Slant Riemannian submersions from Sasakian manifolds

==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 <math display="inline">F</math> be a <math display="inline">C^{\infty }</math>-submersion from a Riemannian manifold <math display="inline">(M,g_M)</math> onto a Riemannian manifold <math display="inline">(N,g_N)</math>. Then according to the conditions on the map <math display="inline">F:(M,g_M)\rightarrow (N,g_N)</math>, <math display="inline">F</math> can be any one of the following types: semi-Riemannian submersion and Lorentzian submersion  [[#br000055|[11]]], Riemannian submersion [[#br000110|[22]]] and [[#br000060|[12]]], slant submersion [[#br000045|[9]]] and [[#br000135|[27]]], almost Hermitian submersion  [[#br000145|[29]]], contact-complex submersion  [[#br000065|[13]]], quaternionic submersion  [[#br000070|[14]]], almost <math display="inline">h</math>-slant submersion and <math display="inline">h</math>-slant submersion  [[#br000120|[24]]], semi-invariant submersion  [[#br000140|[28]]], <math display="inline">h</math>-semi-invariant submersion  [[#br000125|[25]]], etc.

As we know, Riemannian submersions are related to physics and have their applications in the Yang–Mills theory [[#br000030|[6]]] and [[#br000150|[30]]], Kaluza–Klein theory [[#br000035|[7]]] and [[#br000075|[15]]], supergravity and superstring theories [[#br000080|[16]]] and [[#br000105|[21]]]. In  [[#br000130|[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  [[#br000135|[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 <math display="inline">K</math>-contact manifold is an anti invariant submanifold if and only if <math display="inline">\nabla Q=0</math> (see Proposition 4.1 of  [[#br000040|[8]]]). We get a result similar to [[#e000125|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  [[#br000085|[17]]] and [[#br000100|[20]]] and  [[#br000090|[18]]] studied anti-invariant Riemannian submersions from almost contact manifolds independently of each other.

The paper is organized as follows: In Section  [[#s000010|2]], we present the basic information about Riemannian submersions needed throughout this paper. In Section  [[#s000015|3]], we mention about Sasakian manifolds. In Section  [[#s000020|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 <math display="inline">(kerF_{{_\ast}})</math> and <math display="inline">(kerF_{{_\ast}})^{\perp }</math>. We give an example of slant submersions such that the characteristic vector field <math display="inline">\xi </math> is vertical. Moreover, we find a sharp inequality between the scalar curvature and squared mean curvature of fibres.

<span id='s000010'></span>
==2. Riemannian submersions==

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

Let <math display="inline">(M,g_M)</math> be an <math display="inline">m</math>-dimensional Riemannian manifold and let <math display="inline">(N,g_N)</math> be an <math display="inline">n</math>-dimensional Riemannian manifold. A Riemannian submersion is a smooth map <math display="inline">F:M\rightarrow N</math> which is onto and satisfies the following axioms:

<math display="inline">S1</math>. <math display="inline">F</math> has maximal rank.

<math display="inline">S2</math>. The differential <math display="inline">F_{{_\ast}}</math> preserves the lengths of horizontal vectors.

The fundamental tensors of a submersion were defined by O’Neill [[#br000110|[22]]], [[#br000115|[23]]]. They are <math display="inline">(1,2)</math>-tensors on <math display="inline">M</math>, given by the following formulas:

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T(E,F)=T_EF=H\nabla _{VE}VF+V\nabla _{VE}HF\mbox{,}</math>
|}
| style="text-align: right;" | (2.1)
|}

for any vector fields <math display="inline">E</math> and <math display="inline">F</math> on <math display="inline">M</math>. Here <math display="inline">\nabla </math> denotes the Levi-Civita connection of <math display="inline">(M,g_M)</math>. These tensors are called integrability tensors for the Riemannian submersions. Here we denote the projection morphism on the distributions ker<math display="inline">F_{{_\ast}}</math> and <math display="inline">(kerF_{{_\ast}})^{\perp }</math> by <math display="inline">V</math> and <math display="inline">H</math>, respectively. The following lemmas are well known [[#br000110|[22]]] and [[#br000115|[23]]]:

<span id='e000005'></span>

==Lemma 1.                     ==

''For any  ''<math display="inline">U,W</math>''vertical and  ''<math display="inline">X,Y</math>''horizontal vector fields, the tensor fields  ''<math display="inline">T</math>''and  ''<math display="inline">A</math>''satisfy''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(i)T_UW=T_WU\mbox{,}</math>
|}
| style="text-align: right;" | (2.3)
|}

It is easy to see that <math display="inline">T</math> is vertical, <math display="inline">T_E=T_{VE}</math>, <math display="inline">A</math> is horizontal and <math display="inline">A\, =\, A_{HE}</math>.

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

==Lemma 2.                     ==

''Let  ''<math display="inline">F:(M,g_M)\rightarrow (N,g_N)</math>''be a Riemannian submersion. If  ''<math display="inline">X</math>,<math display="inline">Y</math>''are basic vector fields on  ''<math display="inline">M</math>'', then''

(i) <math display="inline">g_M(X,Y)=g_N(X_{{_\ast}},Y_{{_\ast}})\circ F</math>,

(ii) <math display="inline">H[X,Y]</math> is basic and <math display="inline">F</math>-related to <math display="inline">[X_{{_\ast}},Y_{{_\ast}}]</math>,

(iii) <math display="inline">H(\nabla _XY)</math> is a basic vector field corresponding to <math display="inline">\nabla _{X_{{_\ast}}}^{^{{_\ast}}}Y_{{_\ast}}</math> where <math display="inline">\nabla ^{{_\ast}}</math> is the connection on <math display="inline">N</math>,

(iv) for any vertical vector field <math display="inline">V</math>, <math display="inline">[X,V]</math> is vertical.

Moreover, if <math display="inline">X</math> is basic and <math display="inline">U</math> is vertical, then <math display="inline">H(\nabla _UX)=H(\nabla _XU)=A_XU</math>. On the other hand, from  and  we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\nabla _VW=T_VW+\overset{\mbox{ˆ}}{\nabla }_VW\mbox{,}</math>
|}
| style="text-align: right;" | (2.5)
|}

for <math display="inline">X,Y\in \Gamma ((kerF_{{_\ast}})^{\perp })</math> and <math display="inline">V,W\in \Gamma (kerF_{{_\ast}})</math>, where <math display="inline">\overset{\mbox{ˆ}}{\nabla }_VW=V\nabla _VW</math>. On any fibre <math display="inline">F^{-1}(q)</math>, <math display="inline">q\in N,\overset{\mbox{ˆ}}{\nabla }</math> coincides with the Levi-Civita connection with respect to the metric induced by <math display="inline">g_M</math>. This induced metric on fibre <math display="inline">F^{-1}(q)</math> is denoted by <math display="inline">\overset{\mbox{ˆ}}{g}</math>.

Notice that <math display="inline">T</math> 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 <math display="inline">T=0</math> is equivalent to the condition that the fibres are totally geodesic. A Riemannian submersion is called a Riemannian submersion with totally geodesic fibres if <math display="inline">T</math> vanishes identically. Let <math display="inline">U_1,\ldots ,U_{m-n}</math> be an orthonormal frame of <math display="inline">\Gamma (kerF_{{_\ast}})</math>. Then the horizontal vector field <math display="inline">H=\frac{1}{m-n}\sum _{j=1}^{m-n}T_{U_j}U_j</math> is called the mean curvature vector field of the fibre. If <math display="inline">H=0</math>, then the Riemannian submersion is said to be minimal. A Riemannian submersion is called a Riemannian submersion with totally umbilical fibres if

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_UW=g_M(U,W)H</math>
|}
| style="text-align: right;" | (2.9)
|}

for <math display="inline">U,W\in \Gamma (kerF_{{_\ast}})</math>. For any <math display="inline">E\in \Gamma (TM),T_{E\mbox{   }}</math> and  <math display="inline">A_E</math> are skew-symmetric operators on <math display="inline">(\Gamma (TM),g_M)</math> reversing the horizontal and the vertical distributions. By [[#e000005|Lemma 1]], the horizontal distribution <math display="inline">H</math> is integrable if and only if <math display="inline">A\, =0</math>. For any <math display="inline">D,E,G\in \Gamma (TM)</math>, one has

<span id='fd000065'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g(T_DE,G)+g(T_DG,E)=0</math>
|}
| style="text-align: right;" | (2.10)
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g(A_DE,G)+g(A_DG,E)=0\mbox{.}</math>
|}
| style="text-align: right;" | (2.11)
|}

The tensor fields <math display="inline">A</math>, <math display="inline">T</math> and their covariant derivatives play a fundamental role in expressing the Riemannian curvature <math display="inline">R</math> of <math display="inline">(M,g)</math>. By  and , we have

<span id='fd000075'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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_UW\\\displaystyle T_VS)-g(T_VW,T_US)\mbox{,}\end{array}</math>
|}
| style="text-align: right;" | (2.12)
|}

where <math display="inline">\overset{\mbox{ˆ}}{R}</math> is a Riemannian curvature tensor of any fibre <math display="inline">(F^{-1}(q),\overset{\mbox{ˆ}}{g}_q)</math>. Precisely, if <math display="inline">\lbrace U,V\rbrace </math> is an orthonormal basis of the vertical 2-plane, then Eq. [[#fd000075|(2.12)]] implies that

<span id='fd000080'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>K(U\wedge V)=\overset{\mbox{ˆ}}{K}(U\wedge V)+\parallel T_UV\parallel ^2-g(T_UU,T_VV)\mbox{,}</math>
|}
| style="text-align: right;" | (2.13)
|}

where <math display="inline">K</math> and <math display="inline">\overset{\mbox{ˆ}}{K}</math> denote the sectional curvature of <math display="inline">M</math> and fibre <math display="inline">F^{-1}(q)</math>, respectively.

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

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla \varphi _{{_\ast}})(X,Y)=\nabla _X^{\varphi }\varphi _{{_\ast}}(Y)-\varphi _{{_\ast}}(\nabla _X^MY)</math>
|}
| style="text-align: right;" | (2.14)
|}

for <math display="inline">X,Y\in \Gamma (TM)</math>, where <math display="inline">\nabla ^{\varphi }</math> is the pullback connection. It is known that the second fundamental form is symmetric. If <math display="inline">\varphi </math> is a Riemannian submersion, it can be easily proved that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla \varphi _{{_\ast}})(X,Y)=0</math>
|}
| style="text-align: right;" | (2.15)
|}

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

<span id='fd000095'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\tau (\varphi )=div\varphi _{{_\ast}}=\sum_{i=1}^m(\nabla \varphi _{{_\ast}})(e_i,e_i)\mbox{,}</math>
|}
| style="text-align: right;" | (2.16)
|}

where <math display="inline">\lbrace e_1,\ldots ,e_m\rbrace </math> is the orthonormal frame on <math display="inline">M</math>. Then it follows that <math display="inline">\varphi </math> is harmonic if and only if <math display="inline">\tau (\varphi )=0</math>; for details, see  [[#br000010|[2]]].

<span id='s000015'></span>
==3. Sasakian manifolds==

An <math display="inline">n</math>-dimensional differentiable manifold <math display="inline">M</math> is said to have an almost contact structure <math display="inline">(\phi ,\xi ,\eta )</math> if it carries a tensor field <math display="inline">\phi </math> of type <math display="inline">(1,1)</math>, a vector field <math display="inline">\xi </math> and a 1-form <math display="inline">\eta </math> on <math display="inline">M</math> respectively such that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\phi ^2=-I+\eta \otimes \xi \mbox{,}\quad \phi \xi =0\mbox{,}\quad \eta \circ \phi =0\mbox{,}\quad \eta (\xi )=1\mbox{,}</math>
|}
| style="text-align: right;" | (3.1)
|}

where <math display="inline">I</math> denotes the identity tensor.

The almost contact structure is said to be normal if <math display="inline">N+d\eta \otimes \xi =0</math>, where <math display="inline">N</math> is the Nijenhuis tensor of <math display="inline">\phi </math>. Suppose that a Riemannian metric tensor <math display="inline">g</math> is given in <math display="inline">M</math> and satisfies the condition

<span id='fd000105'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y)\mbox{,}\quad \eta (X)=g(X,\xi )\mbox{.}</math>
|}
| style="text-align: right;" | (3.2)
|}

Then the <math display="inline">(\phi ,\xi ,\eta ,g)</math>-structure is called an almost contact metric structure. Define a tensor field <math display="inline">\Phi </math> of type <math display="inline">(0,2)</math> by <math display="inline">\Phi (X,Y)=g(\phi X,Y)</math>. If <math display="inline">d\eta =\Phi </math> 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

<span id='fd000110'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla _X\phi )Y=g(X,Y)\xi -\eta (Y)X\mbox{,}</math>
|}
| style="text-align: right;" | (3.3)
|}

where <math display="inline">\nabla </math> denotes the Levi-Civita connection of <math display="inline">g</math>. For a Sasakian manifold <math display="inline">M=M^{2n+1}</math>, it is known that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>R(\xi ,X)Y=g(X,Y)\xi -\eta (Y)X\mbox{,}</math>
|}
| style="text-align: right;" | (3.4)
|}

and

<span id='fd000130'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\nabla _X\xi =-\phi X\mbox{.}</math>
|}
| style="text-align: right;" | (3.6)
|}

[[#br000025|[5]]].

The curvature tensor <math display="inline">R</math> of a Sasakian space form <math display="inline">M(c)</math> is given by

<span id='fd000135'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | (3.7)
|}

in  [[#br000020|[4]]] for any tangent vector fields <math display="inline">X,Y,Z</math> to <math display="inline">M(c)</math>.

Now we will introduce a well known Sasakian manifold example on <math display="inline">R^{2n+1}</math>.

<span id='e000015'></span>

==Example 1 [[#br000020|[4]]].                     ==

We consider <math display="inline">R^{2n+1}</math> with Cartesian coordinates <math display="inline">(x_i,y_i,z)(i=1,\ldots ,n)</math> and its usual contact form

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\eta =\frac{1}{2}(dz-\sum_{i=1}^ny_idx_i)\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

The characteristic vector field <math display="inline">\xi </math> is given by <math display="inline">2\frac{\partial }{\partial z}</math> and its Riemannian metric <math display="inline">g</math> and tensor field <math display="inline">\phi </math> are given by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="text-align: right;" | 
|}

This gives a contact metric structure on <math display="inline">R^{2n+1}</math>. The vector fields <math display="inline">E_i=2\frac{\partial }{\partial y_i}</math>, <math display="inline">E_{n+i}=2(\frac{\partial }{\partial x_i}+y_i\frac{\partial }{\partial z})</math>, <math display="inline">\xi </math> form a <math display="inline">\phi </math>-basis for the contact metric structure. On the other hand, it can be shown that <math display="inline">R^{2n+1}(\phi ,\xi ,\eta ,g)</math> is a Sasakian manifold.

<span id='s000020'></span>
==4. Slant Riemannian submersions==

==Definition 1.                     ==

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

Now we will give an example.

<span id='e000025'></span>

==Example 2.                     ==

<math display="inline">R^5</math> has got a Sasakian structure as in [[#e000015|Example 1]]. Let <math display="inline">F:R^5\rightarrow R^2</math> be a map defined by <math display="inline">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)</math>. Then, a simple calculation gives

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>kerF_{{_\ast}}=span\lbrace V_1=2E_1+\frac{1}{\sqrt{2}}E_4,V_2=E_2,V_3=\xi =E_5\rbrace </math>
|}
| style="text-align: right;" | 
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(kerF_{{_\ast}})^{\perp }=span\lbrace H_1=2E_1-\frac{1}{\sqrt{2}}E_4,\mbox{   }H_2=E_3\rbrace \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Then it is easy to see that <math display="inline">F</math> is a Riemannian submersion. Moreover, <math display="inline">\phi V_1=2E_3-\frac{1}{\sqrt{2}}E_2</math> and <math display="inline">\phi V_2=E_4</math> imply that <math display="inline">\vert g(\phi V_1,V_2)\vert =\frac{1}{\sqrt{2}}</math>. So <math display="inline">F</math> is a slant submersion with slant angle <math display="inline">\theta =\frac{\pi }{4}</math>.

In [[#e000025|Example 2]], we note that the characteristic vector field <math display="inline">\xi </math> is a vertical vector field. If <math display="inline">\xi </math> is orthogonal to <math display="inline">kerF_{{_\ast}}</math>, we will then give the following theorem.

==Theorem 1.                     ==

''Let  ''<math display="inline">F</math>''be a slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. If  ''<math display="inline">\xi </math>''is orthogonal to  ''<math display="inline">kerF_{{_\ast}}</math>'', then  ''<math display="inline">F</math>''is anti-invariant.''

==Proof.                     ==

By , , [[#fd000065|(2.10)]] and [[#fd000130|(3.6)]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g(\phi U,V)=-g(\nabla _U\xi ,V)=-g(T_U\xi ,V)=g(T_UV,\xi )=g(T_VU\\\displaystyle \xi )=g(U,\phi V)\end{array}</math>
|}
| style="text-align: right;" | 
|}

for any <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>. Using the skew symmetry property of <math display="inline">\phi </math> in the last relation, we complete the proof of the theorem.  □

==Remark 1.                     ==

Lotta  [[#br000095|[19]]] proved that if <math display="inline">M_1</math> is a submanifold of a contact metric manifold of <math display="inline">\tilde{M}_1</math> and <math display="inline">\xi </math> is orthogonal to <math display="inline">M_1</math>, then <math display="inline">M_1</math> is an anti-invariant submanifold. So, our result can be seen as a submersion version of Lotta’s result.

Now, let <math display="inline">F</math> be a slant Riemannian submersion from a Sasakian manifold <math display="inline">M(\phi ,\xi ,\eta ,g_M)</math> onto a Riemannian manifold <math display="inline">(N,g_N)</math>. Then for any <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>, we put

<span id='fd000165'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\phi U=\psi U+\omega U\mbox{,}</math>
|}
| style="text-align: right;" | (4.1)
|}

where <math display="inline">\psi U</math> and <math display="inline">\omega U</math> are vertical and horizontal components of <math display="inline">\phi U</math>, respectively. Similarly, for any <math display="inline">X\in \Gamma (kerF_{{_\ast}})^{\perp }</math>, we have

<span id='fd000170'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\phi X=BX+CX\mbox{,}</math>
|}
| style="text-align: right;" | (4.2)
|}

where <math display="inline">BX</math> (resp. <math display="inline">CX</math>) is the vertical part (resp. horizontal part) of <math display="inline">\phi X</math>.

From [[#fd000105|(3.2)]], [[#fd000165|(4.1)]] and [[#fd000170|(4.2)]], we obtain

<span id='fd000175'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g_M(\psi U,V)=-g_M(U,\psi V)</math>
|}
| style="text-align: right;" | (4.3)
|}

and

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g_M(\omega U,Y)=-g_M(U,BY)\mbox{,}</math>
|}
| style="text-align: right;" | (4.4)
|}

for any <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math> and <math display="inline">Y\in \Gamma ((kerF_{{_\ast}})^{\perp })</math>.

Using , [[#fd000130|(3.6)]] and [[#fd000165|(4.1)]], we obtain

<span id='fd000185'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_U\xi =-\omega U\mbox{,}\quad \overset{\mbox{ˆ}}{\nabla }_U\xi =-\psi U\mbox{,}</math>
|}
| style="text-align: right;" | (4.5)
|}

for any <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>.

Now we will give the following proposition for a Riemannian submersion with two dimensional fibres in a similar way to Proposition 3.2 of  [[#br000005|[1]]].

==Proposition 1.                     ==

''Let  ''<math display="inline">F</math>''be a Riemannian submersion from an almost contact manifold onto a Riemannian manifold. If  ''<math display="inline">dim(kerF_{{_\ast}})=2</math>''and  ''<math display="inline">\xi </math>''is a vertical vector field, then the fibres are anti-invariant.''

As the proof of the following proposition is similar to slant submanifolds (see  [[#br000040|[8]]]), we omit its proof.

==Proposition 2.                     ==

''Let  ''<math display="inline">F</math>''be a Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''such that  ''<math display="inline">\xi \in \Gamma (kerF_{{_\ast}})</math>''. Then  ''<math display="inline">F</math>''is an anti-invariant submersion if and only if  ''<math display="inline">D</math>''is integrable, where  ''<math display="inline">D=kerF_{{_\ast}}-\lbrace \xi \rbrace </math>''.''

==Theorem 2.                     ==

''Let  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''be a Sasakian manifold of dimension  ''<math display="inline">2m+1</math>''and  ''<math display="inline">(N,g_N)</math>''is a Riemannian manifold of dimension  ''<math display="inline">n</math>''. Let  ''<math display="inline">F:M(\phi ,\xi ,\eta ,g_M)\rightarrow (N,g_N)</math>''be a slant Riemannian submersion. Then the fibres are not totally umbilical.''

==Proof.                     ==

Using  and [[#fd000130|(3.6)]], we obtain

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_U\xi =-\omega U\mbox{,}</math>
|}
| style="text-align: right;" | (4.6)
|}

for any <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>. If the fibres are totally umbilical, then we have <math display="inline">T_UV=g_M(U,V)H</math> for any vertical vector fields <math display="inline">U,V</math> where <math display="inline">H</math> is the mean curvature vector field of any fibre. Since <math display="inline">T_{\xi }\xi =0</math>, we have <math display="inline">H=0</math>, which shows that fibres are minimal. Hence the fibres are totally geodesic, which is a contradiction to the fact that <math display="inline">T_U\xi =-\omega U\not =0</math>.  □

By , , [[#fd000165|(4.1)]] and [[#fd000170|(4.2)]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla _U\omega )V=CT_UV-T_U\psi V\mbox{,}</math>
|}
| style="text-align: right;" | (4.7)
|}

where

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla _U\omega )V=H\nabla _U\omega V-\omega \overset{\mbox{ˆ}}{\nabla }_UV</math>
|}
| style="text-align: right;" | (4.9)
|}

for <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>. Now we will characterize slant submersions in the following theorem.

<span id='e000065'></span>

==Theorem 3.                     ==

''Let  ''<math display="inline">F</math>''be a Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''such that  ''<math display="inline">\xi \in \Gamma (kerF_{{_\ast}})</math>''. Then,  ''<math display="inline">F</math>''is a slant Riemannian submersion if and only if there exists a constant  ''<math display="inline">\lambda \in [0,1]</math>''such that''

<span id='fd000225'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\psi ^2=-\lambda (I-\eta \otimes \xi )\mbox{.}</math>
|}
| style="text-align: right;" | (4.11)
|}

''Furthermore, in such a case, if  ''<math display="inline">\theta </math>''is the slant angle of  ''<math display="inline">F</math>'', it satisfies that  ''<math display="inline">\lambda =cos^2\theta </math>.

==Proof.                     ==

Firstly we suppose that <math display="inline">F</math> is not an anti-invariant Riemannian submersion. Then, for <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>,

<span id='fd000230'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="text-align: right;" | (4.12)
|}

Since <math display="inline">\phi U\perp \xi </math>, we have <math display="inline">g(\psi U,\xi )=0</math>. Now, substituting <math display="inline">U</math> by <math display="inline">\psi U</math> in [[#fd000230|(4.12)]] and using [[#fd000105|(3.2)]] we obtain

<span id='fd000235'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>cos\theta =\frac{\vert \psi ^2U\vert }{\vert \phi \psi U\vert }=\frac{\vert \psi ^2U\vert }{\vert \psi U\vert }\mbox{.}</math>
|}
| style="text-align: right;" | (4.13)
|}

From [[#fd000230|(4.12)]] and [[#fd000235|(4.13)]], we have

<span id='fd000240'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\vert \psi U\vert ^2=\vert \psi ^2U\vert \vert \phi U\vert \mbox{.}</math>
|}
| style="text-align: right;" | (4.14)
|}

On the other hand, one can get the following

<span id='fd000245'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g_M(\psi ^2U,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}</math>
|}
| style="text-align: right;" | (4.15)
|}

Using [[#fd000240|(4.14)]] and [[#fd000245|(4.15)]], we get

<span id='fd000250'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g_M(\psi ^2U,U)=-\vert \psi ^2U\vert \vert \phi U\vert =-\vert \psi ^2U\vert \vert \phi ^2U\vert \mbox{.}</math>
|}
| style="text-align: right;" | (4.16)
|}

Also, one can easily get

<span id='fd000255'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g_M(\psi ^2U,\phi ^2U)=-g_M(\psi ^2U,U)\mbox{.}</math>
|}
| style="text-align: right;" | (4.17)
|}

So, by means of [[#fd000250|(4.16)]] and [[#fd000255|(4.17)]], we obtain <math display="inline">g_M(\psi ^2U,\phi ^2U)=\vert \psi ^2U\vert \vert \phi ^2U\vert </math> and it follows that <math display="inline">\psi ^2U</math> and <math display="inline">\phi ^2U</math> are collinear, that is <math display="inline">\psi ^2U=\lambda \phi ^2U=-\lambda (I-\eta \otimes \xi )</math>. Using the last relation together with [[#fd000230|(4.12)]] and [[#fd000235|(4.13)]] we obtain that <math display="inline">cos\theta =\sqrt{\lambda }</math> is constant and so <math display="inline">F</math> is a slant Riemannian submersion.

If <math display="inline">F</math> is an anti-invariant Riemannian submersion then <math display="inline">\phi U</math> is normal, <math display="inline">\psi U=0</math> and it is equivalent to <math display="inline">\psi ^2U=0</math>. In this case <math display="inline">\theta =\frac{\pi }{2}</math> and so Eq. [[#fd000230|(4.12)]] is again satisfied.  □

By using [[#fd000105|(3.2)]], [[#fd000165|(4.1)]], [[#fd000175|(4.3)]] and [[#fd000225|(4.11)]], we have the following lemma.

==Lemma 3.                     ==

''Let  ''<math display="inline">F</math>''be a slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''with slant angle  ''<math display="inline">\theta </math>''. Then the following relations are valid for any  ''<math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>'':''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>g_M(\psi U,\psi V)=cos^2\theta (g_M(U,V)-\eta (U)\eta (V))\mbox{,}</math>
|}
| style="text-align: right;" | (4.18)
|}

We denote the complementary orthogonal distribution to <math display="inline">\omega (kerF_{{_\ast}})</math> in <math display="inline">(kerF_{{_\ast}})^{\perp }</math> by <math display="inline">\mu </math>. Then we have

<span id='fd000275'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(kerF_{{_\ast}})^{\perp }=\omega (kerF_{{_\ast}})\oplus \mu \mbox{.}</math>
|}
| style="text-align: right;" | (4.20)
|}

==Lemma 4.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. Then  ''<math display="inline">\mu </math>''is an invariant distribution of  ''<math display="inline">(kerF_{{_\ast}})^{\perp }</math>''under the endomorphism  ''<math display="inline">\phi </math>''.''

==Proof.                     ==

For <math display="inline">X\in \Gamma (\mu )</math> and <math display="inline">V\in \Gamma (kerF_{{_\ast}})</math>, from [[#fd000105|(3.2)]] and [[#fd000165|(4.1)]], we obtain

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

In a similar way, we have <math display="inline">g_M(\phi X,U)=-g_M(X,\phi U)=0</math> due to <math display="inline">\phi U\in \Gamma ((kerF_{{_\ast}})\oplus \omega (kerF_{{_\ast}}))</math> for <math display="inline">X\in \Gamma (\mu )</math> and <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>. Thus the proof of the lemma is completed. □

By means of [[#fd000270|(4.19)]], we can give the following result:

<span id='e000090'></span>

==Corollary 1.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M^{2m+1}(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N^n,g_N)</math>''. Let''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\lbrace e_1,e_2,\ldots ,e_{2m-n},\xi \rbrace </math>
|}
| style="text-align: right;" | 
|}

''be a local orthonormal frame of  ''<math display="inline">(kerF_{{_\ast}})</math>'', then  ''<math display="inline">\lbrace csc\theta \omega e_1,csc\theta \omega e_2,\ldots ,csc\theta \omega e_{2m-n}\rbrace </math>''is a local orthonormal frame of  ''<math display="inline">\omega (kerF_{{_\ast}})</math>.

By using [[#fd000275|(4.20)]] and [[#e000090|Corollary 1]], one can easily prove the following proposition:

<span id='e000095'></span>

==Proposition 3.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M^{2m+1}(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N^n,g_N)</math>''. Then  ''<math display="inline">dim(\mu )=2(n-m)</math>''. If  ''<math display="inline">\mu =\lbrace 0\rbrace </math>'', then  ''<math display="inline">n=m</math>.

By [[#fd000175|(4.3)]] and , we have

<span id='e000100'></span>

==Lemma 5.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M^{2m+1}(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N^n,g_N)</math>''. If  ''<math display="inline">e_1,e_2,\ldots ,e_k,\xi </math>''are orthogonal unit vector fields in  ''<math display="inline">(kerF_{{_\ast}})</math>'', then''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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 </math>
|}
| style="text-align: right;" | 
|}

''is a local orthonormal frame of  ''<math display="inline">(kerF_{{_\ast}})</math>''. Moreover  ''<math display="inline">dim(kerF_{{_\ast}})=2m-n+1=2k+1</math>''and  ''<math display="inline">dimN=n=2(m-k)</math>.

==Lemma 6.                     ==

''Let  ''<math display="inline">F</math>''be a slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. If  ''<math display="inline">\omega </math>''is parallel, then we have''

<span id='fd000295'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_{\psi U}\psi U=-cos^2\theta (T_UU+\eta (U)\omega U)\mbox{.}</math>
|}
| style="text-align: right;" | (4.21)
|}

==Proof.                     ==

If <math display="inline">\omega </math> is parallel, from [[#fd000200|(4.7)]], we obtain <math display="inline">CT_UV=T_U\psi V</math> for <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>. Antisymmetrizing with respect to <math display="inline">U</math>, <math display="inline">V</math> and using [[#fd000025|(2.3)]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_U\psi V=T_V\psi U\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Substituting <math display="inline">V</math> by <math display="inline">\psi U</math> in the above equation and using [[#e000065|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  [[#br000135|[27]]].

==Theorem 4.                     ==

''Let  ''<math display="inline">F</math>''be a slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. If  ''<math display="inline">\omega </math>''is parallel, then  ''<math display="inline">F</math>''is a harmonic map.''

==Proof.                     ==

From  [[#br000050|[10]]] we know that <math display="inline">F</math> is harmonic if and only if <math display="inline">F</math> has minimal fibres. Thus <math display="inline">F</math> is harmonic if and only if <math display="inline">\sum _{i=1}^{n_1}T_{e_i}e_i=0</math>. Hence using the adapted frame for slant Riemannian submersion and by the help of [[#fd000095|(2.16)]] and [[#e000100|Lemma 5]], we can write

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

Regarding <math display="inline">T_{\xi }\xi =0</math>, we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

Using [[#fd000295|(4.21)]] in the above equation, we obtain

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}</math>
|}
| style="text-align: right;" | 
|}

So we prove that <math display="inline">F</math> is harmonic.  □

Now setting <math display="inline">Q=\psi ^2</math>, we define <math display="inline">\nabla Q</math> by

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>(\nabla _UQ)V=V\nabla _UQV-Q\overset{\mbox{ˆ}}{\nabla }_UV</math>
|}
| style="text-align: right;" | 
|}

for any <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>.We give a characterization for a slant Riemannian submersion from a Sasakian manifold <math display="inline">M(\phi ,\xi ,\eta ,g_M)</math> onto a Riemannian manifold <math display="inline">(N,g_N)</math> by using the value of <math display="inline">\nabla Q</math>.

<span id='e000125'></span>

==Proposition 4.                     ==

''Let  ''<math display="inline">F</math>''be a slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. Then,  ''<math display="inline">\nabla Q=0</math>''if and only if  ''<math display="inline">F</math>''is an anti-invariant submersion.''

==Proof.                     ==

By using [[#fd000225|(4.11)]],

<span id='fd000325'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>Q\overset{\mbox{ˆ}}{\nabla }_UV=-cos^2\theta (\overset{\mbox{ˆ}}{\nabla }_UV-\eta (\overset{\mbox{ˆ}}{\nabla }_UV)\xi )</math>
|}
| style="text-align: right;" | (4.22)
|}

for each <math display="inline">U,V\in \Gamma (kerF_{{_\ast}})</math>, where <math display="inline">\theta </math> is the slant angle.

On the other hand, it follows that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>V(\nabla _UQV)=-cos^2\theta (\overset{\mbox{ˆ}}{\nabla }_UV-\eta (\overset{\mbox{ˆ}}{\nabla }_UV)\xi +g(V,\psi U)\xi +\eta (V)\psi U)\mbox{.}</math>
|}
| style="text-align: right;" | (4.23)
|}

So, from [[#fd000325|(4.22)]] and <math display="inline">\nabla Q=0</math> if and only if <math display="inline">cos^2\theta (g(V,\psi U)\xi +\eta (V)\psi U)=0</math> which implies that <math display="inline">\psi U=0</math> or <math display="inline">\theta =\frac{\pi }{2}</math>. Both the cases verify that <math display="inline">F</math> is an anti-invariant submersion.  □

We now investigate the geometry of leaves of <math display="inline">(kerF_{{_\ast}})^{\perp }</math> and <math display="inline">kerF_{{_\ast}}</math>.

==Theorem 5.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>. ''Then the distribution  ''<math display="inline">(kerF_{{_\ast}})^{\perp }</math>''defines a totally geodesic foliation on  ''<math display="inline">M</math>''if and only if''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g_M(H\nabla _XY,\omega \psi U)-sin^2\theta g_M(Y,\phi X)\eta (U)=g_M(A_XBY\\\displaystyle \omega U)+g_M(H\nabla _XCY,\omega U)\end{array}</math>
|}
| style="text-align: right;" | 
|}

''for any  ''<math display="inline">X,Y\in \Gamma ((kerF_{{_\ast}})^{\perp })</math>''and  ''<math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>.

==Proof.                     ==

From [[#fd000110|(3.3)]] and [[#fd000165|(4.1)]], we have

<span id='fd000340'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g_M(\nabla _XY,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}</math>
|}
| style="text-align: right;" | (4.24)
|}

for any <math display="inline">X,Y\in \Gamma ((kerF_{{_\ast}})^{\perp })</math> and <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math>.

Using [[#fd000110|(3.3)]] and [[#fd000165|(4.1)]] in [[#fd000340|(4.24)]], we obtain

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g_M(\nabla _XY,U)=-g_M(\nabla _XY,\psi ^2U)-g_M(\nabla _XY,\omega \psi U)+g_M(Y\\\displaystyle \phi X)\eta (U)+g_M(\nabla _X\phi Y,\omega U)\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | (4.25)
|}

By [[#fd000170|(4.2)]] and [[#fd000225|(4.11)]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle g_M(\nabla _XY,U)=cos^2\theta g_M(\nabla _XY,U)-cos^2\theta \eta (U)\eta (\nabla _XY)-g_M(\nabla _XY\\\displaystyle \omega \psi U)+g_M(Y,\phi X)\eta (U)+g_M(\nabla _XBY,\omega U)+g_M(\nabla _XCY,\omega U)\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | (4.26)
|}

Using ,  and [[#fd000130|(3.6)]] in the last equation, we obtain

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\begin{array}{l}\displaystyle sin^2\theta g_M(\nabla _XY,U)=sin^2\theta g_M(Y,\phi X)\eta (U)-g_M(H\nabla _XY\\\displaystyle \omega \psi U)+g_M(A_XBY,\omega U)+g_M(H\nabla _XCY,\omega U)\end{array}</math>
|}
| style="text-align: right;" | 
|}

which proves the theorem. □

==Proposition 5.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian manifold  ''<math display="inline">M(\phi ,\xi ,\eta ,g_M)</math>''onto a Riemannian manifold  ''<math display="inline">(N,g_N)</math>''. If the distribution  ''<math display="inline">kerF_{{_\ast}}</math>''defines a totally geodesic foliation on  ''<math display="inline">M</math>'', then  ''<math display="inline">F</math>''is an invariant submersion.''

==Proof.                     ==

By [[#fd000185|(4.5)]], if the distribution <math display="inline">kerF_{{_\ast}}</math> defines a totally geodesic foliation on <math display="inline">M</math>, then we conclude that <math display="inline">\omega U=0</math> for any <math display="inline">U\in \Gamma (kerF_{{_\ast}})</math> which shows that <math display="inline">F</math> is an invariant submersion. □

Now we establish a sharp inequality between norm squared mean curvature <math display="inline">\Vert H\Vert ^2</math> and the scalar curvature <math display="inline">\overset{\mbox{ˆ}}{\tau }</math> of fibre through <math display="inline">p\in M^5(c)</math>.

==Theorem 6.                     ==

''Let  ''<math display="inline">F</math>''be a proper slant Riemannian submersion from a Sasakian space form  ''<math display="inline">M^5(c)</math>''onto a Riemannian manifold  ''<math display="inline">(N^2,g_N)</math>''. Then, we have''

<span id='fd000360'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Vert H\Vert ^2\geq \frac{8}{9}\overset{\mbox{ˆ}}{\tau }-\frac{2}{9}[c+3+(3c+5)cos^2\theta ]</math>
|}
| style="text-align: right;" | (4.27)
|}

''where  ''<math display="inline">H</math>''denotes the mean curvature of fibres. Moreover, the equality sign of''    [[#fd000360|(4.27)]]   ''holds at a point  ''<math display="inline">p</math>''of a fibre if and only if with respect to some suitable slant orthonormal frame  ''<math display="inline">\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 </math>''at  ''<math display="inline">p</math>'', we have''

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_{11}^4=3T_{22}^4\mbox{,}\quad T_{12}^4=0\quad \mbox{   and   }\quad T_{11}^5=0\mbox{,}</math>
|}
| style="text-align: right;" | 
|}

''where  ''<math display="inline">T_{ij}^{\alpha }=g(T(e_i,e_j),e_{\alpha })</math>''for  ''<math display="inline">1\leq i,j\leq 3</math>''and  ''<math display="inline">\alpha =4,5</math>.

==Proof.                     ==

By [[#e000090|Corollary 1]], [[#e000100|Lemma 5]] and [[#e000095|Proposition 3]] we construct a slant orthonormal frame <math display="inline">\lbrace e_1,e_2,e_3,e_4,e_5\rbrace </math> defined by

<span id='fd000370'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{,}</math>
|}
| style="text-align: right;" | (4.28)
|}

where <math display="inline">e_1,e_2,e_3=\xi \in \Gamma (kerF_{{_\ast}})</math> and <math display="inline">e_4,e_5\in \Gamma ((kerF_{{_\ast}})^{\perp })</math>. Let <math display="inline">\overset{\mbox{ˆ}}{\tau }</math> be scalar curvature of fibre <math display="inline">F^{-1}(q)</math>. We choose an arbitrary point <math display="inline">p</math> of the fibre <math display="inline">F^{-1}(q)</math>. We obtain

<span id='fd000375'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | (4.29)
|}

By [[#fd000075|(2.12)]], [[#fd000080|(2.13)]] and [[#fd000135|(3.7)]], we get

<span id='fd000380'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}^4T_{22}^4+T_{11}^5T_{22}^5-\\\displaystyle -(T_{12}^4)^2-(T_{12}^5)^2\mbox{,}\end{array}</math>
|}
| style="text-align: right;" | (4.30)
|}

where <math display="inline">T_{ij}^{\alpha }=g(T(e_i,e_j),e_{\alpha })</math> for <math display="inline">1\leq i,j\leq 3</math> and <math display="inline">\alpha =4,5</math>. Using [[#e000065|Theorem 3]]  and the relation [[#fd000270|(4.19)]], one has

<span id='fd000385'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\psi e_2=-cos\theta e_1\quad \mbox{and}\quad \omega e_2=sin\theta e_5\mbox{.}</math>
|}
| style="text-align: right;" | (4.31)
|}

From [[#fd000205|(4.8)]], we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>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{.}</math>
|}
| style="text-align: right;" | 
|}

Using [[#fd000165|(4.1)]], [[#fd000170|(4.2)]], , [[#fd000370|(4.28)]] and [[#fd000385|(4.31)]] in the last relation, we obtain

<span id='fd000395'></span>
{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>sin\theta [g(T_{e_2}e_2,e_4)-g(T_{e_2}e_1,e_5)]=0\mbox{.}</math>
|}
| style="text-align: right;" | (4.32)
|}

Since the submersion is proper, Eq. [[#fd000395|(4.32)]] implies that

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>T_{22}^4=T_{12}^5\mbox{.}</math>
|}
| style="text-align: right;" | 
|}

Now we choose the unit normal vector <math display="inline">e_4\in \Gamma (ker(F_{{_\ast}}))^{\perp }</math> parallel to the mean curvature vector <math display="inline">H(p)</math> of fibre. Then we have

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | 
|}

So the relation [[#fd000380|(4.30)]] becomes

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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}^4T_{22}^4-(T_{11}^5)^2-\\\displaystyle -(T_{12}^4)^2-(T_{22}^4)^2\mbox{.}\end{array}</math>
|}
| style="text-align: right;" | (4.33)
|}

From the trivial inequality <math display="inline">(\mu -3\lambda )^2\geq 0</math>, one has <math display="inline">(\mu +\lambda )^2\geq 8(\lambda \mu -\lambda ^2)</math>. Putting <math display="inline">\mu =T_{11}^4</math> and <math display="inline">\lambda =T_{22}^4</math> in the last inequality, we find

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\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{.}</math>
|}
| style="text-align: right;" | (4.34)
|}

Using [[#fd000080|(2.13)]], we get

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\overset{\mbox{ˆ}}{K}(e_1\wedge e_3)=\overset{\mbox{ˆ}}{K}(e_2\wedge e_3)=cos^2\theta \mbox{.}</math>
|}
| style="text-align: right;" | 
|}

By [[#fd000080|(2.13)]], [[#fd000375|(4.29)]] and the last relation, we get the required inequality. Moreover, the equality sign of [[#fd000360|(4.27)]] holds at a point <math display="inline">p</math> of a fibre if and only if <math display="inline">T_{11}^4=3T_{22}^4</math>, <math display="inline">T_{12}^4=0</math> and <math display="inline">T_{11}^5=0</math>. □

==Open Problem:==

Let <math display="inline">F</math> be a slant Riemannian submersion from a Sasakian manifold <math display="inline">M(\phi ,\xi ,\eta ,g_M)</math> onto a Riemannian manifold <math display="inline">(N,g_N)</math>. In  [[#br000015|[3]]], Barrera et al. defined and studied the Maslov form of non-invariant slant submanifolds of <math display="inline">S</math>-space form <math display="inline">\tilde{M}(c)</math>. They find conditions for it to be closed. By a similar discussion in  [[#br000015|[3]]], we can define Maslov form <math display="inline">\Omega H</math> of <math display="inline">M</math> as the dual form of the vector field <math display="inline">BH</math>, that is,

{| class="formulaSCP" style="width: 100%; text-align: center;" 
|-
| 
{| style="text-align: center; margin:auto;" 
|-
| <math>\Omega H(U)=g_M(U,BH)</math>
|}
| style="text-align: right;" | 
|}

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

<math display="inline">\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 </math> is a local orthonormal frame of <math display="inline">(kerF_{{_\ast}})</math>.

==Acknowledgement==

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

==References==

<ol style='list-style-type: none;margin-left: 0px;'><li><span id='br000005'></span>
[[#br000005|[1]]] P. Alegre; Slant submanifolds of Lorentzian Sasakian and para Sasakian manifolds; Taiwanese J. Math., 17 (2) (2012), pp. 629–659</li>
<li><span id='br000010'></span>
[[#br000010|[2]]] P. Baird, J.C. Wood; Harmonic Morphisms Between Riemannian Manifolds, London Mathematical Society Monographs, vol. 29, , Oxford University Press, The Clarendon Press, Oxford (2003)</li>
<li><span id='br000015'></span>
[[#br000015|[3]]] j. Barrera, A. Carriazo, L.M. Fernandez, A. Prieto-Martin; The Maslow form in non-invariant slant submanifolds of S-space-forms; Ann. Mat. Pura Appl., 4 (2012), pp. 803–818</li>
<li><span id='br000020'></span>
[[#br000020|[4]]] D.E. Blair; Contact Manifolds in Riemannian Geometry, Lectures Notes in Mathematics, vol. 509, , Springer-Verlag, Berlin (1976), p. 146</li>
<li><span id='br000025'></span>
[[#br000025|[5]]] D.E. Blair; Riemannian geometry of contact and symplectic manifolds; Progress in Mathematics, vol. 203, Birkhauser Boston, Basel, Berlin (2002)</li>
<li><span id='br000030'></span>
[[#br000030|[6]]] J.P. Bourguignon, H.B. Lawson; Stability and isolation phenomena for Yang-mills fields; Comm. Math. Phys., 79 (1981), pp. 189–230</li>
<li><span id='br000035'></span>
[[#br000035|[7]]] J.P. Bourguignon, H.B. Lawson; A Mathematician’s visit to Kaluza-Klein theory; Rend. Semin. Mat. Torino Fasc. Spec. (1989), pp. 143–163</li>
<li><span id='br000040'></span>
[[#br000040|[8]]] J.L. Cabrerizo, A. Carriazo, L.M. Fernandez, M. Fernandez; Slant submanifolds in Sasakian manifolds; Glasg. Math. J., 42 (2000), pp. 125–138</li>
<li><span id='br000045'></span>
[[#br000045|[9]]] B.Y. Chen; Geometry of Slant Submanifolds; Katholieke Universiteit Leuven, Leuven (1990)</li>
<li><span id='br000050'></span>
[[#br000050|[10]]] J. Eells, J.H. Sampson; Harmonic mappings of Riemannian manifolds; Amer. J. Math., 86 (1964), pp. 109–160</li>
<li><span id='br000055'></span>
[[#br000055|[11]]] M. Falcitelli, S. Ianus, A.M. Pastore; Riemannian Submersions and Related Topics; World Scientific Publishing Co (2004)</li>
<li><span id='br000060'></span>
[[#br000060|[12]]] A. Gray; Pseudo-Riemannian almost product manifolds and submersions; J. Math. Mech., 16 (1967), pp. 715–737</li>
<li><span id='br000065'></span>
[[#br000065|[13]]] S. Ianus, A.M. Ionescu, R. Mazzocco, G.E. Vilcu; Riemannian submersions from almost contact metric manifolds; Abh. Math. Semin. Univ. Hambg., 81 (2011), pp. 101–114</li>
<li><span id='br000070'></span>
[[#br000070|[14]]] S. Ianus, R. Mazzocco, G.E. Vilcu; Riemannian submersions from quaternionic manifolds; Acta Appl. Math., 104 (2008), pp. 83–89</li>
<li><span id='br000075'></span>
[[#br000075|[15]]] S. Ianus, M. Visinescu; Kaluza-Klein theory with scalar fields and generalized Hopf manifolds; Class. Quantum Gravity, 4 (1987), pp. 1317–1325</li>
<li><span id='br000080'></span>
[[#br000080|[16]]] S. Ianus, M. Visinescu; ,in: G. Rassias (Ed.), Space–time Compactification and Riemannian Submersions, The Mathematical Heritage of C.F. Gauss, World Scientific, River Edge (1991), pp. 358–371</li>
<li><span id='br000085'></span>
[[#br000085|[17]]] I. Küpeli Erken, C. Murathan, Anti-Invariant Riemannian Submersions from Sasakian manifolds, arXiv:1302.4906  [math. DG].</li>
<li><span id='br000090'></span>
[[#br000090|[18]]] J.W. Lee; Anti-invariant <math display="inline">\xi ^{\perp }</math> Riemannian submersions from almost contact manifolds; Hacet. J. Math. Stat., 42 (2) (2013), pp. 231–241</li>
<li><span id='br000095'></span>
[[#br000095|[19]]] A. Lotta; Slant submanifolds in contact geometry; Bull. Math. Soc. Roumanie, 39 (1996), pp. 183–198</li>
<li><span id='br000100'></span>
[[#br000100|[20]]] C. Murathan, I. Küpeli Erken; Anti-invariant Riemannian submersions from cosymplectic manifolds; Filomat, 29 (7) (2015), pp. 1429–1444</li>
<li><span id='br000105'></span>
[[#br000105|[21]]] M.T. Mustafa; Applications of harmonic morphisms to gravity; J. Math. Phys., 41 (10) (2000), pp. 6918–6929</li>
<li><span id='br000110'></span>
[[#br000110|[22]]] B. O’Neill; The fundamental equations of submersion; Michigan Math. J., 13 (1966), pp. 459–469</li>
<li><span id='br000115'></span>
[[#br000115|[23]]] B. O’Neill; Semi-Riemannian Geometry with Applications to Relativity; Academic Press, New York-London (1983)</li>
<li><span id='br000120'></span>
[[#br000120|[24]]] K.S. Park; H–slant submersions; Bull. Korean Math. Soc., 49 (2) (2012), pp. 329–338</li>
<li><span id='br000125'></span>
[[#br000125|[25]]] K.S. Park; H-semi-invariant submersions; Taiwanese J. Math., 16 (5) (2012), pp. 1865–1878</li>
<li><span id='br000130'></span>
[[#br000130|[26]]] B. Şahin; Anti-invariant Riemannian submersions from almost Hermitian manifolds; Cent. Eur. J. Math., 8 (3) (2010), pp. 437–447</li>
<li><span id='br000135'></span>
[[#br000135|[27]]] B. Şahin; Slant submersions from almost Hermitian manifolds; Bull. Math. Soc. Sci. Math. Roumanie Tome, 54(102) (1) (2011), pp. 93–105</li>
<li><span id='br000140'></span>
[[#br000140|[28]]] B. Şahin; Semi-invariant submersions from almost Hermitian manifolds; Canad. Math. Bull., 54 (3) (2011)</li>
<li><span id='br000145'></span>
[[#br000145|[29]]] B. Watson; Almost Hermitian submersions; J. Differential Geom., 11 (1) (1976), pp. 147–165</li>
<li><span id='br000150'></span>
[[#br000150|[30]]] B. Watson; G, G<math display="inline">^{{'}}</math>- Riemannian submersions and nonlinear gauge field equations of general relativity; ,in: T. Rassias (Ed.), Global Analysis - Analysis on Manifolds, dedicated M. Morse, Teubner-Texte Math, vol. 57, , Teubner, Leipzig (1983), pp. 324–349</li>
</ol>