On semi-slant $\xi^\perp-$Riemannian submersions

TL;DR

This paper introduces and studies semi-slant 0-0-Riemannian submersions from Sasakian manifolds, generalizing existing submersion types, and explores their geometric properties, conditions for base manifolds, and examples.

Contribution

It defines semi-slant 0-0-Riemannian submersions, characterizes their geometry, and establishes conditions for base manifolds to be locally product, totally umbilical, or totally geodesic.

Findings

Characterization of semi-slant 0-0-Riemannian submersions

Conditions for base manifold to be locally product

Examples illustrating the submersions

Abstract

The aim of the present paper to define and study semi-slant $\xi^\perp-$Riemannian submersions from Sasakian manifolds onto Riemannian manifolds as a generalization of anti-invariant $\xi^\perp-$Riemannian submersions, semi-invariant $\xi^\perp-$Riemannian submersions and slant Riemannian submersions. We obtain characterizations, investigate the geometry of foliations which arise from the definition of this new submersion. After we investigate the geometry of foliations, we obtain necessary and sufficient condition for base manifold to be a locally product manifold and proving new conditions to be totally umbilical and totally geodesicness, respectively. Moreover, some examples of such submersions are mentioned.

Equations

$$g_2({\pi }_{\ast }{X}_1,{\pi }_{\ast }{X}_2)$$

$$\mathcal{T}(E_1,E_2)=\mathcal{H}{\nabla }_{\mathcal{V}{E}_1}^{{}^{M_1}}\mathcal{V}{E}_2+\mathcal{V}{\nabla }_{\mathcal{V}{E}_1}^{{}^{M_1}}\mathcal{H}{E}_2$$

$$\mathcal{A}(E_1,E_2)=\mathcal{H}{\nabla }_{\mathcal{H}{E}_1}^{{}^{M_1}}\mathcal{V}{E}_2+\mathcal{V}{\nabla }_{\mathcal{H}{E}_1}^{{}^{M_1}}\mathcal{H}{E}_2$$

$${\mathcal{T}}_{U}V={\mathcal{T}}_{V}U\text{and}{\mathcal{A}}_{X}Y=-{\mathcal{A}}_{Y}X=\frac{1}{2}\mathcal{V}[X,Y],$$

$${\nabla }_V^{{}^M}W={\mathcal{T}}_{V}W+{\hat{\nabla }}_{V}W;$$

$${\nabla }_V^{{}^M}X={\mathcal{T}}_{V}X+\mathcal{H}({\nabla }_V^{{}^M}X);$$

$${\nabla }_X^{{}^M}V=\mathcal{V}({\nabla }_X^{{}^M}V)+{\mathcal{A}}_{X}V;$$

$${\nabla }_X^{{}^M}Y={\mathcal{A}}_{X}Y+\mathcal{H}({\nabla }_X^{{}^M}Y),$$

$$(\nabla {\pi }_{\ast })(X,Y)={\nabla }_X^{{}^{\pi }}{\pi }_{\ast }Y-{\pi }_{\ast }({\nabla }_{X}Y)$$

$${\phi }^2=-I+\eta \otimes \xi ,\eta (\xi )=1$$

$$\phi \xi =0,\eta o\phi =0\text{and}{g}_M(\phi X,\phi Y)=g_M(X,Y)-\eta (X)\eta (Y),$$

$$\Phi (X,Y)=d\eta (X,Y)$$

$$[\phi ,\phi ]+2d\eta \otimes \xi =0,$$

$$({\nabla }_X^{{}^{M_1}}\phi )Y=g_M(X,Y)\xi -\eta (Y)X\text{and}{\nabla }_X^{{}^{M_1}}\xi =-\phi X,$$

$$ker{\pi }_{\ast }=D_1\oplus D_2,\phi (D_1)=D_1,$$

$$U=\mathcal{P}U+\mathcal{Q}U$$

$$Z=\mathcal{V}Z+\mathcal{H}Z$$

$$\phi V=\varphi V+\omega V$$

$$\phi X=\mathcal{B}X+\mathcal{C}X$$

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

$$g_1(\varphi U_1,V_1)=-g_1(U_1,\varphi V_1)$$

$$g_1(\omega U_1,X)=-g_1(U_1,\mathcal{B}X)$$

$$(a)\varphi D_1=D_1,(b)\omega D_1=0,$$

$$(c)\varphi D_2\subset D_2,(d)\mathcal{B}(ker{\pi }_{\ast }{)}^{\perp }=D_2,$$

$$(e){\mathcal{T}}_{U_1}\xi =\varphi U_1,(f){\hat{\nabla }}_{U_1}\xi =-\omega U_1,$$

$$(i){\varphi }^2+\mathcal{B}\omega =-id,(ii){\mathcal{C}}^2+\omega \mathcal{B}=-id,$$

$$(iii)\omega \varphi +\mathcal{C}\omega =0,(iv)\mathcal{BC}+\varphi \mathcal{B}=0,$$

$$\mathcal{B}{\mathcal{T}}_{U}V+\varphi {\hat{\nabla }}_{U}V$$

$$g_1(U,V)\xi +\mathcal{C}{\mathcal{T}}_{U}V+\omega {\hat{\nabla }}_{U}V$$

$$\varphi {\mathcal{T}}_{U}X+\mathcal{B}{\nabla }_U^{{}^{M_1}}X-\eta (X)U$$

$$\omega {\mathcal{T}}_{U}X+\mathcal{C}{\nabla }_U^{{}^{M_1}}X$$

$$g_1(X,Y)\xi -\omega {\mathcal{A}}_{X}Y+\mathcal{C}\mathcal{H}{\nabla }_X^{{}^{M_1}}Y$$

$$\varphi {\mathcal{A}}_{X}Y+\mathcal{B}\mathcal{H}{\nabla }_X^{{}^{M_1}}Y$$

Full text

1. Introduction

A differentiable map $\pi:(M_{1},g_{1})\longrightarrow(M_{2},g_{2})$ between Riemannian manifolds $(M_{1},g_{1})$ and $(M_{2},g_{2})$ is called a Riemannian submersion if $\pi_{*}$ is onto and it satisfies

[TABLE]

for $X_{1},X_{2}$ vector fields tangent to $M_{1}$, where $\pi_{*}$ denotes the derivative map. The study of Riemannian submersions were studied by O’Neill [26] and Gray [16] see also [15]. Later such submersions according to the conditions on the map $\pi:(M_{1},g_{1})\longrightarrow(M_{2},g_{2})$, we have the following submersions: Riemannian submersions [24], almost Hermitian submersions [42], invariant submersions ([11, 37, 38]), anti-invariant submersions ([2, 5, 12, 17, 34, 37, 38]), lagrangian submersions ([43, 44]), semi-invariant submersions ([27, 35]), slant submersions ([13, 14, 18, 30, 36]), semi-slant submersions [1, 19, 28, 29], quaternionic submersions [40, 41], hemi-slant submersions ([3, 39]), pointwise slant submersions [21, 32] etc. We know that Riemannian submersions have several applications both in mathematics and in physics. Indeed, Riemannian submersions have their applications in the Kaluza-Klein theory ([9], [22]), supergravity and superstring theories ([23], [25]) and Yang-Mills theory ([8], [45]). Recently, in [20], Lee defined anti-invariant $\xi^{\perp}$-Riemannian submersions from almost contact metric manifolds and then he studied the geometry of such maps. Then, in [12], Erken and Murathan introduced the notion of slant submersions from Sasakian manifolds.

On the other hand, as a generalization of anti-invariant $\xi^{\perp}$-Riemannian submersions, Akyol et.al in [4] defined the notion of semi-invariant $\xi^{\perp}-$Riemannian submersions from almost contact metric manifolds and investigate the geometry of such maps. In this paper, as a generalization of anti-invariant $\xi^{\perp}-$Riemannian submersions, semi-invariant $\xi^{\perp}-$Riemannian submersions and slant Riemannian submersions, we define semi-slant $\xi^{\perp}-$Riemannian submersions from Sasakian manifolds onto Riemannian manifolds and investigate the geometry of the total space and the base space for the existence of such submersions.

The paper is organized as follows. In Sect. 2, we give some basic informations and notions about Riemannian submersions, the second fundamental form of a map and Sasakian manifolds. In Sect. 3, we define semi-slant $\xi^{\perp}-$Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. In Sect. 4, we investigate the geometry of leaves of the horizontal distribution and the vertical distribution and show that there are certain product structures on total space of a semi-slant $\xi^{\perp}-$Riemannian submersion. In Sect. 5, we find new conditions for a semi-slant $\xi^{\perp}-$Riemannian submersion to be totally umbilical and totally geodesicness, respectively. In Sect. 6, we give lots of examples of such submersions.

2. Riemannian submersions

Let $({M_{1}}^{m_{1}},g_{1})$ and $({M_{2}}^{m_{2}},g_{2})$ be Riemannian manifolds, where $\dim(M_{1})=m_{1},$ $\dim(M_{2})=m_{2}$ and $m_{1}>m_{2}.$ A Riemannian submersion $\pi:M_{1}\longrightarrow M_{2}$ is a map of $M_{1}$ onto $M_{2}$ satisfying the following axioms:

(i) $\pi$ has maximal rank, and

(ii)The differential $\pi_{\ast}$ preserves the lenghts of horizontal vectors, that is $\pi_{*}$ is a linear isometry.

The geometry of Riemannian submersion is characterized by O’Neill’s two $(1,2)$ tensor $\mathcal{T}$ and $\mathcal{A}$ defined as follows:

[TABLE]

and

[TABLE]

for any $E_{1},E_{2}\in\Gamma(M_{1}),$ where $\nabla^{{}^{M_{1}}}$ is the Levi-Civita connection on $g_{1}.$ Note that we denote the projection morphisms on the vertical distribution and the horizontal distribution by $\mathcal{V}$ and $\mathcal{H}$, respectively. One can easily see that $\mathcal{T}$ is vertical, $\mathcal{T}_{E_{1}}=\mathcal{T}_{\mathcal{V}E_{1}}$ and $\mathcal{A}$ is horizontal, $\mathcal{A}_{E_{1}}=\mathcal{A}_{\mathcal{H}E_{1}}.$ We also note that

[TABLE]

for $X,Y\in\Gamma((ker\pi_{*})^{\bot})$ and $U,V\in\Gamma(ker\pi_{*}).$

On the other hand, from (2.1) and (2.2), we obtain

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for any $X,Y\in\Gamma((ker\pi_{\ast})^{\bot})$ and $V,W\in\Gamma(ker\pi_{\ast}).$ Moreover, if $X$ is basic then $\mathcal{H}(\nabla^{{}^{M}}_{V}X)=\mathcal{A}_{X}V.$ It is easy to see that for $U,V\in\Gamma(ker\pi_{*}),$ $\mathcal{T}_{U}V$ coincides with the fibers as the second fundamental form and $\mathcal{A}_{X}Y$ reflecting the complete integrability of the horizontal distribution.

For each $p\in M_{2}$, $\pi^{-1}(p)$ is an $(m_{1}-m_{2})$ dimensional submanifold of $M_{1}$ called a fiber. A vector field on $M_{1}$ is called vertical (resp. hozirontal) if it is always tangent to fibres. A vector field on $M_{1}$ is called horizontal if it is always orthogonal to fibres. A vector field $Z$ on $M_{1}$ is called basic if $Z$ is horizontal and $\pi-$related to a vector field $\bar{Z}$ on $M_{2}$, i.e., $\pi_{\ast}Z_{p}=\bar{Z}_{\pi_{\ast}(p)}$ for all $p\in M_{1}$.

Lemma 2.1**.**

(see [15], [26]). Let $\pi:M_{1}\longrightarrow M_{2}$ be a Riemannian submersion. If $X$ and $Y$ basic vector fields on $M_{1},$ then we get:

1. (i)

$g_{1}(X,Y)=g_{2}(\bar{X},\bar{Y})\circ\pi,$** 2. (ii)

$\mathcal{H}[X,Y]$* is a basic and $\pi_{*}\mathcal{H}[X,Y]=[\bar{X},\bar{Y}]\circ\pi;$* 3. (iii)

$\mathcal{H}(\nabla^{{}^{M_{1}}}_{X}Y)$* is a basic, *$\pi-$related to $(\nabla^{{}^{M_{2}}}_{\bar{X}}\bar{Y}),$ where $\nabla^{{}^{M_{1}}}$ and $\nabla^{{}^{M_{2}}}$ are the Levi-Civita connection on $M_{1}$ and $M_{2};$ 4. (iv)

$[X,V]\in\Gamma(ker\pi_{*})$* is vertical, for any $V\in\Gamma(ker\pi_{*}).$*

Let $(M_{1},g_{1})$ and $(M_{2},g_{2})$ be Riemannian manifolds and $\pi:M_{1}\longrightarrow M_{2}$ is a differentiable map. Then the second fundamental form of $\pi$ is given by

[TABLE]

for $X,Y\in\Gamma(TM_{1}),$ where $\nabla^{{}^{\pi}}$ is the pull back connection and $\nabla$ the Levi-Civita connections of the metrics $g_{1}$ and $g_{2}.$

Finally, let $(M_{1},g_{1})$ be a $(2m+1)-$dimensional Riemannian manifold and $TM_{1}$ denote the tangent bundle of $M_{1}.$ Then $M_{1}$ is called an almost contact metric manifold if there exists a tensor $\varphi$ of type $(1,1)$ and global vector field $\xi$ and $\eta$ is a $1-$form of $\xi$, then we have

[TABLE]

[TABLE]

where $X,Y$ are any vector fields on $M_{1}.$ In this case, $(\varphi,\xi,\eta,g_{1})$ is called the almost contact metric structure of $M_{1}.$ The almost contact metric manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ is called a contact metric manifold if

[TABLE]

for any $X,Y\in\Gamma(TM_{1}),$ where $\Phi$ is a $2-$form in $M_{1}$ defined by $\Phi(X,Y)=g_{1}(X,\varphi Y).$ The $2-$form $\Phi$ is called the fundamental $2-$form of $M_{1}.$ A contact metric structure of $M_{1}$ is said to be normal if

[TABLE]

where $[\varphi,\varphi]$ is Nijenhuis tensor of $\varphi$. Any normal contact metric manifold is called a Sasakian manifold. Moreover, if $M_{1}$ is Sasakian [7, 31], then we have

[TABLE]

where $\nabla^{{}^{M_{1}}}$ is the connection of Levi-Civita covariant differentiation.

3. Semi-slant $\xi^{\perp}-$Riemannian submersions

Definition 3.1**.**

Let $(M_{1},\varphi,\xi,\eta,g_{1})$ be a Sasakian manifold and $(M_{2},g_{2})$ be a Riemannian manifold. Suppose that there exists a Riemannian submersion $\pi:M_{1}\longrightarrow M_{2}$ such that $\xi$ is normal to $kerF_{*}$. Then $\pi:M_{1}\longrightarrow M_{2}$ is called semi-slant $\xi^{\perp}-$Riemannian submersion if there is a distribution $D_{1}\subseteq\ker\pi_{*}$ such that

[TABLE]

and the angle $\theta=\theta(U)$ between $\varphi U$ and the space $(D_{2})_{p}$ is constant for nonzero $U\in(D_{2})_{p}$ and $p\in M$, where $D_{2}$ is the orthogonal complement of $D_{1}$ in $ker\pi_{*}$. As it is, the angle $\theta$ is called the semi-slant angle of the submersion.

Now, let $\pi$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$. Then, for $U\in\Gamma(\ker\pi_{*})$, we put

[TABLE]

where $\mathcal{P}U\in\Gamma(D_{1})$ and $\mathcal{Q}U\in\Gamma(D_{2}).$ For $Z\in\Gamma(TM),$ we have

[TABLE]

where $\mathcal{V}Z\in\Gamma(ker\pi_{*})$ and $\mathcal{H}Z\in\Gamma((ker\pi_{*})^{\perp}).$ For $V\in\Gamma(ker\pi_{*}),$ we get

[TABLE]

where $\phi V$ and $\omega V$ are vertical (resp. horizontal) components of $\varphi V,$ respectively. Similarly, for any $X\in\Gamma((ker\pi_{*})^{\perp}),$ we have

[TABLE]

where $\mathcal{B}X$ (resp. $\mathcal{C}X$) is the vertical part (resp. horizontal part) of $\varphi X.$ Then, the horizontal distribution $(ker\pi_{*})^{\perp}$ is decomposed as

[TABLE]

here $\mu$ is the orthogonal complementary distribution of $\omega D_{2}$ and it is both invariant distribution of $((ker\pi_{*})^{\perp})$ with respect to $\varphi$ and contains $\xi.$ By (2.9), (3.4) and (3.5), we have

[TABLE]

and

[TABLE]

for $U_{1},V_{1}\in\Gamma(\ker\pi_{*})$ and $X\in\Gamma((\ker\pi_{*})^{\perp}).$ From (3.4), (3.5) and (3.6), we get:

Lemma 3.1**.**

Let $\pi$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$. Then we obtain

[TABLE]

for $U_{1}\in\Gamma(\ker\pi_{*})$ and $\xi\in\Gamma((ker\pi_{*})^{\perp}).$

On the other hand, using (3.4), (3.5) and the fact that $\varphi^{2}=-I+\eta\otimes\xi,$ we obtain:

Lemma 3.2**.**

Let $\pi$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$. Then we get

[TABLE]

where $I$ is the identity operator on the space of $\pi.$

Let $(M_{1},\varphi,\xi,\eta,g_{1})$ be a Sasakian manifold and $(M_{2},g_{2})$ be a Riemannian manifold. Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion. We now examine how the Sasakian structure on $M_{1}$ effects the tensor fields $\mathcal{T}$ and $\mathcal{A}$ of a semi-slant $\xi^{\perp}-$Riemannian submersion $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$.

Lemma 3.3**.**

Let $(M_{1},\varphi,\xi,\eta,g_{1})$ be a Sasakian manifold and $(M_{2},g_{2})$ a Riemannian manifold. Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion. Then we have

[TABLE]

[TABLE]

[TABLE]

for all $X,Y\in\Gamma((\ker\pi_{*})^{\perp})$ and $U,V\in\Gamma(\ker\pi_{*})$.

Proof.

Given $U,V\in\Gamma(ker\pi_{*})$, by virtue of (2.10) and (3.4), we have

[TABLE]

By using (2.3), (2.4), (3.4) and (3.5), we get

[TABLE]

In (3.15), comparing horizontal and vertical parts, we get (3.9) and (3.10). The other assertions can be obtained in a similar method.

∎

As the proof of the following theorem is similar to semi-slant submanifolds (Theorem 5.1 of [10]), we omit it.

Theorem 3.1**.**

Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2}).$ Then we have

[TABLE]

where $\theta$ denotes the semi-slant angle of $D_{2}$.

By using (2.9), (3.4), (3.7), (3.8) we get:

Lemma 3.4**.**

Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then we have

[TABLE]

for any $W_{1},W_{2}\in\Gamma(D_{2}).$

4. Integrability, Totally Geodesicness and Decomposition Theorems

In this section, we shall study the integrability and totally geodesicness of the distributions which are involved in the definition of a semi-slant $\xi^{\perp}-$Riemannian submersions and obtain decomposition theorems of such submersions.

Theorem 4.1**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then

1. (i)

$D_{1}$* is integrable $\Leftrightarrow$ $\begin{aligned} (\nabla\pi_{\ast})(U,\varphi V)-(\nabla\pi_{\ast})(V,\varphi U)\notin\Gamma(\pi_{\ast}\mu)\end{aligned}$* 2. (ii)

$D_{2}$* is integrable $\Leftrightarrow$ $\begin{aligned} g_{2}(\pi_{\ast}\omega W,(\nabla\pi_{\ast})(Z,\varphi U))+g_{2}(\pi_{\ast}\omega Z,(\nabla\pi_{\ast})(W,\varphi U))&=g_{1}(\phi W,\hat{\nabla}_{Z}\varphi U)\\ &+g_{1}(\phi Z,\hat{\nabla}_{W}\varphi U)\end{aligned}$*

for $U,V\in\Gamma(D_{1})$ and $Z,W\in\Gamma(D_{2}).$

Proof.

Given $U,V\in\Gamma(D_{1})$ and $X\in\Gamma((ker\pi_{*})^{\perp})$, since $[U,V]\in\Gamma(ker\pi_{*})$, we have $g_{1}([U,V],X)=0.$ Thus $D_{1}$ is integrable $\Leftrightarrow$ $g_{1}([U,V],Z)=0$ for $Z\in\Gamma(D_{2}).$ Since $M_{1}$ is a Sasakian manifold, by (2.9) and (2.10), we have

[TABLE]

Using (3.4) in (4.1), we get

[TABLE]

Then, by (3.16) and (2.7), we conclude that

[TABLE]

After some calculations, we obtain

[TABLE]

which proves (i). The other assertion can be obtained in a similar method. ∎

We now investigate the geometry of leaves of $D_{1}$ and $D_{2}.$

Theorem 4.2**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the distribution $D_{1}$ is parallel if and only if

[TABLE]

and

[TABLE]

for $U,V\in\Gamma(D_{1}),Z\in\Gamma(D_{2})$ and $X\in\Gamma((\ker\pi_{\ast})^{\bot})$.

Proof.

By virtue of (4.1), (3.4) and (2.3), we have

[TABLE]

for $U,V\in\Gamma(D_{1})$ and $Z\in\Gamma(D_{2}).$ Since $\pi$ is a semi-slant $\xi^{\perp}-$Riemannian submersion, using (2.3) and (2.7), we arrive

[TABLE]

or

[TABLE]

which gives (4.2). On the other hand, from (2.9) and (2.10), we get

[TABLE]

for $U,V\in\Gamma(D_{1})$ and $X\in\Gamma((\ker\pi_{\ast})^{\bot}).$ Then using (3.4) and (3.5), we conlude that

[TABLE]

Also, using (2.3), (2.4) and the character of $\pi$, we get

[TABLE]

Then (2.7) imply

[TABLE]

which gives (4.3). ∎

For $D_{2}$, we get:

Theorem 4.3**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the distribution $D_{2}$ is parallel if and only if

[TABLE]

and

[TABLE]

for any $Z,W\in\Gamma(D_{2}),U\in\Gamma(D_{1})$ and $X\in\Gamma((ker\pi_{*})^{\perp}).$

Proof.

Given $Z,W\in\Gamma(D_{2})$ and $U\in\Gamma(D_{1}),$ using (4.1), (3.4) and (2.3), we get

[TABLE]

Since $\pi$ is a semi-slant $\xi^{\perp}-$Riemannian submersion, using (2.3) and (2.7), we arrive

[TABLE]

which completes (4.4). On the other hand, by using (4.1) and(3.4) we get

[TABLE]

for all $Z,W\in\Gamma(D_{2}),U\in\Gamma(D_{1})$ and $X\in\Gamma((ker\pi_{*})^{\perp}).$ Then (2.4) and (3.16) imply that

[TABLE]

Now, using the character of $\pi$ and (2.4), we get

[TABLE]

After some calculations, we have

[TABLE]

which gives (4.5). ∎

Since $(ker\pi_{*})$ is integrable, we only study the integrability of the distribution $(ker\pi_{*})^{\perp}$ and then we investigate the geometry of leaves of $ker\pi_{*}$ and $(ker\pi_{*})^{\perp}$.

Theorem 4.4**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the distribution $(\ker\pi_{\ast})^{\bot}$ is integrable if and only if

[TABLE]

and

[TABLE]

for $X,Y\in\Gamma((ker\pi_{\ast})^{\bot}),V\in\Gamma(D_{1})$ and $W\in\Gamma(D_{2}).$

Proof.

From (4.1), (2.9) and (2.10), we have

[TABLE]

for $X,Y\in\Gamma((\ker\pi_{\ast})^{\bot})$ and $V\in\Gamma(D_{1}).$ Then by (3.5), we have

[TABLE]

From (2.4), we get

[TABLE]

Using (2.7), we obtain

[TABLE]

which gives (4.6). In a similar way, by virtue of (4.1), (2.9) and (2.10), we have

[TABLE]

for $X,Y\in\Gamma((\ker\pi_{\ast})^{\bot})$ and $W\in\Gamma(D_{2}).$ Then by (3.16) and (2.6), we arrive

[TABLE]

Now, using the character of $\pi$, (2.6) and (2.7) imply that

[TABLE]

so with some elementary calculations, we find

[TABLE]

which completes (4.7). ∎

For the geometry of leaves $(\ker\pi_{\ast})^{\bot}$, we obtain:

Theorem 4.5**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the distribution $(\ker\pi_{\ast})^{\bot}$ is parallel if and only if

[TABLE]

and

[TABLE]

for $X,Y\in\Gamma((\ker\pi_{*})^{\bot}),V\in\Gamma(D_{1})$ and $W\in\Gamma(D_{2}).$

Proof.

Given $X,Y\in\Gamma((\ker\pi_{\ast})^{\bot})$ and $V\in\Gamma(D_{1})$, by (2.9) and (2.10), we have

[TABLE]

Thus, from (3.5), we find

[TABLE]

Taking into account that $\pi$ is a semi-slant $\xi^{\perp}-$Riemannian submersion, we get

[TABLE]

By (2.7), we obtain

[TABLE]

which gives (4.8). On the other hand, for $X,Y\in\Gamma((\ker\pi_{*})^{\perp}),$ $W\in\Gamma(D_{2})$, by virtue of (2.9), (2.10), we obtain

[TABLE]

From (3.4) and (3.5), we arrive

[TABLE]

Since $\pi$ is a semi-slant $\xi^{\perp}-$Riemannian submersion, using (3.16) and the character of $\pi$ we get

[TABLE]

Now, using (2.7), we obtain

[TABLE]

which gives (4.9). ∎

Similarly, we get:

Theorem 4.6**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the distribution $(ker\pi_{*})$ is parallel if and only if

[TABLE]

for any $U\in\Gamma(D_{1}),V\in\Gamma(D_{2})$ and $X\in\Gamma((ker\pi_{\ast})^{\bot}).$

Proof.

By virtue of (2.9) and (2.10), we have

[TABLE]

for $U\in\Gamma(D_{1}),V\in\Gamma(D_{2})$ and $X\in\Gamma((ker\pi_{\ast})^{\bot}).$ Then, from (3.4), we arrive

[TABLE]

Now, by (3.4), (3.5) and (3.16), we obtain

[TABLE]

By virtue of (2.7), (2.3) and the character of $\pi,$ we arrive

[TABLE]

so with some elementary calculations, we get

[TABLE]

which completes (4.9). ∎

We now recall the following characterization for locally (usual) product Riemannian manifold from [33]. Let $g$ be a Riemannian metric tensor on the manifold $M=M_{1}\times M_{2}$ and assume that the canonical foliations $D_{M_{1}}$ and $D_{M_{2}}$ intersect perpendicularly everywhere. Then $g$ is the metric tensor of a usual product of Riemannian manifolds if and only if $D_{M_{1}}$ and $D_{M_{2}}$ are totally geodesic foliations.

By virtue of Theorem 4.2, Theorem 4.3 and Theorem 4.5, we have the following theorem;

Theorem 4.7**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the total space $M_{1}$ is a locally product manifold of the leaves of $D_{1}$, $D_{2}$ and $(ker\pi_{*})^{\perp},$ i.e., $M_{1}=M_{1}{{}_{D_{1}}}\times M_{1}{{}_{D_{2}}}\times M_{1}{{}_{(ker\pi_{*})^{\perp}}},$ if and only if

[TABLE]

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

[TABLE]

for $X,Y\in\Gamma((\ker\pi_{*})^{\bot}),$ $U,V\in\Gamma(D_{1})$ and $Z,W\in\Gamma(D_{2}).$

From Theorem 4.5 and Theorem 4.6, we have the following theorem;

Theorem 4.8**.**

Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then the total space $M_{1}$ is a locally (usual) product manifold of the leaves of $ker\pi_{*}$ and $(ker\pi_{*})^{\perp},$ i.e., $M_{1}=M_{1}{{}_{ker\pi_{*}}}\times M_{1}{{}_{(ker\pi_{*})^{\perp}}},$ if and only if

[TABLE]

[TABLE]

and

[TABLE]

for $X,Y\in\Gamma((\ker\pi_{*})^{\bot}),U,V\in\Gamma(D_{1})$ and $W\in\Gamma(D_{2}).$

5. Totally umbilical and Totally geodesicness of $\pi$

In this section, we are going to examine the totally umbilical fibres and the totally geodesicness of a semi-slant $\xi^{\perp}-$ Riemannian submersion. First we give a new condition for a semi-slant $\xi^{\perp}-$Riemannian submersion to be totally umbilical. Let $\pi$ be a Riemannian submersion from a Riemannian manifold $(M_{1},g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$. $\pi$ is called a Riemannian submersion with totally umbilical fibres if

[TABLE]

for $V,W\in\Gamma(ker\pi_{*}),$ where $H$ is mean curvature vector field of the fibres [35]. Then we have the following result.

Theorem 5.1**.**

Let $\pi:(M_{1},\varphi,\xi,\eta,g_{1})\longrightarrow(M_{2},g_{2})$ be a semi-slant $\xi^{\perp}-$Riemannian submersion with totally umbilical fibres from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2}),$ then $H\in\Gamma(\omega D_{2}).$

Proof.

From (2.3), we have

[TABLE]

for any $U_{1},U_{2}\in\Gamma(D_{1}).$ Now, using (3.4) and (3.5), we get

[TABLE]

By virtue of (2.9) and (2.10) imply that

[TABLE]

Taking inner product in above equation with $Z\in\Gamma(\mu)$, we obtain

[TABLE]

From (5.1), we have

[TABLE]

Interchanging $U_{1}$ and $U_{2}$ in (5.2), we get

[TABLE]

Subtracting (5.2) and (5.3), we get $g_{1}(H,Z)=0$ which shows that $H\in\Gamma(\omega D_{2}).$ ∎

Now, we give some conditions for a semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold to be totally geodesic map. Recall that a differential map $\pi$ between two Riemannian manifolds is called totally geodesic if $\nabla\pi_{*}=0$ [6]. It is known that the second fundamental form is symmetric.

Theorem 5.2**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then $\pi$ is a totally geodesic map if

[TABLE]

for any $X\in\Gamma((ker\pi_{*})^{\perp})$ and $Z=Z_{1}+Z_{2}\in\Gamma(TM_{1}),$ where $Z_{1}\in\Gamma(ker\pi_{\ast})$ and $Z_{2}\in\Gamma((ker\pi_{\ast})^{\perp}).$

Proof.

By virtue of (2.5), (2.9) and (2.10), we have

[TABLE]

for any $Z\in\Gamma((ker\pi_{*})^{\perp})$ and $X\in\Gamma(TM_{1})$. Now, from (2.7), we arrive

[TABLE]

By using (2.5), (2.6), (3.4) and (3.5) we get

[TABLE]

for any $Z=Z_{1}+Z_{2}\in\Gamma(TM_{1})$, where $Z_{1}\in\Gamma(ker\pi_{\ast})$ and $Z_{2}\in\Gamma((ker\pi_{\ast})^{\perp}).$ Thus taking into account the vertical parts, we obtain

[TABLE]

which gives our assertion. ∎

Theorem 5.3**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then $\pi$ is a totally geodesic map if and only if

1. (i)

$\begin{aligned} g_{1}(\hat{\nabla}_{U_{1}}\varphi V_{1},\mathcal{B}Z)=g_{1}(\mathcal{T}_{U_{1}}\mathcal{C}Z,\varphi V_{1})-g_{1}(V_{1},\phi U_{1})\eta(Z),\end{aligned}$** 2. (ii)

$\begin{aligned} g_{2}(\nabla\pi_{*}(U_{2},\omega\phi V_{2}),\pi_{*}Z)+g_{2}(\nabla\pi_{*}(U_{2},\omega V_{2}),\pi_{*}Z)&=g_{1}(\mathcal{T}_{U_{2}}\omega V_{2},\mathcal{B}Z)+g_{1}(V_{2},\phi U_{2})\eta(Z)\end{aligned}$** 3. (iii)

$\begin{aligned} g_{2}(\nabla\pi_{*}(U,\mathcal{C}X),\pi_{*}\mathcal{C}Y)-g_{2}(\nabla\pi_{*}(U,\omega\mathcal{B}X),\pi_{*}Y)&=g_{1}(\mathcal{T}_{U}\phi\mathcal{B}X,Y)-g_{1}(\mathcal{T}_{U}\mathcal{C}X,\mathcal{B}Y)\\ &\!\!+\!\!\eta(X)g_{1}(QU,\varphi Y)\!\!-\!\!\eta(Y)[U\eta(X)\!\!+\!\!g_{1}(X,\omega U)]\end{aligned}$**

for any $U_{1},V_{1}\in\Gamma(D_{1}),\ U_{2},V_{2}\in\Gamma(D_{2}),\ U\in\Gamma(ker\pi_{*})$ and $X,Y,Z\in\Gamma((ker\pi_{*})^{\perp})$.

Proof.

(i) Given $U_{1},V_{1}\in\Gamma(D_{1})$ and $Z\in\Gamma((ker\pi_{*})^{\perp})$, from (2.7), (2.9) and (2.10), we have

[TABLE]

By using (2.5) and (2.10), we obtain

[TABLE]

From (2.3), we get

[TABLE]

which gives (i).

(ii) By virtue of (2.7), (2.9) and (2.10), we get

[TABLE]

for $U_{2},V_{2}\in\Gamma(D_{2}).$ Then using (3.4), (3.5) and (2.10), we obtain

[TABLE]

Taking into account that $\pi$ is a semi-slant $\xi^{\perp}-$Riemannian submersion, using (2.3) imply that

[TABLE]

Now, from (2.7) and (3.16), we obtain

[TABLE]

so with some elementary calculations, we arrive

[TABLE]

which completes (ii).

(iii) If $X,Y\in\Gamma((\ker\pi_{\ast})^{\bot}),$ $U\in\Gamma(\ker\pi_{\ast})$, then by using (2.7), (2.9) and (2.10), we have

[TABLE]

Using (2.9), (3.4) and (3.5), we get

[TABLE]

By virtue of (2.4), (3.2), (3.4) and (3.5), we obtain

[TABLE]

By (2.7) and (2.3), we get

[TABLE]

From the above equation, we obtain (iii). ∎

In a similar way, we obtain the following lemma:

Theorem 5.4**.**

Let $\pi$ be a semi slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold $(M_{1},\varphi,\xi,\eta,g_{1})$ onto a Riemannian manifold $(M_{2},g_{2})$ with a semi-slant angle $\theta.$ Then $\pi$ is a totally geodesic map if and only if

1. (i)

$\mathcal{C}(\mathcal{T}_{U}\phi V+\nabla^{{}^{M_{1}}}_{U}\omega V)+\omega(\hat{\nabla}_{U}\phi V+\mathcal{T}_{U}\omega V)+g_{1}(\mathcal{P}V,\phi U)\xi=0.$** 2. (ii)

$\mathcal{C}(\mathcal{A}_{X}\phi U+\mathcal{H}\nabla^{{}^{M_{1}}}_{X}\omega U)+\omega(\mathcal{A}_{X}\omega U+\mathcal{V}\nabla^{{}^{M_{1}}}_{X}\phi U)+g_{1}(QU,\mathcal{B}X)\xi=0.$** 3. (iii)

$\mathcal{C}(\mathcal{T}_{U_{1}}\phi V_{1}+\mathcal{H}\nabla^{{}^{M_{1}}}_{U_{1}}\phi V_{1})+\omega(\mathcal{T}_{U_{1}}\omega V_{1}+\mathcal{V}\nabla^{{}^{M_{1}}}_{U_{1}}\phi V_{1})=0,$**

for $U_{1}\in\Gamma(D_{1}),\ V_{1}\in\Gamma(D_{2}),\ U,V\in\Gamma(ker\pi_{*})$ and $X\in\Gamma((ker\pi_{*})^{\perp})$.

6. Examples

Example 6.1**.**

Every invariant submersion from a Sasakian manifold to a Riemannian manifold is a semi-slant $\xi^{\perp}-$Riemannian submersion with $D_{2}=\{0\}$ and $\theta={0}$.

Example 6.2**.**

Every slant Riemannian submersion from a Sasakian manifold to a Riemannian manifold is a semi-slant $\xi^{\perp}-$Riemannian submersion with $D_{1}=\{0\}$.

Now, we construct some non-trivial examples of semi-slant $\xi^{\perp}-$Riemannian submersion from a Sasakian manifold. Let $(\mathbb{R}^{2n+1},g,\varphi,\xi,\eta)$ denote the manifold $\mathbb{R}^{2n+1}$ with its usual Sasakian structure given by

[TABLE]

[TABLE]

[TABLE]

where $(x^{1},...,x^{n},y^{1},...,y^{n},z)$ are the Cartesian coordinates. Throughout this section, we will use this notation.

Example 6.3**.**

Let $F$ be a submersion defined by

[TABLE]

with $\alpha\in(0,\frac{\pi}{2}).$ Then it follows that

[TABLE]

and

[TABLE]

Hence we have $\varphi Z_{1}=-Z_{2}$, $\varphi Z_{2}=Z_{1}$. Thus it follows that $D_{1}=span\{Z_{1},Z_{2}\}$ and $D_{2}=span\{Z_{3},Z_{4}\}$ is a slant distribution with slant angle $\theta=\alpha.$ Thus $F$ is a semi-slant submersion with semi-slant angle $\theta.$ Also by direct computations, we obtain

[TABLE]

[TABLE]

where $g_{1}$ and $g_{2}$ denote the standard metrics (inner products) of $\mathbb{R}^{9}$ and $\mathbb{R}^{5}$. Thus $F$ is a semi-slant $\xi^{\perp}-$Riemannian submersion.

Example 6.4**.**

Let $F$ be a submersion defined by

[TABLE]

Then the submersion $F$ is a semi-slant $\xi^{\perp}-$ Riemannian submersion such that $D_{1}=<\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial y_{1}}>$ and $D_{2}=<\frac{\partial}{\partial x_{2}}+\frac{\partial}{\partial y_{3}},\frac{\partial}{\partial x_{3}}>$ with semi-slant angle $\alpha=\frac{\pi}{4}.$

Example 6.5**.**

Let $F$ be a submersion defined by

[TABLE]

with $\alpha\in(0,\frac{\pi}{2}).$ Then the submersion $F$ is a semi-slant $\xi^{\perp}-$ Riemannian submersion such that $D_{1}=<\frac{\partial}{\partial x_{1}},\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial y_{1}},\frac{\partial}{\partial y_{2}}>$ and $D_{2}=<-\cos\alpha\frac{\partial}{\partial x_{3}}-\sin\alpha\frac{\partial}{\partial x_{4}},\frac{\partial}{\partial y_{3}}>$ with semi-slant angle $\theta=\alpha.$

Example 6.6**.**

Let $F$ be a submersion defined by

[TABLE]

*Then the submersion $F$ is a semi-slant $\xi^{\perp}-$ Riemannian submersion such that $D_{1}=<\frac{\partial}{\partial x_{1}}+\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial y_{1}}+\frac{\partial}{\partial y_{2}},\frac{\partial}{\partial x_{3}}-\frac{\partial}{\partial x_{2}},\frac{\partial}{\partial y_{3}}-\frac{\partial}{\partial y_{4}}>$ and $D_{2}=<\frac{\partial}{\partial x_{5}}+\frac{\partial}{\partial x_{6}},\frac{\partial}{\partial y_{6}}>$ with semi-slant angle $\alpha=\frac{\pi}{4}.$

**Acknowledgement

**This paper is supported by Bingöl University research project (BAP-FEF.2016.00.011).