{\ast}$ are

$A_{P}\varphi_{\ast}Q+{\mathcal{V}_{1}}\nabla_{P}^{m}\alpha
Q\in\Gamma(D^{\ast}_{2}),$

and

$\displaystyle g_{n}(\nabla_{P}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta
Q)=$ $\displaystyle\lambda_{\ast}^{2}g_{m}(-A_{P}\alpha Q+\beta
Q(ln\lambda_{\ast})P-g_{m}(P,\beta Q)({grad}\ln\lambda_{\ast})$
$\displaystyle-P(ln\lambda_{\ast})\beta Q+\eta_{\ast}(Q)P,\varphi_{\ast}W)$

for $P,Q\in\Gamma(ker\Psi_{\ast})^{\perp}$, $U\in\Gamma(D^{\ast}_{1})$ and
$W\in\Gamma(D^{\ast}_{2})$.

Proof. Here, it is to be noted that the distribution
$(ker\Psi_{\ast})^{\perp}$ characterizes a completely geodesic foliation on
$M$ if and only if $g_{m}(\nabla_{P}^{m}Q,U)=0$ and
$g_{m}(\nabla_{P}^{m}Q,W)=0$ for $P,Q\in\Gamma(ker\Psi_{\ast})^{\perp}$,
$U\in\Gamma(D^{\ast}_{1})$ and $W\in\Gamma(D^{\ast}_{2})$. Then with the help
of (2.1),(2.2), (2.4), (2.10), (2.11) and (3.4), we infer

$\displaystyle g_{m}(\nabla_{P}^{m}Q,U)$
$\displaystyle=g_{m}(\nabla_{P}^{m}\varphi_{\ast}Q,\varphi_{\ast}U)-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}U)$
$\displaystyle=-g_{m}(\varphi_{\ast}\left(A_{P}\varphi_{\ast}Q+{\mathcal{V}_{1}}\nabla_{P}^{m}\alpha
Q\right),U),$

which implies that $A_{P}\varphi_{\ast}Q+{\mathcal{V}_{1}}\nabla_{P}^{m}\alpha
Q\in\Gamma(D^{\ast}_{2})$.
Using (2.1),(2.2), (2.4) and (3.4), we observe that

$\displaystyle g_{m}(\nabla_{P}^{m}Q,W)$
$\displaystyle=g_{m}(\nabla_{P}^{m}\alpha Q+\nabla_{P}^{m}\beta
Q,\varphi_{\ast}W)$ $\displaystyle-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W),$
$\displaystyle=-g_{m}(\alpha Q,\nabla_{P}^{m}\varphi_{\ast}W)-g_{m}(\beta
Q,\nabla_{P}^{m}\varphi_{\ast}W)$
$\displaystyle-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W).$

Since $\Psi$ is a conformal Riemannian submersion, in view of (2.11), (2.13)
and Lemma 2.5, we gain

$\displaystyle g_{m}(\nabla_{P}^{m}Q,U)$ $\displaystyle=-g_{m}(\alpha
Q,A_{P}\varphi_{\ast}W)+\frac{1}{\lambda_{\ast}^{2}}g_{m}({grad}\ln\lambda_{\ast},\varphi_{\ast}W)g_{n}(\Psi_{\ast}P,\Psi_{\ast}\beta
Q)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{m}({grad}\ln\lambda_{\ast},P)g_{n}(\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta
Q,)$
$\displaystyle-\frac{1}{\lambda_{\ast}^{2}}g_{n}(\Psi_{\ast}({grad}\ln\lambda_{\ast}),\Psi_{\ast}\beta
Q)g_{m}(P,\varphi_{\ast}W)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{P}^{\Psi}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta
Q)-\eta_{\ast}(Q)g_{m}(P,\varphi_{\ast}W).$

Using the definition of sic $\zeta^{\perp}$-Rs, we obtain

$\displaystyle g_{m}(\nabla_{P}^{m}Q,U)$ $\displaystyle=g_{m}(A_{P}\alpha
Q-\beta Q(ln\lambda_{\ast})P+g_{m}(P,\beta Q)({grad}\ln\lambda_{\ast})$
$\displaystyle+P(ln\lambda_{\ast})\beta
Q-\eta_{\ast}(Q)P,\varphi_{\ast}W)+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{P}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta
Q).$

Thus, we obtain the proof from (3) and (3).

Now, we state the following definition for further result.

###### Definition 3.10

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. The distribution $D_{2}^{\ast}$ is parallel along
$(ker\Psi_{\ast})^{\perp}$ if $\nabla_{P}^{m}W\in\Gamma(D_{2}^{\ast})$ for
$P\in\Gamma((ker\Psi_{\ast})^{\perp})$ and $W\in\Gamma(D_{2}^{\ast})$.

###### Corollary 3.11

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$ such that $D^{\ast}_{2}$ is parallel along
$(ker\Psi_{\ast})^{\perp}$. Then the ns condition for the $\Psi$ to be
horizontally homothetic map is

$\lambda_{\ast}^{2}g_{m}(A_{P}\alpha
Q,\varphi_{\ast}W)=g_{n}(\nabla_{P}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta
Q)$ (3.18)

for $P,Q\in\Gamma(ker\Psi_{\ast})^{\perp}$ and $W\in\Gamma(D^{\ast}_{2})$.

Proof. Using (2.11), (3.18) and Lemma 2.6, we obtain

$-g_{m}(grad\ ln\lambda_{\ast},\beta Q)g_{m}(P,\varphi_{\ast}W)+g_{m}(P,\beta
Q)g_{m}(grad\ ln\lambda_{\ast},\varphi_{\ast}W)=0.$ (3.19)

Now, replacing $P$ by $\varphi_{\ast}W$ for $W\in\Gamma(D^{\ast}_{2})$ in
3.19, we get

$g_{m}(grad\ ln\lambda_{\ast},\beta
Q)g_{m}(\varphi_{\ast}W,\varphi_{\ast}W)=0.$

Since $||\varphi_{\ast}W||\neq 0$, Thus $\lambda_{\ast}$ is a constant on
$\Gamma(\mu)$.
Now again replace $P$ by $\beta Q$ for $Q\in\Gamma(ker\Psi_{\ast})^{\perp}$ in
(3.19), we get

$g_{m}(\beta Q,\beta Q)g_{m}(grad\ ln\lambda_{\ast},\varphi_{\ast}W)=0.$

Since $||\beta Q||\neq 0$, Thus $\lambda_{\ast}$ is a constant on
$\Gamma(\varphi_{\ast}D^{\ast}_{2})$. Therefore $\lambda_{\ast}$ is constant
on $\Gamma(ker\Psi_{\ast})^{\perp}$.
The converse easily follows from (3.19).

Now, we explore the geometric investigation of leaves of the distribution
$(ker\Psi_{\ast})$.

###### Theorem 3.12

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the $(ker\Psi_{\ast})$ to be
completely geodesic foliation on $M^{\ast}$ are

$\displaystyle g_{n}(\nabla_{\omega V}\Psi_{\ast}Z,\Psi_{\ast}\omega U)$
$\displaystyle=\lambda_{\ast}^{2}\\{g_{m}(\beta T_{U}\phi V+A_{\omega V}\phi
U+g_{m}(\omega V,\omega U)grad\ ln\lambda_{\ast}$ $\displaystyle-g_{m}(\phi
U,V)\zeta,Z)\\},$

and

$T_{U}\omega V+\hat{\nabla}_{U}\phi V\in\Gamma(D^{\ast}_{1}).$

for $U,V\in\Gamma(ker\Psi_{\ast})$, $Z\in\Gamma(\mu)$ and
$W\in\Gamma(D^{\ast}_{2})$.

Proof. It is to be noted that the ns conditions for the $(ker\Psi_{\ast})$ to
be completely geodesic foliation on $M^{\ast}$ are
$g_{m}(\nabla_{U}^{m}V,Z)=0$ and $g_{m}(\nabla_{U}^{m}V,\varphi_{\ast}W)=0$
for $U,V\in\Gamma(ker\Psi_{\ast})$, $Z\in\Gamma(\mu)$ and
$W\in\Gamma(D^{\ast}_{2})$. Then with the help of (2.1), (2.2), (2.4) and
(3.3), we infer

$\displaystyle g_{m}(\nabla_{U}^{m}V,Z)$
$\displaystyle=g_{m}(\nabla_{U}^{m}\phi V,\varphi_{\ast}Z)+g_{m}(\phi
U,\nabla_{\omega V}^{m}Z)+g_{m}(\omega U,\nabla_{\omega V}^{m}Z)$
$\displaystyle+\eta_{\ast}(Z)g_{m}(\varphi_{\ast}U,V).$

Since $\Psi$ is a sic $\zeta^{\perp}$-Rs, from (2.8), (2.11), (2.13) and Lemma
2.6, we get

$\displaystyle g_{m}(\nabla_{U}^{m}V,Z)$ $\displaystyle=g_{m}(T_{U}\phi
V,\varphi_{\ast}Z)+g_{m}(\phi U,A_{\omega V}Z)$
$\displaystyle-\frac{1}{\lambda_{\ast}^{2}}g_{m}(grad\
ln\lambda_{\ast},Z)g_{n}(\Psi_{\ast}\omega V,\Psi_{\ast}\omega U)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{\omega
V}^{\Psi}\Psi_{\ast}Z,\Psi_{\ast}\omega
U)+\eta_{\ast}(Z)g_{m}(\varphi_{\ast}U,V).$

From the previous equation, we have

$\displaystyle g_{m}(\nabla_{U}^{m}V,Z)$ $\displaystyle=g_{m}(-\beta T_{U}\phi
V-A_{\omega V}\phi U-g_{m}(\omega V,\omega U)(grad\ ln\lambda_{\ast}),Z)$
$\displaystyle+g_{m}(\phi
U,V)\zeta,Z)+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{\omega
V}^{\Psi}\Psi_{\ast}Z,\Psi_{\ast}\omega U).$

Now, using (2.1), (2.2) (2.8), (2.9) and (3.3), we infer

$\displaystyle g_{m}(\nabla_{U}^{m}V,\varphi_{\ast}W)$ $\displaystyle=$
$\displaystyle-g_{m}(\varphi_{\ast}(T_{U}\omega V+\hat{\nabla}_{U}\phi
V),\varphi_{\ast}W)$ (3.21) $\displaystyle=$ $\displaystyle-
g_{m}(\omega(T_{U}\omega V+\hat{\nabla}_{U}\phi V),\varphi_{\ast}W)$

So the results follows from (3) and (3.21).

Next, we offer certain conditions for dilation $\lambda_{\ast}$ is a constant
on $\mu$. For which, we need the following convenient definition.

###### Definition 3.13

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M,g_{m},J,\zeta,\eta_{\ast})$ onto Riemannian manifolds $(N^{\ast},g_{n})$.
Then we call that $\mu$ is horizontal along $(ker\Psi_{\ast})$ if
$\nabla_{U}^{m}Z\in\Gamma(\mu)$ for $Z\in\Gamma(\mu)$ and
$U\in\Gamma(ker\Psi_{\ast})$.

###### Corollary 3.14

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$ such that $\mu$ is horizontal along $(ker\Psi_{\ast})$.
Then the ns condition for the $\Psi$ to be constant on $\mu$ is

$\lambda_{\ast}^{2}g_{m}(\beta T_{U}\phi V+A_{\omega V}\phi U-g_{m}(\phi
U,V)\zeta,Z)=g_{n}(\nabla_{\omega V}\Psi_{\ast}Z,\Psi_{\ast}\omega U),$ (3.22)

for $U,V\in(ker\Psi_{\ast})$ and $Z\in\Gamma(\mu)$.

Proof. Let $U,V\in\Gamma(ker\Psi_{\ast})$, $Z\in\Gamma(\mu)$. Then
$g_{m}(\nabla_{U}^{m}V,Z)=-g_{m}(V,\nabla_{U}^{m}Z)$. Since $\mu$ is
horizontal along $(ker\Psi_{\ast})$, so $g_{m}(V,\nabla_{U}^{m}Z)=0$.

Using (3) and (3.22), we infer

$g_{m}(\omega V,\omega U)g_{m}(grad\ ln\lambda_{\ast},Z)=0.$ (3.23)

Replacing $U$ by $V$ in the foregoing equation, we may say that
$\lambda_{\ast}$ is a constant on $\Gamma(\mu)$.
The converse follows from (3).

###### Theorem 3.15

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for $M^{\ast}$ to be a locally
product manifold of the form
$M_{\ker\Psi_{\ast}}\times_{\lambda_{\ast}}M_{\ker\Psi_{\ast}^{\perp}}$ are

$\displaystyle g_{n}(\nabla_{\omega V}\Psi_{\ast}Z,\Psi_{\ast}\omega U)$
$\displaystyle=\lambda_{\ast}^{2}\\{g_{m}(\beta T_{U}\phi V+A_{\omega V}\phi
U+g_{m}(\omega V,\omega U)grad\ ln\lambda_{\ast}$ $\displaystyle-g_{m}(\phi
U,V)\zeta,Z)\\},$ $T_{U}\omega V+\hat{\nabla}_{U}\phi
V\in\Gamma(D^{\ast}_{1}).$

and

$A_{P}\varphi_{\ast}Q+{\mathcal{V}_{1}}\nabla_{P}^{m}\alpha
Q\in\Gamma(D^{\ast}_{2}),$ $\displaystyle
g_{n}(\nabla_{P}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\beta Q)=$
$\displaystyle\lambda_{\ast}^{2}g_{m}(-A_{P}\alpha Q+\beta
Q(ln\lambda_{\ast})P-g_{m}(P,\beta Q)({grad}\ln\lambda_{\ast})$
$\displaystyle-P(ln\lambda_{\ast})\beta Q+\eta_{\ast}(Q)P,\varphi_{\ast}W)$

for $P,Q\in\Gamma(ker\Psi_{\ast})^{\perp}$, $U,V,W\in\Gamma(ker\Psi_{\ast})$.

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. For $U\in\Gamma(D^{\ast}_{2})$ and $Q\in\Gamma(\mu)$, we
have

$\frac{1}{\lambda_{\ast}^{2}}g_{n}(\Psi_{\ast}\varphi_{\ast}U,\Psi_{\ast}Q)=g_{m}(\varphi_{\ast}U,Q)=0,$

which shows that the distributions $\Psi_{\ast}(\varphi_{\ast}D^{\ast}_{2})$
and $\Psi_{\ast}(\mu)$ are orthogonal.

Now, we explore the geometric properties of the leaves of the distributions
$D^{\ast}_{1}$ and $D^{\ast}_{2}$.

###### Theorem 3.16

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the distribution $D^{\ast}_{1}$
to be completely geodesic foliations on $M$ are

$(\nabla\Psi_{\ast})(U,\varphi_{\ast}V)\in\Gamma(\Psi_{\ast}\mu)$

and

$\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla\Psi_{\ast})(U,\varphi_{\ast}V),\Psi_{\ast}\beta
Z)=g_{m}(V,T_{U}\omega\alpha Z+\hat{\nabla}_{U}\phi\alpha Z)-\eta_{\ast}(\beta
Z)g_{m}(U,V)$

for $U,V\in\Gamma(D^{\ast}_{1})$ and $Z\in\Gamma(ker\Psi_{\ast})^{\perp}$.

Proof. The ns conditions for the distribution $D^{\ast}_{1}$ to be completely
geodesic foliations on $M$ are $g_{m}(\nabla_{U}^{m}V,W)=0$ and
$g_{m}(\nabla_{U}^{m}V,Z)=0$ for $U,V\in\Gamma(D^{\ast}_{1})$,
$W\in\Gamma(D^{\ast}_{2})$ and $Z\in\Gamma(\ker\Psi_{\ast})^{\perp}$. Using
(2.1), (2.2), (2.4) and (3.3), we obtain

$\displaystyle g_{m}(\nabla_{U}^{m}V,W)$
$\displaystyle=g_{m}(\varphi_{\ast}\nabla_{U}^{m}V,\varphi_{\ast}W)-\eta_{\ast}(\nabla_{U}^{m}V)\eta_{\ast}(W)$
$\displaystyle=g_{m}(\nabla_{U}^{m}\varphi_{\ast}V,\varphi_{\ast}W)$

From (2.13) and using the fact that $\Psi$ is a sic $\zeta^{\perp}$-Rs, we
have

$g_{m}(\nabla_{U}^{m}V,W))=-\frac{1}{\lambda_{\ast}^{2}}g_{n}((\nabla\Psi_{\ast})(U,\varphi_{\ast}V),\Psi_{\ast}\varphi_{\ast}W)$
(3.24)

Using (2.1), (2.2), (2.4), (2.8) and (3.4), we have

$\displaystyle g_{m}(\nabla_{U}^{m}V,Z)$ $\displaystyle=$ $\displaystyle-
g_{m}(V,\nabla_{U}^{m}Z)=g_{m}(V,\nabla_{U}^{m}\varphi_{\ast}^{2}Z)$
$\displaystyle=$ $\displaystyle g_{m}(V,\nabla_{U}^{m}\varphi_{\ast}\alpha
Z)+g_{m}(V,\nabla_{U}^{m}\varphi_{\ast}\beta Z)$ $\displaystyle=$
$\displaystyle g_{m}(V,\nabla_{U}^{m}\varphi_{\ast}\alpha
Z)+g_{m}(\nabla_{U}^{m}\varphi_{\ast}V,\beta Z)-\eta_{\ast}(\beta
Z)g_{m}(U,V)$

Now, using (2.9), (2.13) and (3.3), we get

$\displaystyle g_{m}(\nabla_{U}^{m}V,Z)$
$\displaystyle=g_{m}(V,T_{U}\omega\alpha Z)+g_{m}(V,\hat{\nabla}_{U}\phi\alpha
Z)$ (3.25)
$\displaystyle-\frac{1}{\lambda_{\ast}^{2}}g_{m}((\nabla\Psi_{\ast})(U,\varphi_{\ast}V),\Psi_{\ast}\beta
Z)$ $\displaystyle-\eta_{\ast}(\beta Z)g_{m}(U,V).$

The proof of the theorem comes from (3.24) and (3.25).

###### Theorem 3.17

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the distribution $D^{\ast}_{2}$
to be a completely geodesic foliations on $M^{\ast}$ are

$(\nabla\Psi_{\ast})(W,\varphi_{\ast}U)\in\Gamma(\Psi_{\ast}\mu)$

and

$\displaystyle\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{\varphi_{\ast}\xi}^{\Psi}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\varphi_{\ast}\beta
Z)$ $\displaystyle=$ $\displaystyle g_{m}(\varphi_{\ast}\xi,T_{W}\alpha
Z)-g_{m}(W,\xi)g_{m}({\mathcal{H}_{1}}grad\
ln\lambda_{\ast},\varphi_{\ast}\beta Z)$ $\displaystyle+\eta_{\ast}(\beta
Z)g_{m}(W,\xi).$

for $W,\xi\in\Gamma(D^{\ast}_{2})$, $U\in\Gamma(D^{\ast}_{1})$ and
$Z\in\Gamma(ker\Psi_{\ast}^{\perp})$.

Proof. The ns conditions for the distribution $D^{\ast}_{2}$ to be a
completely geodesic foliations on $M^{\ast}$ are
$g_{m}(\nabla_{W}^{m}\xi,U)=0$ and $g_{m}(\nabla_{W}^{m}\xi,Z)=0$ for
$W,\xi\in\Gamma(D^{\ast}_{2})$, $U\in\Gamma(D^{\ast}_{1})$ and
$Z\in\Gamma((\ker\Psi_{\ast})^{\perp})$. Using (2.1), (2.2), (2.4) and (3.3),
we come to the next equation

$\displaystyle g_{m}(\nabla_{W}^{m}\xi,U))$
$\displaystyle=g_{m}(\varphi_{\ast}\nabla_{W}^{m}\xi,\varphi_{\ast}U)+\eta_{\ast}(\nabla_{W}^{m}\xi)\eta_{\ast}(U)$
$\displaystyle=g_{m}(\nabla_{W}^{m}\varphi_{\ast}\xi,\varphi_{\ast}U)=-g_{m}(\varphi_{\ast}\xi,\nabla_{W}^{m}\varphi_{\ast}U)$

From (2.13) and definition of sic $\zeta^{\perp}$-Rs, we get

$g_{m}(\nabla_{W}^{m}\xi,U))=\frac{1}{\lambda_{\ast}^{2}}g_{m}((\nabla\Psi_{\ast})(W,\varphi_{\ast}U),\Psi_{\ast}\varphi_{\ast}\xi).$
(3.26)

With the help of (2.1), (2.2), (2.4), (2.8) and (3.4), we obtain

$g_{m}(\nabla_{W}^{m}\xi,Z))=-g_{m}(\varphi_{\ast}\xi,T_{W}\alpha
Z)+g_{m}(\nabla_{\varphi_{\ast}\xi}^{m}W,\beta Z).$
$g_{m}(\nabla_{W}^{m}\xi,Z)=-g_{m}(\varphi_{\ast}\xi,T_{W}\alpha
Z)+g_{m}(\nabla_{\varphi_{\ast}\xi}^{m}\varphi_{\ast}W,\varphi_{\ast}\beta
Z)-\eta_{\ast}(\beta Z)g_{m}(W,\xi).$

Then from (2.13), (3.4) and Lemma 2.6, we obtain

$\displaystyle g_{m}(\nabla_{W}^{m}\xi,Z)$
$\displaystyle=-g_{m}(\varphi_{\ast}\xi,T_{W}\alpha
Z)+g_{m}(W,\xi)g_{m}({\mathcal{H}_{1}}grad\
ln\lambda_{\ast},\varphi_{\ast}\beta Z)$
$\displaystyle+\frac{1}{\lambda_{\ast}^{2}}g_{n}(\nabla_{\varphi_{\ast}\xi}^{\Psi}\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\varphi_{\ast}\beta
Z)-\eta_{\ast}(\beta Z)g_{m}(W,\xi).$

The proof of the theorem comes from (3.26) and (3).

From Theorem 3.16 and 3.17, we have the following theorem:

###### Theorem 3.18

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the fibers of $\Psi$ to be
locally product manifold are

$(\nabla\Psi_{\ast})(U,\varphi_{\ast}V)\in\Gamma(\Psi_{\ast}\mu)$

and

$(\nabla\Psi_{\ast})(W,\varphi_{\ast}U)\in\Gamma(\Psi_{\ast}\mu)$

for any $U,V\in\Gamma(D^{\ast}_{1})$ and $W\in\Gamma(D^{\ast}_{2})$.

##  4 Harmonicity of sic $\zeta^{\perp}$-RS

In this part, we provide the ns conditions for a sic $\zeta^{\perp}$-Rs to be
harmonic. We also look into the ns conditions for such submersions to be
completely geodesic. By decomposition of all over space of sic
$\zeta^{\perp}$-Rs, we have the following Lemma.

###### Lemma 4.1

Let $\Psi:(M^{2(p+q+r)+1},g_{m},\varphi_{\ast})\rightarrow(N^{q+2r+1},g_{n})$
be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the tension field $\tau$ of $\Psi$ is

$\tau(\Psi)=-(2p+q)\Psi_{{\ast}}(\mu^{\ker\Psi_{{\ast}}})+(2-q-2r)\Psi_{{\ast}}({grad}\ln\lambda_{\ast}),$
(4.1)

where $\mu^{\ker\Psi_{{\ast}}}$ is the mean curvature vector field of the
distribution of $\ker\Psi_{{\ast}}$.

Proof: It is easy to prove this lemma with the help of the method [1].

In view of Lemma 4.1, we have the following result.

###### Theorem 4.2

Let $\Psi:(M^{2(p+q+r)+1},g_{m},\varphi_{\ast})\rightarrow(N^{q+2r+1},g_{n})$
be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$ such that $q+2r\neq 2$. Then any two of the conditions
imply the third:

(i) $\Psi$ is harmonic

(ii) The fibers are minimal.

(iii) $\Psi$ is a horizontally homothetic map.

Proof: Using (4.1), we can easily find the results.

###### Corollary 4.3

Let $\Psi:(M^{2(p+q+r)+1},g_{m},\varphi_{\ast})\rightarrow(N^{q+2r+1},g_{n})$
be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. If $q+2r=2$ then the ns condition for the $\Psi$ to be
harmonic is the minimal fiber.

A map is claimed to be totally geodesic map if it maps every geodesic in the
total manifold into a geodesic within the base manifold in proportion to arc
lengths. We recall that a differentiable map $\Psi$ between two Riemannian
manifolds is called completely geodesic if $(\nabla\Psi_{\ast})(U,V)=0\ \
\forall U,V\in\Gamma(TM^{\ast})$.

We now, present the subsequent definition:

###### Definition 4.4

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then $\Psi$ is called a
$(\varphi_{\ast}D^{\ast}_{2},\mu)$-completely geodesic map if

$(\nabla\Psi_{\ast})(\varphi_{\ast}W,Q)=0\;\ {\rm for}\
W\in\Gamma(D^{\ast}_{2})\ {\rm and}\ Q\in\Gamma(\mu).$

In this sequence, we have the following important result.

###### Theorem 4.5

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from acm manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. The ns condition for $\Psi$ to be a
$(\varphi_{\ast}D^{\ast}_{2},\mu)$-completely geodesic map if $\Psi$ is a
horizontally homothetic map.

Proof. For $W\in\Gamma(D^{\ast}_{2})$ and $Q\in\Gamma(\mu)$, from Lemma 2.6,
we infer

$(\nabla\Psi_{\ast})(\varphi_{\ast}W,Q)=\varphi_{\ast}W(\ln\lambda_{\ast})\Psi_{\ast}Q+Q(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}W-g_{m}(\varphi_{\ast}W,Q)\Psi_{\ast}({grad}\ln\lambda_{\ast}).$

Since $\Psi$ is a horizontally homothetic therefore
$(\nabla\Psi_{\ast})(\varphi_{\ast}W,Q)=0$.
Conversely, if $(\nabla\Psi_{\ast})(\varphi_{\ast}W,Q)=0,$ we have

$\varphi_{\ast}W(\ln\lambda_{\ast})\Psi_{\ast}Q+Q(\ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}W=0$
(4.2)

Taking scaler product in (4.2) with $\Psi_{\ast}\varphi_{\ast}W$ and since
$\Psi$ is a sic $\zeta^{\perp}$-Rs, we infer

$g_{m}({grad}\ln\lambda_{\ast},\varphi_{\ast}W)g_{n}(\Psi_{\ast}Q,\Psi_{\ast}\varphi_{\ast}W)+g_{m}({grad}\ln\lambda_{\ast},Q)g_{n}(\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}\varphi_{\ast}W)=0.$

Since $||\varphi_{\ast}W||\neq 0$, then the above equation reduces to

$g_{m}({\mathcal{H}_{1}}{grad}\ln\lambda_{\ast},Q)=0,$

which implies that $\lambda_{\ast}$ is a constant on $\Gamma(\mu)$.
Taking scaler product in (4.2) with $\Psi_{\ast}Q$ ,we get

$g_{m}({grad}\ln\lambda_{\ast},\varphi_{\ast}W)g_{n}(\Psi_{\ast}Q,\Psi_{\ast}Q)+g_{m}({grad}\ln\lambda_{\ast},Q)g_{n}(\Psi_{\ast}\varphi_{\ast}W,\Psi_{\ast}Q)=0.$

Since $||Q||\neq 0$, then the above equation reduces to

$g_{m}({grad}\ln\lambda_{\ast},\varphi_{\ast}W)=0,$

it follows that $\lambda_{\ast}$ is a constant on
$\Gamma(\varphi_{\ast}D^{\ast}_{2})$. Thus $\lambda_{\ast}$ is a constant on
$\Gamma((ker\Psi_{\ast})^{\perp})$, which over the proof.
At last, we infer the ns conditions for sic $\zeta^{\perp}$-Rs to be
completely geodesic.

###### Theorem 4.6

Let $\Psi$ be a sic $\zeta^{\perp}$-Rs from Sasakian manifolds
$(M^{\ast},\varphi_{\ast},\zeta,\eta_{\ast},g_{m})$ onto Riemannian manifolds
$(N^{\ast},g_{n})$. Then the ns conditions for the $\Psi$ to be completely
geodesic map are

(a) $\beta
T_{U}\varphi_{\ast}V+\omega\hat{\nabla}_{U}\varphi_{\ast}V-\eta_{\ast}(T_{U}V)\zeta=0,\
U,V\in\Gamma(D^{\ast}_{1})$,

(b) $\omega T_{U}\varphi_{\ast}W+\beta
H\hat{\nabla}_{U}\varphi_{\ast}W-\eta_{\ast}(T_{U}^{m}W)\zeta=0,\
U\in\Gamma(ker\Psi_{\ast}),\ W\in\Gamma(D^{\ast}_{2})$,

(c) $\Psi$ is a horizontally homothetic map,

(d) $\omega{\mathcal{V}_{1}}\nabla_{P}\phi U+\beta A_{P}\phi
U+\beta{\mathcal{H}_{1}}\nabla_{P}\omega U+\omega A_{P}\omega
U-\eta_{\ast}\left(A_{P}U\right)\zeta=0$, $U\in\Gamma(ker\Psi_{\ast})$,
$P\in\Gamma((ker\Psi_{\ast})^{\perp})$.

Proof. (a) For $U,V\in\Gamma(D_{1}^{\ast})$, using (2.1), (2.2), (2.4), (2.8)
and (2.13), we have

$\displaystyle(\nabla{\Psi_{\ast}})(U,V)$
$\displaystyle=\nabla_{U}\Psi_{\ast}V-\Psi_{\ast}(\nabla_{U}^{m}V)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}^{2}\nabla_{U}^{m}V-\eta_{\ast}(\nabla_{U}^{m}V)\zeta)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}\nabla_{U}^{m}\varphi_{\ast}V-\eta_{\ast}(\nabla_{U}^{m}V)\zeta)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}(T_{U}\varphi_{\ast}V+\hat{\nabla}_{U}\varphi_{\ast}V)-\eta_{\ast}(T_{U}V)\zeta).$

With the help of (3.3) and (3.4) in the above expression turns into

$(\nabla{\Psi_{\ast}})(U,V)=\Psi_{\ast}(\alpha T_{U}\varphi_{\ast}V+\beta
T_{U}\varphi_{\ast}V+\phi\hat{\nabla}_{U}\varphi_{\ast}V+\omega\hat{\nabla}_{U}\varphi_{\ast}V-\eta_{\ast}(\nabla_{U}^{m}V)\zeta),$

since $\alpha
T_{U}\varphi_{\ast}V+\phi\hat{\nabla}_{U}\varphi_{\ast}V\in\Gamma((ker\Psi_{\ast})$,
we infer

$(\nabla{\Psi_{\ast}})(U,V)=\Psi_{\ast}(\beta
T_{U}\varphi_{\ast}V+\omega\hat{\nabla}_{U}\varphi_{\ast}V-\eta_{\ast}(\nabla_{U}^{m}V)\zeta).$

Since $\Psi$ is a linear isomorphism between $(ker\Psi_{\ast})^{\perp}$ and
$\Gamma(TN)$, so $(\nabla{\Psi_{\ast}})(U,V)=0$ if and only if $\beta
T_{U}\varphi_{\ast}V+\omega\hat{\nabla}_{U}\varphi_{\ast}V-\eta_{\ast}(\nabla_{U}^{m}V)\zeta=0.$
(b) For $U\in\Gamma(ker\Psi_{\ast})$ and $W\in\Gamma(D_{2}^{\ast})$, using
(2.1), (2.2), (2.4) and 2.13), we come to the next equation

$\displaystyle(\nabla{\Psi_{\ast}})(U,W)$
$\displaystyle=\nabla_{U}\Psi_{\ast}W-\Psi_{\ast}(\nabla_{U}^{m}W)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}^{2}\nabla_{U}^{m}W-\eta_{\ast}(\nabla_{U}^{m}W)\zeta)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}\nabla_{U}^{m}\varphi_{\ast}W-\eta_{\ast}(T_{U}^{m}W)\zeta)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}(T_{U}\varphi_{\ast}W+{\mathcal{H}_{1}}\nabla_{U}\varphi_{\ast}W)-\eta_{\ast}(T_{U}^{m}W)\zeta).$

In view of (3.3) and (3.4) in the foregoing equation, we gain

$(\nabla{\Psi_{\ast}})(U,W)=\Psi_{\ast}(\phi T_{U}\varphi_{\ast}W+\omega
T_{U}\varphi_{\ast}W+\alpha{\mathcal{H}_{1}}\nabla_{U}\varphi_{\ast}W+\beta{\mathcal{H}_{1}}\nabla_{U}\varphi_{\ast}W-\eta_{\ast}(T_{U}^{m}W)\zeta)$

Since $\phi
T_{U}\varphi_{\ast}W+\alpha{\mathcal{H}_{1}}\nabla_{U}\varphi_{\ast}W\in\Gamma(\ker\Psi_{\ast})$,
we infer

$(\nabla{\Psi_{\ast}})(U,W)=\Psi_{\ast}(\omega
T_{U}\varphi_{\ast}W+\beta{\mathcal{H}_{1}}\nabla_{U}\varphi_{\ast}W-\eta_{\ast}(T_{U}^{m}W)\zeta).$

Since $\Psi$ is a linear isomorphism between $(ker\Psi_{\ast})^{\perp}$ and
$\Gamma(TN)$, so $(\nabla{\Psi_{\ast}})(U,W)=0$ if and only if $\omega
T_{U}\varphi_{\ast}W+\beta
H\hat{\nabla}_{U}\varphi_{\ast}W-\eta_{\ast}(T_{U}^{m}W)\zeta=0.$
(c) For $U,V\in\Gamma(\mu)$, from Lemma 2.6, we get

$(\nabla\Psi_{\ast})(U,V)=U(\ln\lambda_{\ast})\Psi_{\ast}V+V(\ln\lambda_{\ast})\Psi_{\ast}U-g_{m}(U,V)\Psi_{\ast}({grad}\ln\lambda_{\ast}).$

In the above equation, replacing $V$ by $\varphi_{\ast}U$, we infer

$\displaystyle(\nabla\Psi_{\ast})(U,\varphi_{\ast}U)$
$\displaystyle=U(\ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U+\varphi_{\ast}U(\ln\lambda_{\ast})\Psi_{\ast}U-g_{m}(U,\varphi_{\ast}U)\Psi_{\ast}({grad}\ln\lambda_{\ast}).$
$\displaystyle=U(\ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U+\varphi_{\ast}U(\ln\lambda_{\ast})\Psi_{\ast}U.$

If $(\nabla\Psi_{\ast})(U,\varphi_{\ast}U)=0$, we have

$U(\ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U+\varphi_{\ast}U(\ln\lambda_{\ast})\Psi_{\ast}U=0.$
(4.3)

Taking scaler product of the above equation with $\Psi_{\ast}\varphi_{\ast}U$,
we infer

$g_{m}({grad}\ln\lambda_{\ast},U)g_{n}(\Psi_{\ast}\varphi_{\ast}U,\Psi_{\ast}\varphi_{\ast}U)+g_{m}(gradln\lambda_{\ast},\varphi_{\ast}U)g_{n}(\Psi_{\ast}U,\Psi_{\ast}\varphi_{\ast}U)=0,$

which shows that $\lambda_{\ast}$ is a constant $\Gamma(\mu)$.
In a similar pattern, for $U,V\in\Gamma(D_{2}^{\ast})$, using Lemma 2.6, we
get

$\displaystyle(\nabla\Psi_{\ast})(\varphi_{\ast}U,\varphi_{\ast}V)$
$\displaystyle=\varphi_{\ast}U(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}V+\varphi_{\ast}V(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U$
$\displaystyle-g_{m}(\varphi_{\ast}U,\varphi_{\ast}V)\Psi_{\ast}(grad\
ln\lambda_{\ast}).$

In the above equation, replacing $V$ by $U$, we get

$\displaystyle(\nabla\Psi_{\ast})(\varphi_{\ast}U,\varphi_{\ast}U)$
$\displaystyle=\varphi_{\ast}U(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U+\varphi_{\ast}U(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U$
(4.4) $\displaystyle-g_{m}(\varphi_{\ast}U,\varphi_{\ast}U)\Psi_{\ast}(grad\
ln\lambda_{\ast})$
$\displaystyle=2\varphi_{\ast}U(ln\lambda_{\ast})\Psi_{\ast}\varphi_{\ast}U-g_{m}(\varphi_{\ast}U,\varphi_{\ast}U)\Psi_{\ast}(grad\
ln\lambda_{\ast}).$

Consider $(\nabla\Psi_{\ast})(\varphi_{\ast}U,\varphi_{\ast}U)=0$ and taking
scalar product of above equation with $\Psi_{\ast}\varphi_{\ast}U$ and using
the fact that $\Psi$ is sic $\zeta^{\perp}$-Rs, we infer

$2g_{m}(grad\
ln\lambda_{\ast},\varphi_{\ast}U)g_{n}(\Psi_{\ast}\varphi_{\ast}U,\Psi_{\ast}\varphi_{\ast}U)-g_{m}(\varphi_{\ast}U,\varphi_{\ast}U)g_{n}(\Psi_{\ast}(grad\
ln\lambda_{\ast}),\Psi_{\ast}\varphi_{\ast}U).$

From the above expression, it follows that $\lambda_{\ast}$ is a constant on
$\Gamma(\varphi_{\ast}D_{2}^{\ast})$.
Thus $\lambda_{\ast}$ is a constant on $\Gamma((ker\Psi_{\ast})^{\perp})$.

If $\Psi$ is a horizontally homothetic map, then ${\mathcal{H}_{1}}{grad}\
ln\lambda_{\ast}$ vanishes.
Therefore $(\nabla\Psi_{\ast})(U,V)=0$ for
$U,V\in\Gamma((ker\Psi_{\ast})^{\perp})$.
(d) For $U\in\Gamma(ker\Psi_{\ast})$ and
$P\in\Gamma((ker\Psi_{\ast})^{\perp})$, using (2.1), (2.2), (2.4), (2.9) and
(2.13), we acquire

$\displaystyle(\nabla{\Psi_{\ast}})(P,U)$
$\displaystyle=\nabla_{P}\Psi_{\ast}U-\Psi_{\ast}(\nabla_{P}^{m}U)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}^{2}\nabla_{P}^{m}U-\eta_{\ast}(\nabla_{P}^{m}U)\zeta)$
$\displaystyle=\Psi_{\ast}(\varphi_{\ast}\nabla_{P}^{m}\varphi_{\ast}U-\eta_{\ast}(\nabla_{P}^{m}U)\zeta).$

Then from (2.8), (2.9) and (3.4), we arrive at

$\displaystyle(\nabla{\Psi_{\ast}})(P,U)$ $\displaystyle=$
$\displaystyle\Psi_{\ast}(\varphi_{\ast}\nabla_{P}^{m}(\phi+\omega)U-\eta_{\ast}\left(A_{P}U+{\mathcal{V}_{1}}\nabla_{P}U\right)\zeta)$
$\displaystyle=$ $\displaystyle\Psi_{\ast}(\varphi_{\ast}(A_{P}\phi
U+{\mathcal{V}_{1}}\nabla_{P}\phi
U)+\varphi_{\ast}({\mathcal{H}_{1}}\nabla_{P}\omega U+A_{P}\omega U))$
$\displaystyle-\Psi_{\ast}(\eta_{\ast}\left(A_{P}U\right)\zeta).$

Using (3.3)and (3.4) in the above equation, we have

$\displaystyle(\nabla{\Psi_{\ast}})(P,U)$ $\displaystyle=$
$\displaystyle\Psi_{\ast}(\phi{\mathcal{V}_{1}}\nabla_{P}\phi
U+\omega{\mathcal{V}_{1}}\nabla_{P}\phi U+\alpha A_{P}\phi U+\beta A_{P}\phi
U$ $\displaystyle+\alpha{\mathcal{H}_{1}}\nabla_{P}\omega
U+\beta{\mathcal{H}_{1}}\nabla_{P}\omega U+\phi A_{P}\omega U+\omega
A_{P}\omega U-\eta_{\ast}\left(A_{P}U\right)\zeta)$

Since $\phi{\mathcal{V}_{1}}\nabla_{P}\phi U+\alpha A_{P}\phi
U+\alpha{\mathcal{H}_{1}}\nabla_{P}\omega U+\phi A_{P}\omega
U\in\Gamma(ker\Psi_{\ast})$, we infer

$(\nabla{\Psi_{\ast}})(P,U)=\Psi_{\ast}(\omega{\mathcal{V}_{1}}\nabla_{P}\phi
U+\beta A_{P}\phi U+\beta{\mathcal{H}_{1}}\nabla_{P}\omega U+\omega
A_{P}\omega U-\eta_{\ast}\left(A_{P}U\right)\zeta).$

Since $\Psi$ is a linear isomorphism between $(ker\Psi_{\ast})^{\perp}$ and
$\Gamma(TN)$, so $(\nabla{\Psi_{\ast}})(P,U)=0$ if and only if
$\omega{\mathcal{V}_{1}}\nabla_{P}\phi U+\beta A_{P}\phi
U+\beta{\mathcal{H}_{1}}\nabla_{P}\omega U+\omega A_{P}\omega
U-\eta_{\ast}\left(A_{P}U\right)\zeta=0$.

Conflict of interest: The authors declare that they have no conflict of
interest.

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Department of Mathematics
University of Calcutta
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West Bengal, India.
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Department of Mathematics and Astronomy
University of Lucknow
Lucknow, Uttar Pradesh, 226007, India.
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Department of Mathematics and Statistics
Dr. Harisingh Gour University
Sagar-470003, Madhya Pradesh, India.
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