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Abstract

We introduce and characterize slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results of slant Riemannian submersions defined on Sasakian manifolds. We give a sufficient condition for a slant Riemannian submersion from Sasakian manifolds onto Riemannian manifolds to be harmonic. We also give an example of such slant submersions. Moreover, we find a sharp inequality between the scalar curvature and norm squared mean curvature of fibres.

2010 Mathematics Subject Classification

primary, 53C25, 53C43, 53C55; secondary, 53D15

Keywords

Riemannian submersion; Sasakian manifold; Anti-invariant submersion; Slant submersion

1. Introduction

Let ${\textstyle F}$ be a ${\textstyle C^{\infty }}$-submersion from a Riemannian manifold ${\textstyle (M,g_{M})}$ onto a Riemannian manifold ${\textstyle (N,g_{N})}$. Then according to the conditions on the map ${\textstyle F:(M,g_{M})\rightarrow (N,g_{N})}$, ${\textstyle F}$ can be any one of the following types: semi-Riemannian submersion and Lorentzian submersion  [11], Riemannian submersion [22] and [12], slant submersion [9] and [27], almost Hermitian submersion  [29], contact-complex submersion  [13], quaternionic submersion  [14], almost ${\textstyle h}$-slant submersion and ${\textstyle h}$-slant submersion  [24], semi-invariant submersion  [28], ${\textstyle h}$-semi-invariant submersion  [25], etc.

As we know, Riemannian submersions are related to physics and have their applications in the Yang–Mills theory [6] and [30], Kaluza–Klein theory [7] and [15], supergravity and superstring theories [16] and [21]. In  [26], Şahin introduced anti-invariant Riemannian submersions from almost Hermitian manifolds onto Riemannian manifolds. He gave a generalization of Hermitian submersions and anti-invariant submersions by defining and studying slant submersions from almost Hermitian manifolds onto Riemannian manifolds  [27].

The present work is another step in this direction, more precisely from the point of view of slant Riemannian submersions from Sasakian manifolds. We also want to carry anti-invariant submanifolds of Sasakian manifolds to anti-invariant Riemannian submersion theory and to prove dual results for submersions. For instance, a slant submanifold of a ${\textstyle K}$-contact manifold is an anti invariant submanifold if and only if ${\textstyle \nabla Q=0}$ (see Proposition 4.1 of  [8]). We get a result similar to Proposition 4. Although slant submanifolds of contact metric manifolds were studied by several different authors and are considered a well-established topic in contact Riemannian geometry, only little about slant submersions are known. So, we study slant Riemannian submersions from almost contact metric manifolds onto Riemannian manifolds. Recently, the authors in  [17] and [20] and  [18] studied anti-invariant Riemannian submersions from almost contact manifolds independently of each other.

The paper is organized as follows: In Section  2, we present the basic information about Riemannian submersions needed throughout this paper. In Section  3, we mention about Sasakian manifolds. In Section  4, we give the definition of slant Riemannian submersions and introduce slant Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We survey main results on slant submersions defined on Sasakian manifolds. We give a sufficient condition for a slant Riemannian submersion from Sasakian manifolds onto Riemannian manifolds to be harmonic. Moreover, we investigate the geometry of leaves of ${\textstyle (kerF_{_{\ast }})}$ and ${\textstyle (kerF_{_{\ast }})^{\perp }}$. We give an example of slant submersions such that the characteristic vector field ${\textstyle \xi }$ is vertical. Moreover, we find a sharp inequality between the scalar curvature and squared mean curvature of fibres.

2. Riemannian submersions

In this section we recall several notions and results which will be needed throughout the paper.

Let ${\textstyle (M,g_{M})}$ be an ${\textstyle m}$-dimensional Riemannian manifold and let ${\textstyle (N,g_{N})}$ be an ${\textstyle n}$-dimensional Riemannian manifold. A Riemannian submersion is a smooth map ${\textstyle F:M\rightarrow N}$ which is onto and satisfies the following axioms:

${\textstyle S1}$. ${\textstyle F}$ has maximal rank.

${\textstyle S2}$. The differential ${\textstyle F_{_{\ast }}}$ preserves the lengths of horizontal vectors.

The fundamental tensors of a submersion were defined by O’Neill [22], [23]. They are ${\textstyle (1,2)}$-tensors on ${\textstyle M}$, given by the following formulas:

${\displaystyle T(E,F)=T_{E}F=H\nabla _{VE}VF+V\nabla _{VE}HF{\mbox{,}}}$

(2.1)

for any vector fields ${\textstyle E}$ and ${\textstyle F}$ on ${\textstyle M}$. Here ${\textstyle \nabla }$ denotes the Levi-Civita connection of ${\textstyle (M,g_{M})}$. These tensors are called integrability tensors for the Riemannian submersions. Here we denote the projection morphism on the distributions ker${\textstyle F_{_{\ast }}}$ and ${\textstyle (kerF_{_{\ast }})^{\perp }}$ by ${\textstyle V}$ and ${\textstyle H}$, respectively. The following lemmas are well known [22] and [23]:

Lemma 1.

For any  ${\textstyle U,W}$vertical and  ${\textstyle X,Y}$horizontal vector fields, the tensor fields  ${\textstyle T}$and  ${\textstyle A}$satisfy

${\displaystyle (i)T_{U}W=T_{W}U{\mbox{,}}}$

(2.3)

It is easy to see that ${\textstyle T}$ is vertical, ${\textstyle T_{E}=T_{VE}}$, ${\textstyle A}$ is horizontal and ${\textstyle A\,=\,A_{HE}}$.

For each ${\textstyle q\in N}$, ${\textstyle F^{-1}(q)}$ is an ${\textstyle (m-n)}$-dimensional submanifold of ${\textstyle M}$. The submanifolds ${\textstyle F^{-1}(q)}$, ${\textstyle q\in N}$, are called fibres. A vector field on ${\textstyle M}$ is called vertical if it is always tangent to fibres. A vector field on ${\textstyle M}$ is called horizontal if it is always orthogonal to fibres. A vector field ${\textstyle X}$ on ${\textstyle M}$ is called basic if ${\textstyle X}$ is horizontal and ${\textstyle F}$-related to a vector field ${\textstyle X}$ on ${\textstyle N}$, i. e., ${\textstyle F_{_{\ast }}X_{p}=X_{{_{\ast }}F(p)}}$ for all ${\textstyle p\in M}$.

Lemma 2.

Let  ${\textstyle F:(M,g_{M})\rightarrow (N,g_{N})}$be a Riemannian submersion. If  ${\textstyle X}$,${\textstyle Y}$are basic vector fields on  ${\textstyle M}$, then

(i) ${\textstyle g_{M}(X,Y)=g_{N}(X_{_{\ast }},Y_{_{\ast }})\circ F}$,

(ii) ${\textstyle H[X,Y]}$ is basic and ${\textstyle F}$-related to ${\textstyle [X_{_{\ast }},Y_{_{\ast }}]}$,

(iii) ${\textstyle H(\nabla _{X}Y)}$ is a basic vector field corresponding to ${\textstyle \nabla _{X_{_{\ast }}}^{^{_{\ast }}}Y_{_{\ast }}}$ where ${\textstyle \nabla ^{_{\ast }}}$ is the connection on ${\textstyle N}$,

(iv) for any vertical vector field ${\textstyle V}$, ${\textstyle [X,V]}$ is vertical.

Moreover, if ${\textstyle X}$ is basic and ${\textstyle U}$ is vertical, then ${\textstyle H(\nabla _{U}X)=H(\nabla _{X}U)=A_{X}U}$. On the other hand, from  and  we have

${\displaystyle \nabla _{V}W=T_{V}W+{\overset {\mbox{ˆ}}{\nabla }}_{V}W{\mbox{,}}}$

(2.5)

for ${\textstyle X,Y\in \Gamma ((kerF_{_{\ast }})^{\perp })}$ and ${\textstyle V,W\in \Gamma (kerF_{_{\ast }})}$, where ${\textstyle {\overset {\mbox{ˆ}}{\nabla }}_{V}W=V\nabla _{V}W}$. On any fibre ${\textstyle F^{-1}(q)}$, ${\textstyle q\in N,{\overset {\mbox{ˆ}}{\nabla }}}$ coincides with the Levi-Civita connection with respect to the metric induced by ${\textstyle g_{M}}$. This induced metric on fibre ${\textstyle F^{-1}(q)}$ is denoted by ${\textstyle {\overset {\mbox{ˆ}}{g}}}$.

Notice that ${\textstyle T}$ acts on the fibres as the second fundamental form of the submersion and restricted to vertical vector fields and it can be easily seen that ${\textstyle T=0}$ is equivalent to the condition that the fibres are totally geodesic. A Riemannian submersion is called a Riemannian submersion with totally geodesic fibres if ${\textstyle T}$ vanishes identically. Let ${\textstyle U_{1},\ldots ,U_{m-n}}$ be an orthonormal frame of ${\textstyle \Gamma (kerF_{_{\ast }})}$. Then the horizontal vector field ${\textstyle H={\frac {1}{m-n}}\sum _{j=1}^{m-n}T_{U_{j}}U_{j}}$ is called the mean curvature vector field of the fibre. If ${\textstyle H=0}$, then the Riemannian submersion is said to be minimal. A Riemannian submersion is called a Riemannian submersion with totally umbilical fibres if

${\displaystyle T_{U}W=g_{M}(U,W)H}$

(2.9)

for ${\textstyle U,W\in \Gamma (kerF_{_{\ast }})}$. For any ${\textstyle E\in \Gamma (TM),T_{E{\mbox{   }}}}$ and  ${\textstyle A_{E}}$ are skew-symmetric operators on ${\textstyle (\Gamma (TM),g_{M})}$ reversing the horizontal and the vertical distributions. By Lemma 1, the horizontal distribution ${\textstyle H}$ is integrable if and only if ${\textstyle A\,=0}$. For any ${\textstyle D,E,G\in \Gamma (TM)}$, one has

${\displaystyle g(T_{D}E,G)+g(T_{D}G,E)=0}$

(2.10)

and

${\displaystyle g(A_{D}E,G)+g(A_{D}G,E)=0{\mbox{.}}}$

(2.11)

The tensor fields ${\textstyle A}$, ${\textstyle T}$ and their covariant derivatives play a fundamental role in expressing the Riemannian curvature ${\textstyle R}$ of ${\textstyle (M,g)}$. By  and , we have

${\displaystyle {\begin{array}{l}\displaystyle R(U,V,S,W)=g(R(S,W)V,U)={\overset {\mbox{ˆ}}{R}}(U,V,S,W)+g(T_{U}W\\\displaystyle T_{V}S)-g(T_{V}W,T_{U}S){\mbox{,}}\end{array}}}$

(2.12)

where ${\textstyle {\overset {\mbox{ˆ}}{R}}}$ is a Riemannian curvature tensor of any fibre ${\textstyle (F^{-1}(q),{\overset {\mbox{ˆ}}{g}}_{q})}$. Precisely, if ${\textstyle \lbrace U,V\rbrace }$ is an orthonormal basis of the vertical 2-plane, then Eq. (2.12) implies that

${\displaystyle K(U\wedge V)={\overset {\mbox{ˆ}}{K}}(U\wedge V)+\parallel T_{U}V\parallel ^{2}-g(T_{U}U,T_{V}V){\mbox{,}}}$

(2.13)

where ${\textstyle K}$ and ${\textstyle {\overset {\mbox{ˆ}}{K}}}$ denote the sectional curvature of ${\textstyle M}$ and fibre ${\textstyle F^{-1}(q)}$, respectively.

We recall the notion of harmonic maps between Riemannian manifolds. Let ${\textstyle (M,g_{M})}$ and ${\textstyle (N,g_{N})}$ be two Riemannian manifolds and suppose that ${\textstyle \varphi :M\rightarrow N}$ is a smooth map between them. Then the differential ${\textstyle \varphi _{_{\ast }}}$ of ${\textstyle \varphi }$ can be viewed as a section of the bundle ${\textstyle Hom(TM,\varphi ^{-1}TN)\rightarrow M}$, where ${\textstyle \varphi ^{-1}TN}$ is the pullback bundle which has fibres ${\textstyle (\varphi ^{-1}TN)_{p}=T_{\varphi (p)}N}$, ${\textstyle p\in M.Hom(TM,\varphi ^{-1}TN)}$ has a connection ${\textstyle \nabla }$ induced from the Levi-Civita connection ${\textstyle \nabla ^{M}}$ and the pullback connection. Then the second fundamental form of ${\textstyle \varphi }$ is given by

${\displaystyle (\nabla \varphi _{_{\ast }})(X,Y)=\nabla _{X}^{\varphi }\varphi _{_{\ast }}(Y)-\varphi _{_{\ast }}(\nabla _{X}^{M}Y)}$

(2.14)

for ${\textstyle X,Y\in \Gamma (TM)}$, where ${\textstyle \nabla ^{\varphi }}$ is the pullback connection. It is known that the second fundamental form is symmetric. If ${\textstyle \varphi }$ is a Riemannian submersion, it can be easily proved that

${\displaystyle (\nabla \varphi _{_{\ast }})(X,Y)=0}$

(2.15)

for ${\textstyle X,Y\in \Gamma ((kerF_{_{\ast }})^{\perp })}$. A smooth map ${\textstyle \varphi :(M,g_{M})\rightarrow (N,g_{N})}$ is said to be harmonic if ${\textstyle trace(\nabla \varphi _{_{\ast }})=0}$. On the other hand, the tension field of ${\textstyle \varphi }$ is the section ${\textstyle \tau (\varphi )}$ of ${\textstyle \Gamma (\varphi ^{-1}TN)}$ defined by

${\displaystyle \tau (\varphi )=div\varphi _{_{\ast }}=\sum _{i=1}^{m}(\nabla \varphi _{_{\ast }})(e_{i},e_{i}){\mbox{,}}}$

(2.16)

where ${\textstyle \lbrace e_{1},\ldots ,e_{m}\rbrace }$ is the orthonormal frame on ${\textstyle M}$. Then it follows that ${\textstyle \varphi }$ is harmonic if and only if ${\textstyle \tau (\varphi )=0}$; for details, see  [2].

3. Sasakian manifolds

An ${\textstyle n}$-dimensional differentiable manifold ${\textstyle M}$ is said to have an almost contact structure ${\textstyle (\phi ,\xi ,\eta )}$ if it carries a tensor field ${\textstyle \phi }$ of type ${\textstyle (1,1)}$, a vector field ${\textstyle \xi }$ and a 1-form ${\textstyle \eta }$ on ${\textstyle M}$ respectively such that

${\displaystyle \phi ^{2}=-I+\eta \otimes \xi {\mbox{,}}\quad \phi \xi =0{\mbox{,}}\quad \eta \circ \phi =0{\mbox{,}}\quad \eta (\xi )=1{\mbox{,}}}$

(3.1)

where ${\textstyle I}$ denotes the identity tensor.

The almost contact structure is said to be normal if ${\textstyle N+d\eta \otimes \xi =0}$, where ${\textstyle N}$ is the Nijenhuis tensor of ${\textstyle \phi }$. Suppose that a Riemannian metric tensor ${\textstyle g}$ is given in ${\textstyle M}$ and satisfies the condition

${\displaystyle g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y){\mbox{,}}\quad \eta (X)=g(X,\xi ){\mbox{.}}}$

(3.2)

Then the ${\textstyle (\phi ,\xi ,\eta ,g)}$-structure is called an almost contact metric structure. Define a tensor field ${\textstyle \Phi }$ of type ${\textstyle (0,2)}$ by ${\textstyle \Phi (X,Y)=g(\phi X,Y)}$. If ${\textstyle d\eta =\Phi }$ then an almost contact metric structure is said to be normal contact metric structure. A normal contact metric structure is called a Sasakian structure, which satisfies

${\displaystyle (\nabla _{X}\phi )Y=g(X,Y)\xi -\eta (Y)X{\mbox{,}}}$

(3.3)

where ${\textstyle \nabla }$ denotes the Levi-Civita connection of ${\textstyle g}$. For a Sasakian manifold ${\textstyle M=M^{2n+1}}$, it is known that

${\displaystyle R(\xi ,X)Y=g(X,Y)\xi -\eta (Y)X{\mbox{,}}}$

(3.4)

and

${\displaystyle \nabla _{X}\xi =-\phi X{\mbox{.}}}$

(3.6)

[5].

The curvature tensor ${\textstyle R}$ of a Sasakian space form ${\textstyle M(c)}$ is given by

${\displaystyle {\begin{array}{l}\displaystyle R(X,Y)Z={\frac {c+3}{4}}(g(Y,Z)X-g(X\\\displaystyle Z)Y)+{\frac {c-1}{4}}(\eta (X)\eta (Z)Y-\eta (Y)\eta (Z)X+g(X,Z)\eta (Y)\xi -g(Y\\\displaystyle Z)\eta (X)\xi +g(\phi Y,Z)\phi X-g(\phi X,Z)\phi Y-2g(\phi X,Y)\phi Z){\mbox{,}}\end{array}}}$

(3.7)

in  [4] for any tangent vector fields ${\textstyle X,Y,Z}$ to ${\textstyle M(c)}$.

Now we will introduce a well known Sasakian manifold example on ${\textstyle R^{2n+1}}$.

Example 1 [4].

We consider ${\textstyle R^{2n+1}}$ with Cartesian coordinates ${\textstyle (x_{i},y_{i},z)(i=1,\ldots ,n)}$ and its usual contact form

${\displaystyle \eta ={\frac {1}{2}}(dz-\sum _{i=1}^{n}y_{i}dx_{i}){\mbox{.}}}$

The characteristic vector field ${\textstyle \xi }$ is given by ${\textstyle 2{\frac {\partial }{\partial z}}}$ and its Riemannian metric ${\textstyle g}$ and tensor field ${\textstyle \phi }$ are given by

${\displaystyle g={\frac {1}{4}}\eta \otimes \eta +\sum _{i=1}^{n}((dx_{i})^{2}+(dy_{i})^{2}){\mbox{,}}\quad \phi =({\begin{array}{ccc}0&\delta _{ij}&0\\-\delta _{ij}&0&0\\0&y_{j}&0\end{array}}){\mbox{,}}\quad i=1,\ldots ,n{\mbox{.}}}$

This gives a contact metric structure on ${\textstyle R^{2n+1}}$. The vector fields ${\textstyle E_{i}=2{\frac {\partial }{\partial y_{i}}}}$, ${\textstyle E_{n+i}=2({\frac {\partial }{\partial x_{i}}}+y_{i}{\frac {\partial }{\partial z}})}$, ${\textstyle \xi }$ form a ${\textstyle \phi }$-basis for the contact metric structure. On the other hand, it can be shown that ${\textstyle R^{2n+1}(\phi ,\xi ,\eta ,g)}$ is a Sasakian manifold.

4. Slant Riemannian submersions

Definition 1.

Let ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$ be a Sasakian manifold and ${\textstyle (N,g_{N})}$ be a Riemannian manifold. A Riemannian submersion ${\textstyle F:M(\phi ,\xi ,\eta ,g_{M})\rightarrow (N,g_{N})}$ is said to be slant if for any nonzero vector ${\textstyle X\in \Gamma (kerF_{_{\ast }})-\lbrace \xi \rbrace }$, the angle ${\textstyle \theta (X)}$ between ${\textstyle \phi X}$ and the space ${\textstyle kerF_{_{\ast }}}$ is a constant (which is independent of the choice of ${\textstyle p\in M}$ and of ${\textstyle X\in \Gamma (kerF_{_{\ast }})-\lbrace \xi \rbrace }$). The angle ${\textstyle \theta }$ is called the slant angle of the slant submersion. Invariant and anti-invariant submersions are slant submersions with ${\textstyle \theta =0}$ and ${\textstyle \theta =\pi /2}$, respectively. A slant submersion which is not invariant nor anti-invariant is called proper submersion.

Now we will give an example.

Example 2.

${\textstyle R^{5}}$ has got a Sasakian structure as in Example 1. Let ${\textstyle F:R^{5}\rightarrow R^{2}}$ be a map defined by ${\textstyle F(x_{1},x_{2},y_{1},y_{2},z)=(x_{1}-2{\sqrt {2}}x_{2}+y_{1},2x_{1}-2{\sqrt {2}}x_{2}+y_{1})}$. Then, a simple calculation gives

${\displaystyle kerF_{_{\ast }}=span\lbrace V_{1}=2E_{1}+{\frac {1}{\sqrt {2}}}E_{4},V_{2}=E_{2},V_{3}=\xi =E_{5}\rbrace }$

and

${\displaystyle (kerF_{_{\ast }})^{\perp }=span\lbrace H_{1}=2E_{1}-{\frac {1}{\sqrt {2}}}E_{4},{\mbox{   }}H_{2}=E_{3}\rbrace {\mbox{.}}}$

Then it is easy to see that ${\textstyle F}$ is a Riemannian submersion. Moreover, ${\textstyle \phi V_{1}=2E_{3}-{\frac {1}{\sqrt {2}}}E_{2}}$ and ${\textstyle \phi V_{2}=E_{4}}$ imply that ${\textstyle \vert g(\phi V_{1},V_{2})\vert ={\frac {1}{\sqrt {2}}}}$. So ${\textstyle F}$ is a slant submersion with slant angle ${\textstyle \theta ={\frac {\pi }{4}}}$.

In Example 2, we note that the characteristic vector field ${\textstyle \xi }$ is a vertical vector field. If ${\textstyle \xi }$ is orthogonal to ${\textstyle kerF_{_{\ast }}}$, we will then give the following theorem.

Theorem 1.

Let  ${\textstyle F}$be a slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. If  ${\textstyle \xi }$is orthogonal to  ${\textstyle kerF_{_{\ast }}}$, then  ${\textstyle F}$is anti-invariant.

Proof.

By , , (2.10) and (3.6), we have

${\displaystyle {\begin{array}{l}\displaystyle g(\phi U,V)=-g(\nabla _{U}\xi ,V)=-g(T_{U}\xi ,V)=g(T_{U}V,\xi )=g(T_{V}U\\\displaystyle \xi )=g(U,\phi V)\end{array}}}$

for any ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$. Using the skew symmetry property of ${\textstyle \phi }$ in the last relation, we complete the proof of the theorem.  □

Remark 1.

Lotta  [19] proved that if ${\textstyle M_{1}}$ is a submanifold of a contact metric manifold of ${\textstyle {\tilde {M}}_{1}}$ and ${\textstyle \xi }$ is orthogonal to ${\textstyle M_{1}}$, then ${\textstyle M_{1}}$ is an anti-invariant submanifold. So, our result can be seen as a submersion version of Lotta’s result.

Now, let ${\textstyle F}$ be a slant Riemannian submersion from a Sasakian manifold ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$ onto a Riemannian manifold ${\textstyle (N,g_{N})}$. Then for any ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$, we put

${\displaystyle \phi U=\psi U+\omega U{\mbox{,}}}$

(4.1)

where ${\textstyle \psi U}$ and ${\textstyle \omega U}$ are vertical and horizontal components of ${\textstyle \phi U}$, respectively. Similarly, for any ${\textstyle X\in \Gamma (kerF_{_{\ast }})^{\perp }}$, we have

${\displaystyle \phi X=BX+CX{\mbox{,}}}$

(4.2)

where ${\textstyle BX}$ (resp. ${\textstyle CX}$) is the vertical part (resp. horizontal part) of ${\textstyle \phi X}$.

From (3.2), (4.1) and (4.2), we obtain

${\displaystyle g_{M}(\psi U,V)=-g_{M}(U,\psi V)}$

(4.3)

and

${\displaystyle g_{M}(\omega U,Y)=-g_{M}(U,BY){\mbox{,}}}$

(4.4)

for any ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$ and ${\textstyle Y\in \Gamma ((kerF_{_{\ast }})^{\perp })}$.

Using , (3.6) and (4.1), we obtain

${\displaystyle T_{U}\xi =-\omega U{\mbox{,}}\quad {\overset {\mbox{ˆ}}{\nabla }}_{U}\xi =-\psi U{\mbox{,}}}$

(4.5)

for any ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$.

Now we will give the following proposition for a Riemannian submersion with two dimensional fibres in a similar way to Proposition 3.2 of  [1].

Proposition 1.

Let  ${\textstyle F}$be a Riemannian submersion from an almost contact manifold onto a Riemannian manifold. If  ${\textstyle dim(kerF_{_{\ast }})=2}$and  ${\textstyle \xi }$is a vertical vector field, then the fibres are anti-invariant.

As the proof of the following proposition is similar to slant submanifolds (see  [8]), we omit its proof.

Proposition 2.

Let  ${\textstyle F}$be a Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$such that  ${\textstyle \xi \in \Gamma (kerF_{_{\ast }})}$. Then  ${\textstyle F}$is an anti-invariant submersion if and only if  ${\textstyle D}$is integrable, where  ${\textstyle D=kerF_{_{\ast }}-\lbrace \xi \rbrace }$.

Theorem 2.

Let  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$be a Sasakian manifold of dimension  ${\textstyle 2m+1}$and  ${\textstyle (N,g_{N})}$is a Riemannian manifold of dimension  ${\textstyle n}$. Let  ${\textstyle F:M(\phi ,\xi ,\eta ,g_{M})\rightarrow (N,g_{N})}$be a slant Riemannian submersion. Then the fibres are not totally umbilical.

Proof.

Using  and (3.6), we obtain

${\displaystyle T_{U}\xi =-\omega U{\mbox{,}}}$

(4.6)

for any ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$. If the fibres are totally umbilical, then we have ${\textstyle T_{U}V=g_{M}(U,V)H}$ for any vertical vector fields ${\textstyle U,V}$ where ${\textstyle H}$ is the mean curvature vector field of any fibre. Since ${\textstyle T_{\xi }\xi =0}$, we have ${\textstyle H=0}$, which shows that fibres are minimal. Hence the fibres are totally geodesic, which is a contradiction to the fact that ${\textstyle T_{U}\xi =-\omega U\not =0}$.  □

By , , (4.1) and (4.2), we have

${\displaystyle (\nabla _{U}\omega )V=CT_{U}V-T_{U}\psi V{\mbox{,}}}$

(4.7)

where

${\displaystyle (\nabla _{U}\omega )V=H\nabla _{U}\omega V-\omega {\overset {\mbox{ˆ}}{\nabla }}_{U}V}$

(4.9)

for ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$. Now we will characterize slant submersions in the following theorem.

Theorem 3.

Let  ${\textstyle F}$be a Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$such that  ${\textstyle \xi \in \Gamma (kerF_{_{\ast }})}$. Then,  ${\textstyle F}$is a slant Riemannian submersion if and only if there exists a constant  ${\textstyle \lambda \in [0,1]}$such that

${\displaystyle \psi ^{2}=-\lambda (I-\eta \otimes \xi ){\mbox{.}}}$

(4.11)

Furthermore, in such a case, if  ${\textstyle \theta }$is the slant angle of  ${\textstyle F}$, it satisfies that  ${\textstyle \lambda =cos^{2}\theta }$.

Proof.

Firstly we suppose that ${\textstyle F}$ is not an anti-invariant Riemannian submersion. Then, for ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$,

${\displaystyle cos\theta ={\frac {g_{M}(\phi U,\psi U)}{\vert \phi U\vert \vert \psi U\vert }}={\frac {\vert \psi U\vert ^{2}}{\vert \phi U\vert \vert \psi U\vert }}={\frac {\vert \psi U\vert }{\vert \phi U\vert }}{\mbox{.}}}$

(4.12)

Since ${\textstyle \phi U\perp \xi }$, we have ${\textstyle g(\psi U,\xi )=0}$. Now, substituting ${\textstyle U}$ by ${\textstyle \psi U}$ in (4.12) and using (3.2) we obtain

${\displaystyle cos\theta ={\frac {\vert \psi ^{2}U\vert }{\vert \phi \psi U\vert }}={\frac {\vert \psi ^{2}U\vert }{\vert \psi U\vert }}{\mbox{.}}}$

(4.13)

From (4.12) and (4.13), we have

${\displaystyle \vert \psi U\vert ^{2}=\vert \psi ^{2}U\vert \vert \phi U\vert {\mbox{.}}}$

(4.14)

On the other hand, one can get the following

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(\psi ^{2}U,U)=g_{M}(\phi \psi U,U)=-g_{M}(\psi U,\phi U)=-g_{M}(\psi U\\\displaystyle \psi U)=-\vert \psi U\vert ^{2}{\mbox{.}}\end{array}}}$

(4.15)

Using (4.14) and (4.15), we get

${\displaystyle g_{M}(\psi ^{2}U,U)=-\vert \psi ^{2}U\vert \vert \phi U\vert =-\vert \psi ^{2}U\vert \vert \phi ^{2}U\vert {\mbox{.}}}$

(4.16)

Also, one can easily get

${\displaystyle g_{M}(\psi ^{2}U,\phi ^{2}U)=-g_{M}(\psi ^{2}U,U){\mbox{.}}}$

(4.17)

So, by means of (4.16) and (4.17), we obtain ${\textstyle g_{M}(\psi ^{2}U,\phi ^{2}U)=\vert \psi ^{2}U\vert \vert \phi ^{2}U\vert }$ and it follows that ${\textstyle \psi ^{2}U}$ and ${\textstyle \phi ^{2}U}$ are collinear, that is ${\textstyle \psi ^{2}U=\lambda \phi ^{2}U=-\lambda (I-\eta \otimes \xi )}$. Using the last relation together with (4.12) and (4.13) we obtain that ${\textstyle cos\theta ={\sqrt {\lambda }}}$ is constant and so ${\textstyle F}$ is a slant Riemannian submersion.

If ${\textstyle F}$ is an anti-invariant Riemannian submersion then ${\textstyle \phi U}$ is normal, ${\textstyle \psi U=0}$ and it is equivalent to ${\textstyle \psi ^{2}U=0}$. In this case ${\textstyle \theta ={\frac {\pi }{2}}}$ and so Eq. (4.12) is again satisfied.  □

By using (3.2), (4.1), (4.3) and (4.11), we have the following lemma.

Lemma 3.

Let  ${\textstyle F}$be a slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$with slant angle  ${\textstyle \theta }$. Then the following relations are valid for any  ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$:

${\displaystyle g_{M}(\psi U,\psi V)=cos^{2}\theta (g_{M}(U,V)-\eta (U)\eta (V)){\mbox{,}}}$

(4.18)

We denote the complementary orthogonal distribution to ${\textstyle \omega (kerF_{_{\ast }})}$ in ${\textstyle (kerF_{_{\ast }})^{\perp }}$ by ${\textstyle \mu }$. Then we have

${\displaystyle (kerF_{_{\ast }})^{\perp }=\omega (kerF_{_{\ast }})\oplus \mu {\mbox{.}}}$

(4.20)

Lemma 4.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. Then  ${\textstyle \mu }$is an invariant distribution of  ${\textstyle (kerF_{_{\ast }})^{\perp }}$under the endomorphism  ${\textstyle \phi }$.

Proof.

For ${\textstyle X\in \Gamma (\mu )}$ and ${\textstyle V\in \Gamma (kerF_{_{\ast }})}$, from (3.2) and (4.1), we obtain

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(\phi X,\omega V)=g_{M}(\phi X,\phi V)-g_{M}(\phi X,\psi V)=g_{M}(X\\\displaystyle V)-\eta (X)\eta (V)-g_{M}(\phi X,\psi V)=-g_{M}(X,\phi \psi V)=0{\mbox{.}}\end{array}}}$

In a similar way, we have ${\textstyle g_{M}(\phi X,U)=-g_{M}(X,\phi U)=0}$ due to ${\textstyle \phi U\in \Gamma ((kerF_{_{\ast }})\oplus \omega (kerF_{_{\ast }}))}$ for ${\textstyle X\in \Gamma (\mu )}$ and ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$. Thus the proof of the lemma is completed. □

By means of (4.19), we can give the following result:

Corollary 1.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M^{2m+1}(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N^{n},g_{N})}$. Let

${\displaystyle \lbrace e_{1},e_{2},\ldots ,e_{2m-n},\xi \rbrace }$

be a local orthonormal frame of  ${\textstyle (kerF_{_{\ast }})}$, then  ${\textstyle \lbrace csc\theta \omega e_{1},csc\theta \omega e_{2},\ldots ,csc\theta \omega e_{2m-n}\rbrace }$is a local orthonormal frame of  ${\textstyle \omega (kerF_{_{\ast }})}$.

By using (4.20) and Corollary 1, one can easily prove the following proposition:

Proposition 3.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M^{2m+1}(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N^{n},g_{N})}$. Then  ${\textstyle dim(\mu )=2(n-m)}$. If  ${\textstyle \mu =\lbrace 0\rbrace }$, then  ${\textstyle n=m}$.

By (4.3) and , we have

Lemma 5.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M^{2m+1}(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N^{n},g_{N})}$. If  ${\textstyle e_{1},e_{2},\ldots ,e_{k},\xi }$are orthogonal unit vector fields in  ${\textstyle (kerF_{_{\ast }})}$, then

${\displaystyle \lbrace e_{1},sec\theta \psi e_{1},e_{2},sec\theta \psi e_{2},\ldots ,e_{k},sec\theta \psi e_{k},\xi \rbrace }$

is a local orthonormal frame of  ${\textstyle (kerF_{_{\ast }})}$. Moreover  ${\textstyle dim(kerF_{_{\ast }})=2m-n+1=2k+1}$and  ${\textstyle dimN=n=2(m-k)}$.

Lemma 6.

Let  ${\textstyle F}$be a slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. If  ${\textstyle \omega }$is parallel, then we have

${\displaystyle T_{\psi U}\psi U=-cos^{2}\theta (T_{U}U+\eta (U)\omega U){\mbox{.}}}$

(4.21)

Proof.

If ${\textstyle \omega }$ is parallel, from (4.7), we obtain ${\textstyle CT_{U}V=T_{U}\psi V}$ for ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$. Antisymmetrizing with respect to ${\textstyle U}$, ${\textstyle V}$ and using (2.3), we get

${\displaystyle T_{U}\psi V=T_{V}\psi U{\mbox{.}}}$

Substituting ${\textstyle V}$ by ${\textstyle \psi U}$ in the above equation and using Theorem 3, we get the required formula.  □

We give a sufficient condition for a slant Riemannian submersion to be harmonic as an analogue of a slant Riemannian submersion from an almost Hermitian manifold onto a Riemannian manifold in  [27].

Theorem 4.

Let  ${\textstyle F}$be a slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. If  ${\textstyle \omega }$is parallel, then  ${\textstyle F}$is a harmonic map.

Proof.

From  [10] we know that ${\textstyle F}$ is harmonic if and only if ${\textstyle F}$ has minimal fibres. Thus ${\textstyle F}$ is harmonic if and only if ${\textstyle \sum _{i=1}^{n_{1}}T_{e_{i}}e_{i}=0}$. Hence using the adapted frame for slant Riemannian submersion and by the help of (2.16) and Lemma 5, we can write

${\displaystyle \tau =-\sum _{i=1}^{m-{\frac {n}{2}}}F_{_{\ast }}(T_{e_{i}}e_{i}+T_{sec\theta \psi e_{i}}sec\theta \psi e_{i})-F_{_{\ast }}(T_{\xi }\xi ){\mbox{.}}}$

Regarding ${\textstyle T_{\xi }\xi =0}$, we have

${\displaystyle \tau =-\sum _{i=1}^{m-{\frac {n}{2}}}F_{_{\ast }}(T_{e_{i}}e_{i}+sec^{2}\theta T_{\psi e_{i}}\psi e_{i}){\mbox{.}}}$

Using (4.21) in the above equation, we obtain

${\displaystyle {\begin{array}{l}\displaystyle \tau =-\sum _{i=1}^{m-{\frac {n}{2}}}F_{_{\ast }}(T_{e_{i}}e_{i}+sec^{2}\theta (-cos^{2}\theta (T_{e_{i}}e_{i}+\\\displaystyle +\eta (e_{i})\omega e_{i})))=-\sum _{i=1}^{m-{\frac {n}{2}}}F_{_{\ast }}(T_{e_{i}}e_{i}-T_{e_{i}}e_{i})=0{\mbox{.}}\end{array}}}$

So we prove that ${\textstyle F}$ is harmonic.  □

Now setting ${\textstyle Q=\psi ^{2}}$, we define ${\textstyle \nabla Q}$ by

${\displaystyle (\nabla _{U}Q)V=V\nabla _{U}QV-Q{\overset {\mbox{ˆ}}{\nabla }}_{U}V}$

for any ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$.We give a characterization for a slant Riemannian submersion from a Sasakian manifold ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$ onto a Riemannian manifold ${\textstyle (N,g_{N})}$ by using the value of ${\textstyle \nabla Q}$.

Proposition 4.

Let  ${\textstyle F}$be a slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. Then,  ${\textstyle \nabla Q=0}$if and only if  ${\textstyle F}$is an anti-invariant submersion.

Proof.

By using (4.11),

${\displaystyle Q{\overset {\mbox{ˆ}}{\nabla }}_{U}V=-cos^{2}\theta ({\overset {\mbox{ˆ}}{\nabla }}_{U}V-\eta ({\overset {\mbox{ˆ}}{\nabla }}_{U}V)\xi )}$

(4.22)

for each ${\textstyle U,V\in \Gamma (kerF_{_{\ast }})}$, where ${\textstyle \theta }$ is the slant angle.

On the other hand, it follows that

${\displaystyle V(\nabla _{U}QV)=-cos^{2}\theta ({\overset {\mbox{ˆ}}{\nabla }}_{U}V-\eta ({\overset {\mbox{ˆ}}{\nabla }}_{U}V)\xi +g(V,\psi U)\xi +\eta (V)\psi U){\mbox{.}}}$

(4.23)

So, from (4.22) and ${\textstyle \nabla Q=0}$ if and only if ${\textstyle cos^{2}\theta (g(V,\psi U)\xi +\eta (V)\psi U)=0}$ which implies that ${\textstyle \psi U=0}$ or ${\textstyle \theta ={\frac {\pi }{2}}}$. Both the cases verify that ${\textstyle F}$ is an anti-invariant submersion.  □

We now investigate the geometry of leaves of ${\textstyle (kerF_{_{\ast }})^{\perp }}$ and ${\textstyle kerF_{_{\ast }}}$.

Theorem 5.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. Then the distribution  ${\textstyle (kerF_{_{\ast }})^{\perp }}$defines a totally geodesic foliation on  ${\textstyle M}$if and only if

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(H\nabla _{X}Y,\omega \psi U)-sin^{2}\theta g_{M}(Y,\phi X)\eta (U)=g_{M}(A_{X}BY\\\displaystyle \omega U)+g_{M}(H\nabla _{X}CY,\omega U)\end{array}}}$

for any  ${\textstyle X,Y\in \Gamma ((kerF_{_{\ast }})^{\perp })}$and  ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$.

Proof.

From (3.3) and (4.1), we have

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(\nabla _{X}Y,U)=-g_{M}(\phi \nabla _{X}\phi Y,U)+g_{M}(Y,\phi X)\eta (U)=g_{M}(\nabla _{X}\phi Y\\\displaystyle \phi U)+g_{M}(Y,\phi X)\eta (U)=g_{M}(\nabla _{X}\phi Y,\psi U)+g_{M}(\nabla _{X}\phi Y\\\displaystyle \omega U)+g_{M}(Y,\phi X)\eta (U)\end{array}}}$

(4.24)

for any ${\textstyle X,Y\in \Gamma ((kerF_{_{\ast }})^{\perp })}$ and ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$.

Using (3.3) and (4.1) in (4.24), we obtain

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(\nabla _{X}Y,U)=-g_{M}(\nabla _{X}Y,\psi ^{2}U)-g_{M}(\nabla _{X}Y,\omega \psi U)+g_{M}(Y\\\displaystyle \phi X)\eta (U)+g_{M}(\nabla _{X}\phi Y,\omega U){\mbox{.}}\end{array}}}$

(4.25)

By (4.2) and (4.11), we have

${\displaystyle {\begin{array}{l}\displaystyle g_{M}(\nabla _{X}Y,U)=cos^{2}\theta g_{M}(\nabla _{X}Y,U)-cos^{2}\theta \eta (U)\eta (\nabla _{X}Y)-g_{M}(\nabla _{X}Y\\\displaystyle \omega \psi U)+g_{M}(Y,\phi X)\eta (U)+g_{M}(\nabla _{X}BY,\omega U)+g_{M}(\nabla _{X}CY,\omega U){\mbox{.}}\end{array}}}$

(4.26)

Using ,  and (3.6) in the last equation, we obtain

${\displaystyle {\begin{array}{l}\displaystyle sin^{2}\theta g_{M}(\nabla _{X}Y,U)=sin^{2}\theta g_{M}(Y,\phi X)\eta (U)-g_{M}(H\nabla _{X}Y\\\displaystyle \omega \psi U)+g_{M}(A_{X}BY,\omega U)+g_{M}(H\nabla _{X}CY,\omega U)\end{array}}}$

which proves the theorem. □

Proposition 5.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian manifold  ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$onto a Riemannian manifold  ${\textstyle (N,g_{N})}$. If the distribution  ${\textstyle kerF_{_{\ast }}}$defines a totally geodesic foliation on  ${\textstyle M}$, then  ${\textstyle F}$is an invariant submersion.

Proof.

By (4.5), if the distribution ${\textstyle kerF_{_{\ast }}}$ defines a totally geodesic foliation on ${\textstyle M}$, then we conclude that ${\textstyle \omega U=0}$ for any ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$ which shows that ${\textstyle F}$ is an invariant submersion. □

Now we establish a sharp inequality between norm squared mean curvature ${\textstyle \Vert H\Vert ^{2}}$ and the scalar curvature ${\textstyle {\overset {\mbox{ˆ}}{\tau }}}$ of fibre through ${\textstyle p\in M^{5}(c)}$.

Theorem 6.

Let  ${\textstyle F}$be a proper slant Riemannian submersion from a Sasakian space form  ${\textstyle M^{5}(c)}$onto a Riemannian manifold  ${\textstyle (N^{2},g_{N})}$. Then, we have

${\displaystyle \Vert H\Vert ^{2}\geq {\frac {8}{9}}{\overset {\mbox{ˆ}}{\tau }}-{\frac {2}{9}}[c+3+(3c+5)cos^{2}\theta ]}$

(4.27)

where  ${\textstyle H}$denotes the mean curvature of fibres. Moreover, the equality sign of   (4.27)   holds at a point  ${\textstyle p}$of a fibre if and only if with respect to some suitable slant orthonormal frame  ${\textstyle \lbrace e_{1},e_{2}=sec\theta \psi e_{1},e_{3}=\xi ,e_{4}=csc\theta we_{1},e_{5}=csc\theta we_{2}\rbrace }$at  ${\textstyle p}$, we have

${\displaystyle T_{11}^{4}=3T_{22}^{4}{\mbox{,}}\quad T_{12}^{4}=0\quad {\mbox{   and   }}\quad T_{11}^{5}=0{\mbox{,}}}$

where  ${\textstyle T_{ij}^{\alpha }=g(T(e_{i},e_{j}),e_{\alpha })}$for  ${\textstyle 1\leq i,j\leq 3}$and  ${\textstyle \alpha =4,5}$.

Proof.

By Corollary 1, Lemma 5 and Proposition 3 we construct a slant orthonormal frame ${\textstyle \lbrace e_{1},e_{2},e_{3},e_{4},e_{5}\rbrace }$ defined by

${\displaystyle e_{1},e_{2}=sec\theta \psi e_{1}{\mbox{,}}\quad e_{3}=\xi {\mbox{,}}\quad e_{4}=csc\theta we_{1}{\mbox{,}}\quad e_{5}=csc\theta we_{2}{\mbox{,}}}$

(4.28)

where ${\textstyle e_{1},e_{2},e_{3}=\xi \in \Gamma (kerF_{_{\ast }})}$ and ${\textstyle e_{4},e_{5}\in \Gamma ((kerF_{_{\ast }})^{\perp })}$. Let ${\textstyle {\overset {\mbox{ˆ}}{\tau }}}$ be scalar curvature of fibre ${\textstyle F^{-1}(q)}$. We choose an arbitrary point ${\textstyle p}$ of the fibre ${\textstyle F^{-1}(q)}$. We obtain

${\displaystyle {\overset {\mbox{ˆ}}{\tau }}(p)={\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{2})+{\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{3})+{\overset {\mbox{ˆ}}{K}}(e_{2}\wedge e_{3}){\mbox{.}}}$

(4.29)

By (2.12), (2.13) and (3.7), we get

${\displaystyle {\begin{array}{l}\displaystyle {\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{2})={\frac {c+3}{4}}+{\frac {3}{4}}(c-1)cos^{2}\theta +T_{11}^{4}T_{22}^{4}+T_{11}^{5}T_{22}^{5}-\\\displaystyle -(T_{12}^{4})^{2}-(T_{12}^{5})^{2}{\mbox{,}}\end{array}}}$

(4.30)

where ${\textstyle T_{ij}^{\alpha }=g(T(e_{i},e_{j}),e_{\alpha })}$ for ${\textstyle 1\leq i,j\leq 3}$ and ${\textstyle \alpha =4,5}$. Using Theorem 3  and the relation (4.19), one has

${\displaystyle \psi e_{2}=-cos\theta e_{1}\quad {\mbox{and}}\quad \omega e_{2}=sin\theta e_{5}{\mbox{.}}}$

(4.31)

From (4.8), we have

${\displaystyle g(({\overset {\mbox{ˆ}}{\nabla }}_{e_{2}}\psi )e_{2},e_{1})=g(BT_{e_{2}}e_{2},e_{1})-g(T_{e_{2}}\omega e_{2},e_{1}){\mbox{.}}}$

Using (4.1), (4.2), , (4.28) and (4.31) in the last relation, we obtain

${\displaystyle sin\theta [g(T_{e_{2}}e_{2},e_{4})-g(T_{e_{2}}e_{1},e_{5})]=0{\mbox{.}}}$

(4.32)

Since the submersion is proper, Eq. (4.32) implies that

${\displaystyle T_{22}^{4}=T_{12}^{5}{\mbox{.}}}$

Now we choose the unit normal vector ${\textstyle e_{4}\in \Gamma (ker(F_{_{\ast }}))^{\perp }}$ parallel to the mean curvature vector ${\textstyle H(p)}$ of fibre. Then we have

${\displaystyle \Vert H(p)\Vert ^{2}={\frac {1}{9}}(T_{11}^{4}+T_{22}^{4})^{2}{\mbox{,}}\quad T_{11}^{5}+T_{22}^{5}=0{\mbox{.}}}$

So the relation (4.30) becomes

${\displaystyle {\begin{array}{l}\displaystyle {\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{2})={\frac {c+3}{4}}+{\frac {3}{4}}(c-1)cos^{2}\theta +T_{11}^{4}T_{22}^{4}-(T_{11}^{5})^{2}-\\\displaystyle -(T_{12}^{4})^{2}-(T_{22}^{4})^{2}{\mbox{.}}\end{array}}}$

(4.33)

From the trivial inequality ${\textstyle (\mu -3\lambda )^{2}\geq 0}$, one has ${\textstyle (\mu +\lambda )^{2}\geq 8(\lambda \mu -\lambda ^{2})}$. Putting ${\textstyle \mu =T_{11}^{4}}$ and ${\textstyle \lambda =T_{22}^{4}}$ in the last inequality, we find

${\displaystyle \Vert H\Vert ^{2}\geq {\frac {8}{9}}[{\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{2})-{\frac {c+3}{4}}-{\frac {3}{4}}(c-1)cos^{2}\theta ]{\mbox{.}}}$

(4.34)

Using (2.13), we get

${\displaystyle {\overset {\mbox{ˆ}}{K}}(e_{1}\wedge e_{3})={\overset {\mbox{ˆ}}{K}}(e_{2}\wedge e_{3})=cos^{2}\theta {\mbox{.}}}$

By (2.13), (4.29) and the last relation, we get the required inequality. Moreover, the equality sign of (4.27) holds at a point ${\textstyle p}$ of a fibre if and only if ${\textstyle T_{11}^{4}=3T_{22}^{4}}$, ${\textstyle T_{12}^{4}=0}$ and ${\textstyle T_{11}^{5}=0}$. □

Open Problem:

Let ${\textstyle F}$ be a slant Riemannian submersion from a Sasakian manifold ${\textstyle M(\phi ,\xi ,\eta ,g_{M})}$ onto a Riemannian manifold ${\textstyle (N,g_{N})}$. In  [3], Barrera et al. defined and studied the Maslov form of non-invariant slant submanifolds of ${\textstyle S}$-space form ${\textstyle {\tilde {M}}(c)}$. They find conditions for it to be closed. By a similar discussion in  [3], we can define Maslov form ${\textstyle \Omega H}$ of ${\textstyle M}$ as the dual form of the vector field ${\textstyle BH}$, that is,

${\displaystyle \Omega H(U)=g_{M}(U,BH)}$

for any ${\textstyle U\in \Gamma (kerF_{_{\ast }})}$. So it will be interesting to give a characterization with respect to ${\textstyle \Omega H}$ for slant submersions, where ${\textstyle H=\sum _{i=1}^{m-{\frac {n}{2}}}T_{e_{i}}e_{i}+T_{sec\theta \psi e_{i}}sec\theta \psi e_{i}}$ and

${\textstyle \lbrace e_{1},sec\theta \psi e_{1},e_{2},sec\theta \psi e_{2},\ldots ,e_{k},sec\theta \psi e_{k},\xi \rbrace }$ is a local orthonormal frame of ${\textstyle (kerF_{_{\ast }})}$.

Acknowledgement

This paper is supported by Uludag University research project (KUAP(F)-2012/57).

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