Semi-Slant $\xi ^\perp$-, Hemi-Slant $\xi ^\perp$-Riemannian Submersions and Quasi Hemi-Slant Submanifolds

Abstract

A differentiable map $\pi : (M, g_M) \longrightarrow (N, g_N)$ between Riemannian manifolds $(M, g_M)$ and $(N, g_N)$ is called a Riemannian submersion if $\pi _*$ is onto and it satisfies

$$\begin{aligned} g_N(\pi _*X_1, \pi _*X_2 )&=g_M(X_1,X_2) \end{aligned}$$

for $X_1, X_2$ vector fields tangent to M, where $\pi _*$ denotes the derivative map.

1 Introduction

A differentiable map $\pi : (M, g_M) \longrightarrow (N, g_N)$ between Riemannian manifolds $(M, g_M)$ and $(N, g_N)$ is called a Riemannian submersion if $\pi _*$ is onto and it satisfies

$$\begin{aligned} g_N(\pi _*X_1, \pi _*X_2 )&=g_M(X_1,X_2) \end{aligned}$$

(1.1)

for $X_1, X_2$ vector fields tangent to M, where $\pi _*$ denotes the derivative map. The study of Riemannian submersions were studied by O’Neill [1] and Gray [2] see also [3]. Riemannian submersions have several applications in mathematical physics. Indeed, Riemannian submersions have their applications in the Yang–Mills theory [42, 43], Kaluza–Klein theory [44, 45], supergravity and superstring theories [46, 47] and more. Later, such submersions according to the conditions on the map $\pi : (M, g_M) \longrightarrow (N, g_N)$, we have the following submersions: Riemannian submersions [4], almost Hermitian submersions [5], invariant submersions [6,7,8], anti-invariant submersions [7,8,9,10,11,12,13], lagrangian submersions [14, 15], semi-invariant submersions [16, 17], slant submersions [18,19,20,21,22], semi-slant submersions [23,24,25,26], quaternionic submersions [27, 28], hemi-slant submersions [29, 30], pointwise slant submersions [31, 32], etc. In [33], Lee defined anti-invariant $\xi ^\perp$-Riemannian submersions from almost contact metric manifolds and studied the geometry of such maps.

As a generalization of anti-invariant $\xi ^\perp$-Riemannian submersions, Akyol et al. in [34] defined the notion of semi-invariant $\xi ^\perp$-Riemannian submersions from almost contact metric manifolds and investigated the geometry of such maps. In 2017, Mehmet et al. [35], as a generalization of anti-invariant $\xi ^\perp$-Riemannian submersions, semi-invariant $\xi ^\perp$-Riemannian submersions and slant Riemannian submersions, defined and studied semi-slant $\xi ^\perp$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. Very recently Ramazan Sari and Mehmet Akif Akyol [36] also introduced and studied Hemi-slant $\xi ^\perp$-submersions and obtained interesting results. On the other hand, in 1996, using Chen’s notion on slant submanifold, Lotta [37] introduced the notion of slant submanifold in almost contact metric manifold which was further generalized as semi-slant, hemi-slant and bi-slant submanifolds. Motivated from these studies, Rajendra Prasad et al. introduced and studied quasi hemi-slant submanifolds of cosymplectic manifolds.

The aim of this chapter is to discuss briefly some results of semi-slant $\xi ^\perp$-submersions [35], hemi-slant $\xi ^\perp$-submersions [36] and quasi hemi-slant submanifolds [38].

2 Riemannian Submersions

Let $({M},g_{M})$ and $({N},g_{N})$ be two Riemannian manifolds. A Riemannian submersion $\pi :M\longrightarrow N$ is a map of M onto N satisfying the following axioms:

• (i) $\pi$ has maximal rank, and

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

The geometry of Riemannian submersion is characterized by O’Neill’s tensors $\mathcal {T}$ and $\mathcal {A}$ defined as follows:

$$\begin{aligned} \mathcal {T}(E_1,E_2)=\mathcal {H}\nabla ^{^{M}}_{\mathcal {V}E_1}\mathcal {V}E_2+\mathcal {V} \nabla ^{^{M}}_{\mathcal {V}E_1}\mathcal {H}E_2 \end{aligned}$$

(2.1)

and

$$\begin{aligned} \mathcal {A}(E_1,E_2)=\mathcal {H}\nabla ^{^{M}}_{\mathcal {H}E_1}\mathcal {V}E_2 +\mathcal {V}\nabla ^{^{M}}_{\mathcal {H}E_1}\mathcal {H}E_2 \end{aligned}$$

(2.2)

for any $E_1, E_2\in \Gamma (M),$ where $\nabla ^{^{M}}$ is the Levi-Civita connection on $g_{M}.$ 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

$$\mathcal {T}_UV=\mathcal {T}_VU\,\, \text {and}\,\,\mathcal {A}_XY=-\mathcal {A}_YX=\frac{1}{2}\mathcal {V}[X,Y], $$

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

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

(2.3)

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

(2.4)

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

(2.5)

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

(2.6)

for any $X, Y\in \Gamma ((ker\pi _{*})^{\bot })$ and $V, W\in \Gamma (ker\pi _{*}).$ Moreover, if X is basic, then $\mathcal {H}(\nabla ^{^M}_{V}X)=\mathcal {A}_{X}V.$ It is easy to see that for $U,V\in \Gamma (ker\pi _*),$ $\mathcal {T}_UV$ coincides with the fibres as the second fundamental form and $\mathcal {A}_XY$ reflecting the complete integrability of the horizontal distribution.A vector field on M is called vertical if it is always tangent to fibres. A vector field on M is called horizontal if it is always orthogonal to fibres. A vector field Z on M is called basic if Z is horizontal and $\pi$-related to a vector field $\bar{Z}$ on N, i.e., $\pi _{*}Z_{p}=\bar{Z}_{\pi _{*}(p)}$ for all $p\in M$.

Lemma 2.1

(see [1, 3]) Let $\pi : M\longrightarrow N$ be a Riemannian submersion. If X and Y basic vector fields on M,  then we get:

• (i) $g_{M}(X,Y)=g_{N}(\bar{X},\bar{Y})\circ \pi ,$

• (ii) $\mathcal {H}[X,Y]$ is a basic and $\pi _*\mathcal {H}[X,Y]= [\bar{X},\bar{Y}]\circ \pi ;$

• (iii) $\mathcal {H}(\nabla ^{^{M}}_{X}Y)$ is a basic, $\pi$-related to $(\nabla ^{^{N}}_{\bar{X}}\bar{Y}),$ where $\nabla ^{^{M}}$ and $\nabla ^{^{N}}$ are the Levi-Civita connection on M and N; 

• (iv) $[X,V]\in \Gamma (ker\pi _*)$ is vertical, for any $V\in \Gamma (ker\pi _*).$

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

$$\begin{aligned} (\nabla \pi _*)(X,Y)=\nabla ^{^\pi }_{X}\pi _*Y-\pi _*(\nabla _{X}Y) \end{aligned}$$

(2.7)

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

Finally, let $(M,g_{M})$ be a $(2m + 1)$-dimensional Riemannian manifold and TM denote the tangent bundle of M. Then M 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

$$\begin{aligned} \varphi ^{2}=-I+\eta \otimes \xi ,\ \ \ \eta (\xi )=1 \end{aligned}$$

(2.8)

$$\begin{aligned} \varphi \xi =0,\ \ \eta o\varphi =0\ \ \ \text {and} \ \ g_{M}(\varphi X,\varphi Y)=g_{M}(X,Y)-\eta (X)\eta (Y), \end{aligned}$$

(2.9)

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

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

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

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

where $[\varphi ,\varphi ]$ is Nijenhuis tensor of $\varphi$. Any normal contact metric manifold is called a Sasakian manifold. Moreover, if M is Sasakian [39, 40], then we have

$$\begin{aligned} (\nabla ^{^{M}}_{X}\varphi )Y=g_{M}(X,Y)\xi -\eta (Y)X \ \ \text {and} \ \ \nabla ^{^{M}}_{X}\xi =-\varphi X, \end{aligned}$$

(2.10)

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

3 Semi-slant $\xi ^\perp$-Riemannian Submersions

In 2017, Mehmet et al. [35], as a generalization of anti-invariant $\xi ^\perp$-Riemannian submersions, semi-invariant $\xi ^\perp$-Riemannian submersions and slant Riemannian submersions, defined and studied semi-slant $\xi ^\perp$-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. In this Sect. 3, we will discuss some results of this paper briefly.

Definition 3.1

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

$$\begin{aligned} ker\pi _{*}=D_{1}\oplus D_{2},\, \, \, \varphi (D_{1})=D_{1}, \end{aligned}$$

(3.1)

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 $(M,\varphi ,\xi ,\eta ,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then, for $U\in \Gamma (\ker \pi _{*})$, we put

$$\begin{aligned} U=\mathcal {P}U+\mathcal {Q}U \end{aligned}$$

(3.2)

where $\mathcal {P}U\in \Gamma (D_1)$ and $\mathcal {Q}U\in \Gamma (D_2).$ For $Z\in \Gamma (TM),$ we have

$$\begin{aligned} Z=\mathcal {V}Z+\mathcal {H}Z \end{aligned}$$

(3.3)

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

$$\begin{aligned} \varphi V=\phi V+\omega V \end{aligned}$$

(3.4)

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

$$\begin{aligned} \varphi X=\mathcal {B}X+\mathcal {C}X \end{aligned}$$

(3.5)

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

$$\begin{aligned} (\ker \pi _*)^\perp =\omega D_2\oplus \mu , \end{aligned}$$

(3.6)

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

$$\begin{aligned} g_{M}(\phi U_1, V_1)=-g_{M}(U_1,\phi V_1) \end{aligned}$$

(3.7)

and

$$\begin{aligned} g_{M}(\omega U_1, X)=-g_{M}(U_1,\mathcal {B}X) \end{aligned}$$

(3.8)

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 have

Lemma 3.2

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

$$\begin{aligned}&(a)\ \ \phi D_1=D_1,\,\,\,\,\,\, (b)\ \ \omega D_1=0,\\&(c)\ \ \phi D_2\subset D_2,\,\,\,\,\,\, (d)\ \ \mathcal {B}(ker\pi _*)^\perp =D_2,\\&(e)\ \ \mathcal {T}_{U_1}\xi =\phi U_1,\,\,\,\,(f)\ \ \hat{\nabla }_{U_1}\xi =-\omega U_1, \end{aligned}$$

for $U_1\in \Gamma (\ker \pi _{*})$ and $\xi \in \Gamma ((ker\pi _*)^\perp ).$

Using (3.4), (3.5) and the fact that $\varphi ^2=-I+\eta \otimes \xi ,$ we have

Lemma 3.3

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

$$\begin{aligned}&(i)\ \ \phi ^2+\mathcal {B}\omega =-id,\,\,\,\,\,\,\, (ii)\ \ \mathcal {C}^2+\omega \mathcal {B}=-id,\\&(iii)\ \ \omega \phi +\mathcal {C}\omega =0,\,\,\,\,\,\,\, (iv)\ \ \mathcal {B}\mathcal {C}+\phi \mathcal {B}=0, \end{aligned}$$

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

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

Lemma 3.4

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

$$\begin{aligned} \mathcal {B}\mathcal {T}_{U}V+\phi \hat{\nabla }_{U}V&=\hat{\nabla }_{U}\phi V+\mathcal {T}_{U}\omega V, \end{aligned}$$

(3.9)

$$\begin{aligned} g_{M}(U,V)\xi +\mathcal {C}\mathcal {T}_{U}V+\omega \hat{\nabla }_{U}V&=\mathcal {T}_{U}\phi V+\mathcal {H}\nabla ^{^{M}}_{U}\omega V, \end{aligned}$$

(3.10)

$$\begin{aligned} \phi \mathcal {T}_{U}X+\mathcal {B}\nabla ^{^{M}}_{U}X-\eta (X)U&=\hat{\nabla }_{U}\mathcal {B}X+\mathcal {T}_{U}\mathcal {C}X, \end{aligned}$$

(3.11)

$$\begin{aligned} \omega \mathcal {T}_{U}X+\mathcal {C}\nabla ^{^{M}}_{U}X&=\mathcal {T}_{U}\mathcal {B}X+\mathcal {H}\nabla ^{^{M}}_{U}\mathcal {C}X, \end{aligned}$$

(3.12)

$$\begin{aligned} g_{M}(X,Y)\xi -\omega \mathcal {A}_{X}Y+\mathcal {C}\mathcal {H}\nabla ^{^{M}}_{X}Y&=\mathcal {A}_{X}\mathcal {B}Y+\nabla ^{^{M}}_{X}\mathcal {C}Y+\eta (Y)X, \end{aligned}$$

(3.13)

$$\begin{aligned} \phi \mathcal {A}_{X}Y+\mathcal {B}\mathcal {H}\nabla ^{^{M}}_{X}Y&=\mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Y+\mathcal {A}_{X}\mathcal {C}Y, \end{aligned}$$

(3.14)

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

$$g_{M}(U,V)\xi -\eta (V)U=\nabla ^{^{M}}_{U}\phi V+\nabla ^{^{M}}_{U}\omega V-\varphi \nabla ^{^{M}}_{U}V.$$

Making use of (2.3), (2.4), (3.4) and (3.5), we have

$$\begin{aligned} g_{M}(U,V)\xi&=\mathcal {T}_{U}\phi V+\hat{\nabla }_{U}\phi V+\mathcal {T}_{U}\omega V+\mathcal {H}\nabla ^{^{M}}_{U}\omega V \nonumber \\&-\mathcal {B}\mathcal {T}_{U}V-\mathcal {C}\mathcal {T}_{U}V-\phi \hat{\nabla }_{U}V-\omega \hat{\nabla }_{U}V. \end{aligned}$$

(3.15)

Comparing horizontal and vertical parts, we get (3.9) and (3.10). The other assertions can be obtained in a similar method.    $\square$

Theorem 3.5

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

$$\begin{aligned} \phi ^{2}W=-\cos ^2{\theta }W,\,\,\ W\in \Gamma (D_2), \end{aligned}$$

(3.16)

where $\theta$ denotes the semi-slant angle of $D_2$.

Lemma 3.6

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

$$\begin{aligned} g_{M}(\phi W_1,\phi W_2)&=\cos ^{2}\theta g_{M}(W_1,W_2), \end{aligned}$$

(3.17)

$$\begin{aligned} g_{M}(\omega W_1,\omega W_2)&=\sin ^{2}\theta g_{M}(W_1,W_2), \end{aligned}$$

(3.18)

for any $W_1, W_2\in \Gamma (D_2).$

3.1 Integrable and Parallel Distributions

In this section, we will discuss integrability conditions of the distributions involved in the definition of a semi-slant $\xi ^\perp$-Riemannian submersion. First, we have

Theorem 3.7

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

• (i) $D_{1}$ is integrable $\Leftrightarrow$ $\begin{aligned} (\nabla \pi _{*})(U,\varphi V)-(\nabla \pi _{*})(V,\varphi U)\notin \Gamma (\pi _{*}\mu ) \end{aligned}$

• (ii) $D_{2}$ is integrable $\Leftrightarrow$$\begin{aligned} g_{N}(\pi _{*}\omega W,(\nabla \pi _{*})(Z,\varphi U))+g_{N}(\pi _{*}\omega Z,(\nabla \pi _{*})(W,\varphi U))&=g_{M}(\phi W,\hat{\nabla }_{Z}\varphi U)\\&+g_{M}(\phi Z,\hat{\nabla }_{W}\varphi U) \end{aligned}$

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

Proof

For $U, V\in \Gamma (D_{1})$ and $X\in \Gamma ((ker\pi _*)^\perp )$, since $[U,V]\in \Gamma (ker\pi _*)$, we have $g_{M}([U,V],X)=0.$ Thus, $D_1$ is integrable $\Leftrightarrow$ $g_{M}([U,V],Z)=0$ for $Z\in \Gamma (D_2).$ Since M is a Sasakian manifold, by (2.9) and (2.10), we have

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,Z)&=g_{M}(\nabla ^{^{M}}_{U}\varphi V-g_{M}(U,V)\xi -\eta (V)U,\varphi Z)\nonumber \\&=g_{M}(\nabla ^{^{M}}_{U}\varphi V,\varphi Z). \end{aligned}$$

(3.19)

Using (3.4) in (3.19), we get

$$\begin{aligned} g_{M}([U,V],Z)&=-g_{M}(\nabla ^{^{M}}_{U}V,\varphi \phi Z)+g_{M}(\mathcal {H}\nabla ^{^{M}}_{U}\varphi V,wZ)-g_{M}(\nabla ^{^{M}}_{V}U,\varphi \phi Z)-g_{M}(\mathcal {H}\nabla ^{^{M}}_{V}\varphi U,wZ). \end{aligned}$$

Now, by using (2.7) and (3.16), we get

$$\begin{aligned} g_{M}([U,V],Z)&=\cos ^{2}\theta g_{M}(\nabla ^{^{M}}_{U}V,Z)-g_{N}((\nabla \pi _{*})(U,\varphi V)+\nabla _{U}^{\pi }\pi _{*}\varphi V,\pi _{*}wZ) \\&-\cos ^{2}\theta g_{M}(\nabla ^{^{M}}_{V}U,Z)+g_{N}((\nabla \pi _{*})(V,\varphi U)+\nabla _{V}^{\pi }\pi _{*}\varphi U,\pi _{*}wZ). \end{aligned}$$

Thus, we have

$$\begin{aligned} (\sin ^{2}\theta )g_{M}([U,V],Z)&=-g_{N}((\nabla \pi _{*})(U,\varphi V)-(\nabla \pi _{*})(V,\varphi U),\pi _{*}wZ), \end{aligned}$$

   $\square$

which completes the proof.

Now for the geometry of leaves of $D_1$, we have

Theorem 3.8

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

$$\begin{aligned} g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}\omega Z)&=g_{M}(\mathcal {T}_{U}\omega \phi Z,V) \end{aligned}$$

(3.20)

and

$$\begin{aligned} -g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}\mathcal {C}X)&=g_{M}(V,\hat{\nabla }_{U}\phi \mathcal {B}X+\mathcal {T}_{U}\omega \mathcal {B}X) +g_{M}(V,\varphi U)\eta (X) \end{aligned}$$

(3.21)

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

Proof

Making use of (3.19), (3.4) and (2.3), for $U, V\in \Gamma (D_{1})$ and $Z\in \Gamma (D_{2})$, we have

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,Z)&=-g_{M}(\nabla ^{^{M}}_{U}V,\phi ^{2}Z)-g_{M}(\nabla ^{^{M}}_{U}V,\omega \phi Z) +g_{M}(\mathcal {H}\nabla ^{^{M}}_{U}\varphi V,\omega Z). \end{aligned}$$

By virtue of (2.7) and (3.16), we get

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,Z)&=\cos ^{2}\theta g_{M}(\nabla ^{^{M}}_{U}V,Z)-g_{M}(\mathcal {T}_{U}V,w\phi Z)+g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}(wZ)) \end{aligned}$$

or

$$\begin{aligned} \sin ^{2}\theta g_{M}(\nabla ^{^{M}}_{U}V,Z)=-g_{M}(\mathcal {T}_{U}w\phi Z,V)+g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}(wZ)), \end{aligned}$$

which gives (3.20). On the other hand, from (2.9) and (2.10), we have

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,X)&=g_{M}(\nabla ^{^{M}}_{U}\varphi V,\varphi X)+g_{M}(V,\varphi U)\eta (X) \end{aligned}$$

for $U,V\in \Gamma (D_{1})$ and $X\in \Gamma ((\ker \pi _{*})^{\bot }).$ By using (3.5), we obtain

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,X)&=g_{M}(V,\nabla ^{^{M}}_{U}\phi \mathcal {B}X)+g_{M}(V,\nabla ^{^{M}}_{U}\omega \mathcal {B}X) +g_{M}(\mathcal {C}X,\mathcal {H}\nabla ^{^{M}}_{U}\varphi V)+g_{M}(V,\varphi U)\eta (X). \end{aligned}$$

Taking into account of (2.3), we write

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,X)&=g_{M}(V,\mathcal {T}_{U}\phi \mathcal {B}X+\hat{\nabla }_{U}\phi \mathcal {B}X)+g_{M}(V,\mathcal {T}_{U}\omega \mathcal {B}X+\mathcal {H}\nabla ^{^{M}}_{U}\omega \mathcal {B}X) \\&-g_{N}(\pi _{*}(\mathcal {C}X),\pi _{*}(\mathcal {H}\nabla ^{^{M}}_{U}\varphi V))+g_{M}(V,\varphi U)\eta (X) \end{aligned}$$

hence,

$$\begin{aligned} g_{M}(\nabla ^{^{M}}_{U}V,X)&=g_{M}(V,\hat{\nabla }_{U}\phi \mathcal {B}X)+g_{M}(V,\mathcal {T}_{U}\omega \mathcal {B}X)+g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}\mathcal {C}X)\\&+g_{M}(V,\varphi U)\eta (X). \end{aligned}$$

which gives (3.21). This completes the assertion.    $\square$

Similarly for $D_{2}$, we have:

Theorem 3.9

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

$$\begin{aligned} g_{N}(\pi _{*}\omega W,(\nabla \pi _{*})(Z,\varphi U))&=g_{M}(\phi W,\hat{\nabla }_{Z}\varphi U) \end{aligned}$$

(3.22)

and

$$\begin{aligned} g_{N}((\nabla \pi _{*})(Z,\omega W),\pi _{*}(X))-g_{N}((\nabla \pi _{*})(Z,\omega \phi W),\pi _{*}(X))&=g_{M}(\mathcal {T}_{Z}\omega W,\mathcal {B}X)+g_{M}(W,\varphi Z)\eta (X) \end{aligned}$$

(3.23)

for any $Z,W\in \Gamma (D_{2}), U\in \Gamma (D_1)$ and $X\in \Gamma ((ker\pi _*)^\perp ).$

Theorem 3.10

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

$$\begin{aligned} g_{N}((\nabla \pi _{*})(Y,\phi V),\pi _{*}(X))+g_{N}((\nabla \pi _{*})(X,\phi V),\pi _{*}(X))&=g_{M}(\phi V,\mathcal {V}(\nabla ^{^{M}}_{X}\mathcal {B}Y+\nabla ^{^{M}}_{Y}\mathcal {B}X)) \end{aligned}$$

(3.24)

and

$$\begin{aligned} g_{N}((\nabla \pi _{*})(X,\mathcal {C}Y)-(\nabla \pi _{*})(Y,\mathcal {C}X),\pi _{*}\omega W)&=g_{M}(\mathcal {A}_{X}\mathcal {B}Y+\mathcal {A}_{Y}\mathcal {B}X,\omega W)\nonumber \\&+\eta (Y)g_{M}(X,\omega W)-\eta (X)g_{M}(Y,\omega W) \end{aligned}$$

(3.25)

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

Proof

Using (3.19), (2.9) and (2.10), we have for $X,Y\in \Gamma ((\ker \pi _{*})^{\bot })$ and $V\in \Gamma (D_{1}).$

$$\begin{aligned} g_{M}([X,Y],V)&=g_{M}(\nabla ^{^{M}}_{X}\varphi Y,\varphi V)-g_{M}(\nabla ^{^{M}}_{Y}\varphi X,\varphi V). \end{aligned}$$

Now, by using (3.5), we obtain

$$\begin{aligned} g_{M}([X,Y],V)&=-g_{M}(\mathcal {B}Y,\nabla ^{^{M}}_{X}\varphi V)-g_{M}(\mathcal {C}Y,\nabla ^{^{M}}_{X}\varphi V)+g_{M}(\mathcal {B}X,\nabla ^{^{M}}_{Y}\varphi V)+g_{M}(\mathcal {C}X,\nabla ^{^{M}}_{Y}\varphi V). \end{aligned}$$

By using (2.5) and taking into account of the property of the map, we have

$$\begin{aligned} g_{M}([X,Y],V)&=g_{M}(\varphi V,\mathcal {A}_{Y}\mathcal {B}X+\mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Y)-g_{N}(\pi _{*}(\mathcal {C}Y),\pi _{*}(\nabla ^{^{M}}_{X}\varphi V))\\&-g_{M}(\varphi V,\mathcal {A}_{X}\mathcal {B}Y+\mathcal {V}\nabla ^{^{M}}_{Y}\mathcal {B}X) -g_{N}(\pi _{*}(\mathcal {C}X),\pi _{*}(\nabla ^{^{M}}_{Y}\varphi V)). \end{aligned}$$

Thus, we have

$$\begin{aligned} g_{M}([X,Y],V)&=g_{M}(\varphi V,\mathcal {V}(\nabla ^{^{M}}_{X}\mathcal {B}Y-\nabla ^{^{M}}_{Y}\mathcal {B}X))+g_{N}(\pi _{*}(\mathcal {C}Y),(\nabla \pi _{*})(X, \varphi V))\\&-g_{N}(\pi _{*}(\mathcal {C}X),(\nabla \pi _{*})(Y,\varphi V)), \end{aligned}$$

which gives (3.24). In a similar way, by virtue of (3.19), (2.9) and (2.10), we have for $X,Y\in \Gamma ((\ker \pi _{*})^{\bot })$ and $W\in \Gamma (D_{2}),$

$$\begin{aligned} g_{M}([X,Y],W)&=g_{M}(\varphi \nabla ^{^{M}}_{X}Y,\phi W)+g_{M}(\varphi \nabla ^{^{M}}_{X}Y,\omega W)+\eta (Y)g_{M}(X,\omega W) \\&-g_{M}(\varphi \nabla ^{^{M}}_{Y}X,\phi W)-g_{M}(\varphi \nabla ^{^{M}}_{Y}X,\omega W)-\eta (X)g_{M}(Y,\omega W). \end{aligned}$$

By virtue of (3.5) and (3.6), we have

$$\begin{aligned} g_{M}([X,Y],W)&=-g_{M}(\nabla ^{^{M}}_{X}Y,\phi ^{2}W)-g_{M}(\nabla ^{^{M}}_{X}Y,\omega \phi W)+g_{M}(\nabla ^{^{M}}_{X}\mathcal {B}Y,\omega W) +g_{M}(\nabla ^{^{M}}_{X}\mathcal {C}Y,\omega W) \\&-g_{M}(\nabla ^{^{M}}_{Y}X,\phi ^{2}W)-g_{M}(\nabla ^{^{M}}_{Y}X,\omega \phi W)+g_{M}(\nabla ^{^{M}}_{Y}\mathcal {B}X,\omega W) +g_{M}(\nabla ^{^{M}}_{Y}\mathcal {C}X,\omega W)\\&+\eta (Y)g_{M}(X,\omega W)-\eta (X)g_{M}(Y,\omega W). \end{aligned}$$

Now, by using (3.16) and the property of the map, we get

$$\begin{aligned} g_{M}([X,Y],W)&=\cos ^{2}\theta g_{M}([X,Y],W)+g_{N}((\nabla \pi _{*})(X,Y),\omega \phi W)+g_{M}(\mathcal {A}_{X}\mathcal {B}Y,\omega W)\\&-g_{N}((\nabla \pi _{*})(X,\mathcal {C}Y),\pi _{*}\omega W)-g_{N}((\nabla \pi _{*})(Y,X),\omega \phi W)+g_{M}(\mathcal {A}_{Y}\mathcal {B}X,\omega W)\\&+g_{N}((\nabla \pi _{*})(Y,\mathcal {C}X),\pi _{*}\omega W)+\eta (Y)g_{M}(X,\omega W)-\eta (X)g_{M}(Y,\omega W). \end{aligned}$$

Thus, we have

$$\begin{aligned} \sin ^{2}\theta g_{M}([X,Y],W)&=g_{N}((\nabla \pi _{*})(Y,\mathcal {C}X)-(\nabla \pi _{*})(X,\mathcal {C}Y),\pi _{*}\omega W)+g_{M}(\mathcal {A}_{X}\mathcal {B}Y+\mathcal {A}_{Y}\mathcal {B}X,\omega W) \\&+\eta (Y)g_{M}(X,\omega W)-\eta (X)g_{M}(Y,\omega W), \end{aligned}$$

which gives (3.25). This completes the proof.    $\square$

For the geometry of leaves $(\ker \pi _{*})^{\bot }$, we have

Theorem 3.11

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

$$\begin{aligned} g_{M}(V,\mathcal {V}\nabla ^{^{M}}_{X}\phi \mathcal {B}Y+\mathcal {A}_{X}\omega \mathcal {B}Y)&=g_{N}(\pi _*(\mathcal {C}Y),(\nabla \pi _*)(X,\varphi V)) \end{aligned}$$

(3.26)

and

$$\begin{aligned} g_{M}(\mathcal {A}_{X}\omega W,\mathcal {B}Y)+\eta (Y)g_{M}(X,\omega W)&=g_{N}((\nabla \pi _*)(X,Y),\pi _*\omega \phi W)\nonumber \\&-g_{N}((\nabla \pi _*)(X,\mathcal {C}Y),\pi _*\omega W), \end{aligned}$$

(3.27)

for $X, Y\in \Gamma ((\ker \pi _*)^{\bot }), V\in \Gamma (D_1)$ and $W\in \Gamma (D_{2}).$

Theorem 3.12

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

$$\begin{aligned} g_{M}(\omega V,\mathcal {T}_{U}\mathcal {B}X)+g_{M}(V,\phi U)\eta (X)&=g_{N}((\nabla \pi _{*})(U,\mathcal {C}X),\pi _{*}\omega V)-g_{N}((\nabla \pi _{*})(U,X),\pi _{*}\omega \phi V) \end{aligned}$$

(3.28)

for any $U\in \Gamma (D_1), V\in \Gamma (D_2)$ and $X\in \Gamma ((ker \pi _{*})^{\bot }).$

By virtue of Theorems 3.8, 3.9 and 3.11, we have the following theorem;

Theorem 3.13

Let $\pi$ be a semi-slant $\xi ^\perp$-Riemannian submersion from a Sasakian manifold $(M,\varphi ,\xi ,\eta ,g_{M})$ onto a Riemannian manifold $(N,g_{N})$ with a semi-slant angle $\theta .$ Then the total space M is a locally product manifold of the leaves of $D_1$, $D_2$ and $(ker\pi _*)^\perp ,$ i.e., $M=M{_{D_1}}\times M{_{D_2}}\times M{_{(ker\pi _*)^\perp }},$ if and only if

$$\begin{aligned} g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}\omega Z)&=g_{M}(\mathcal {T}_{U}\omega \phi Z,V), \end{aligned}$$

$$\begin{aligned} -g_{N}((\nabla \pi _{*})(U,\varphi V),\pi _{*}\mathcal {C}X)=g_{M}(V,\hat{\nabla }_{U}\phi \mathcal {B}X +\mathcal {T}_{U}\omega \mathcal {B}X)+g_{M}(V,\varphi U)\eta (X), \end{aligned}$$

$$\begin{aligned} g_{N}(\pi _{*}\omega W,(\nabla \pi _{*})(Z,\varphi U))&=g_{M}(\phi W,\hat{\nabla }_{Z}\varphi U), \end{aligned}$$

$$\begin{aligned} g_{N}((\nabla \pi _{*})(Z,\omega W),\pi _{*}(X))&\quad -g_{N}((\nabla \pi _{*})(Z,\omega \phi W),\pi _{*}(X))\\&=g_{M}(\mathcal {T}_{Z}\omega W,\mathcal {B}X)\\&\quad +g_{M}(W,\varphi Z)\eta (X) \end{aligned}$$

and

$$\begin{aligned} g_{M}(V,\mathcal {V}\nabla ^{^{M}}_{X}\phi \mathcal {B}Y+\mathcal {A}_{X}\omega \mathcal {B}Y)&=g_{N}(\pi _*(\mathcal {C}Y),(\nabla \pi _*)(X,\varphi V)), \end{aligned}$$

$$\begin{aligned} g_{M}(\mathcal {A}_{X}\omega W,\mathcal {B}Y)+\eta (Y)g_{M}(X,\omega W)&=g_{N}((\nabla \pi _*)(X,Y),\pi _*\omega \phi W)\\&\quad -g_{N}((\nabla \pi _*)(X,\mathcal {C}Y),\pi _*\omega W) \end{aligned}$$

for $X, Y\in \Gamma ((\ker \pi _*)^{\bot }),$ $U, V\in \Gamma (D_1)$ and $Z, W\in \Gamma (D_{2}).$

From Theorems 3.11 to 3.12, we have the following theorem;

Theorem 3.14

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

$$\begin{aligned} g_{M}(V,\mathcal {V}\nabla ^{^{M}}_{X}\phi \mathcal {B}Y+\mathcal {A}_{X}\omega \mathcal {B}Y)&=g_{N}(\pi _*(\mathcal {C}Y),(\nabla \pi _*)(X,\varphi V)), \end{aligned}$$

$$\begin{aligned} g_{M}(\mathcal {A}_{X}\omega W,\mathcal {B}Y)+\eta (Y)g_{M}(X,\omega W)&=g_{N}((\nabla \pi _*)(X,Y),\pi _*\omega \phi W)\\&\quad -g_{N}((\nabla \pi _*)(X,\mathcal {C}Y),\pi _*\omega W) \end{aligned}$$

and

$$\begin{aligned} g_{M}(\omega V,\mathcal {T}_{U}\mathcal {B}X)+g_{M}(V,\phi U)\eta (X)&=g_{N}((\nabla \pi _*)(U,\mathcal {C}X),\pi _*\omega V)\\&\quad -g_{N}((\nabla \pi _*)(U,X),\pi _*\omega \phi V) \end{aligned}$$

for $X, Y\in \Gamma ((\ker \pi _*)^{\bot }), U, V\in \Gamma (D_1)$ and $W\in \Gamma (D_{2}).$

3.2 Totally Geodesic Semi-Slant $\xi ^\perp$-Submersions

Recall that a differential map $\pi$ between two Riemannian manifolds is called totally geodesic if $\nabla \pi _*=0$ [41]. Then we have

Theorem 3.15

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

$$\begin{aligned} -\nabla ^{\pi }_{X}\pi _{*}Z_2&=\pi _{*}(\mathcal {C}(\mathcal {H}\nabla ^{^{M}}_{X}\omega Z_{1}-\mathcal {A}_{X}\phi Z_{1}+\mathcal {A}_{X}\mathcal {B}Z_{2}+\mathcal {H}\nabla ^{^{M}}_{X}\mathcal {C}Z_2) \\&\quad +\omega (\mathcal {A}_{X}\omega Z_{1}-\mathcal {V}\nabla ^{^{M}}_{X}\phi Z_{1} +\mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Z_{2}+\mathcal {A}_{X}\mathcal {C}Z_{2})\nonumber \\&\quad -\eta (Z_2)\mathcal {C}X-\eta (X)\eta (Z_2)-g_{M}(Y,\mathcal {C}X)\xi )\nonumber \end{aligned}$$

(3.29)

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

Proof

Making use of (2.5), (2.9) and (2.10), we have

$$\begin{aligned} \nabla ^{^{M}}_{X}Z=\varphi (\nabla ^{^{M}}_{X}\varphi )Z-\varphi \nabla ^{^{M}}_{X}\varphi Z+\eta (\nabla ^{^{M}}_{X}Z)\xi \end{aligned}$$

for any $Z\in \Gamma ((ker\pi _*)^\perp )$ and $X\in \Gamma (TM)$. Now, from (2.7), we have

$$\begin{aligned} (\nabla \pi _{*})(X,Z)&=\nabla ^{\pi }_{X}\pi _{*}Z +\pi _{*}(\varphi \nabla ^{^{M}}_{X}\varphi Z-\varphi (\nabla ^{^{M}}_{X}\varphi )Z-\eta (\nabla ^{^{M}}_{X}Z)\xi )\\&=\nabla ^{\pi }_{X}\pi _{*}Z+\pi _{*}(\varphi (\nabla ^{^{M}}_{X}\varphi Z_1+\nabla ^{^{M}}_{X}\varphi Z_2)-\eta (Z)\varphi X-\eta (\nabla ^{^{M}}_{X}Z)\xi ). \end{aligned}$$

Or,

$$\begin{aligned} (\nabla \pi _{*})(X,Z)&=\nabla ^{\pi }_{X}\pi _{*}Z_2+\pi _{*}(\mathcal {B}\mathcal {A}_{X}\phi Z_{1}+\mathcal {C}\mathcal {A}_{X}\phi Z_{1}+\phi \mathcal {V}\nabla ^{^{M}}_{X}\phi Z_{1} +\omega \mathcal {V}\nabla ^{^{M}}_{X}\phi Z_{1}\\&\quad +\phi \mathcal {A}_{X}\omega Z_{1}+\omega \mathcal {A}_{X}\omega Z_{1}+\mathcal {B}\mathcal {H}\nabla ^{^{M}}_{X}\omega Z_{1} +\mathcal {C}\mathcal {H}\nabla ^{^{M}}_{X}\omega Z_{1}\\&\quad +\mathcal {B}\mathcal {A}_{X}\mathcal {B}Z_{2}+\mathcal {C}\mathcal {A}_{X}\mathcal {B}Z_{2}+\phi \mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Z_{2} +\omega \mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Z_{2}\\&\quad +\phi \mathcal {A}_{X}\mathcal {C}Z_{2}+\omega \mathcal {A}_{X}\mathcal {C}Z_{2}+\mathcal {B}\mathcal {H}\nabla ^{^{M}}_{X}\mathcal {C}Z_{2} +\mathcal {C}\mathcal {H}\nabla ^{^{M}}_{X}\mathcal {C}Z_{2}\\&\quad -\eta (Z_2)\varphi X-\eta (X)\eta (Z_2)-g_{M}(Z_2,\mathcal {C}X)\xi ) \end{aligned}$$

for any $Z=Z_{1}+Z_{2}\in \Gamma (TM)$, where $Z_{1}\in \Gamma (ker\pi _{*})$ and $Z_{2}\in \Gamma ((ker\pi _{*})^{\perp }).$

$$\begin{aligned} (\nabla \pi _{*})(X,Z)&=\nabla ^{\pi }_{X}\pi _{*}Z_2+\pi _{*}(\mathcal {C}(\mathcal {A}_{X}\phi Z_{1}+\mathcal {H}\nabla ^{^{M}}_{X}\omega Z_{1} +\mathcal {A}_{X}\mathcal {B}Z_{2}+\mathcal {H}\nabla ^{^{M}}_{X}\mathcal {C}Z_{2})\\&\quad +\omega (\mathcal {V}\nabla ^{^{M}}_{X}\phi Z_{1}+\mathcal {A}_{X}\omega Z_{1}+\mathcal {V}\nabla ^{^{M}}_{X}\mathcal {B}Z_{2}+\mathcal {A}_{X}\mathcal {C}Z_{2})\\&\quad -\eta (Z_2)\mathcal {C}X-\eta (X)\eta (Z_2)-g_{M}(Z_2,\mathcal {C}X)\xi ), \end{aligned}$$

which gives (3.29). This completes the assertion.    $\square$

Theorem 3.16

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

• (i) $\begin{aligned} g_{M}(\hat{\nabla }_{U_{1}}\varphi V_{1},\mathcal {B}Z)=g_{M}(\mathcal {T}_{U_{1}}\mathcal {C}Z,\varphi V_{1})-g_{M}(V_{1},\phi U_{1})\eta (Z), \end{aligned}$

• (ii) $\begin{aligned} (g_{N}(\nabla \pi _*(U_{2},\omega \phi V_{2}))+g_{N}(\nabla \pi _*(U_{2},\omega V_{2}))),\pi _*Z&=g_{M}(\mathcal {T}_{U_{2}}\omega V_{2},\mathcal {B}Z)+g_{M}(V_{2},\phi U_{2})\eta (Z) \end{aligned}$

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

for any $U_1, V_1\in \Gamma (D_1),\ U_2,V_2\in \Gamma (D_2),\ U\in \Gamma (ker\pi _*)$ and $X, Y, Z\in \Gamma ((ker\pi _{*})^\perp )$.

Theorem 3.17

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

• (i) $\mathcal {C}(\mathcal {T}_{U}\phi V+\nabla ^{^{M}}_{U}\omega V)+\omega (\hat{\nabla }_{U}\phi V+\mathcal {T}_{U}\omega V) +g_{M}(\mathcal {P}V,\phi U)\xi =0.$

• (ii) $\mathcal {C}(\mathcal {A}_{X}\phi U+\mathcal {H}\nabla ^{^{M}}_{X}\omega U)+\omega (\mathcal {A}_{X}\omega U+\mathcal {V}\nabla ^{^{M}}_{X}\phi U)+g_{M}(QU,\mathcal {B}X)\xi =0.$

• (iii) $\mathcal {C}(\mathcal {T}_{U_1}\phi V_1+\mathcal {H}\nabla ^{^{M}}_{U_1}\phi V_1)+\omega (\mathcal {T}_{U_1}\omega V_1+\mathcal {V}\nabla ^{^{M}}_{U_1}\phi V_1)=0,$

for $U_1\in \Gamma (D_1),\ V_1\in \Gamma (D_2),\ U,V\in \Gamma (ker\pi _*)$ and $X\in \Gamma ((ker\pi _{*})^\perp )$.

3.3 Some Examples

Example 3.18

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 3.19

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

$$ \varphi (\sum _{i=1}^{n}(X_{i}\frac{\partial }{\partial x^{i}}+Y_{i}\frac{\partial }{\partial y^{i}})+Z\frac{\partial }{\partial z})= \sum _{i=1}^{n}(Y_{i}\frac{\partial }{\partial x^{i}}-X_{i}\frac{\partial }{ \partial y^{i}}) $$

$$ g=\eta \otimes \eta +\frac{1}{4}\sum _{i=1}^{n}(dx^{i}\otimes dx^{i}+dy^{i}\otimes dy^{i}), $$

$$ \eta =\frac{1}{2}(dz-\sum _{i=1}^{n}y^{i}dx^{i}),\,\,\,\xi =2\frac{\partial }{ \partial z}, $$

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

Example 3.20

Let F be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} F: &{} \mathbb {R}^{9} &{} \longrightarrow &{} \mathbb {R}^{5} \\ &{} (x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4},z) &{} &{} (\frac{x_{1}+x_{2} }{\sqrt{2}},\frac{y_{1}+y_{2}}{\sqrt{2}},sin\alpha x_{3}-cos\alpha x_{4},y_{4},z) \end{array} \end{aligned}$$

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

$$\begin{aligned} kerF_{*}=span\{&Z_{1}=\frac{\partial }{\partial x^{1}}-\frac{\partial }{ \partial x^{2}},\ Z_{2}=\frac{\partial }{\partial y^{1}}-\frac{\partial }{ \partial y^{2}}, \\&Z_{3}=-\cos \alpha \frac{\partial }{\partial x^{3}}-\sin \alpha \frac{\partial }{\partial x^{4}} ,Z_{4}=\frac{\partial }{\partial y^{3}}\} \end{aligned}$$

and

$$\begin{aligned} (kerF_{*})^{\perp }=span\{&H_{1}=\frac{\partial }{\partial x^{1}}+\frac{ \partial }{\partial x^{2}},\ H_{2}=\frac{\partial }{\partial y^{1}}+\frac{ \partial }{\partial y^{2}},\ H_{3}=\sin \alpha \frac{\partial }{\partial x^{3}}-\cos \alpha \frac{ \partial }{\partial x^{4}}, \\&H_{4}=\frac{\partial }{\partial y^{4}},\ H_{5}=\frac{\partial }{\partial z}=\xi \}. \end{aligned}$$

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

$$\begin{aligned} g_{N}(F_*H_{1},F_*H_{1})=g_{M}(H_{1},H_{1}),\ g_{N}(F_*H_{2},F_*H_{2})=g_{M}(H_{2},H_{2}), \end{aligned}$$

$$\begin{aligned} g_{N}(F_{*}H_{3},F_{*}H_{3})=g_{M}(H_{3},H_{3}),\ g_{N}(F_{*}H_{4},F_{*}H_{4})=g_{M}(H_{4},H_{4}),\ g_{N}(F_{*}\xi ,F_{*}\xi )=g_{M}(\xi ,\xi ) \end{aligned}$$

where $g_{M}$ and $g_{N}$ 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 3.21

Let F be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} F: &{} \mathbb {R}^{7} &{} \longrightarrow &{} \mathbb {R}^{3} \\ &{} (x_{1},x_{2},x_{3},y_{1},y_{2},y_{3},z) &{} &{} (\frac{x_2-y_3}{\sqrt{2}},y_{2},z). \end{array} \end{aligned}$$

Then the submersion F is a semi-slant $\xi ^\perp$-Riemannian submersion such that $D_1=span(\frac{\partial }{\partial x_{1}},\frac{\partial }{\partial y_{1}})$ and $D_2=span(\frac{\partial }{\partial x_{2}}+\frac{\partial }{\partial y_{3}},\frac{\partial }{\partial x_{3}})$ with semi-slant angle $\alpha =\frac{\pi }{4}.$

Example 3.22

Let F be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} F: &{} \mathbb {R}^{9} &{} \longrightarrow &{} \mathbb {R}^{3} \\ &{} (x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4},z) &{} &{} (sin\alpha x_{3}-cos\alpha x_{4},y_{4},z) \end{array} \end{aligned}$$

with $\alpha \in (0,\frac{\pi }{2}).$ Then the submersion F is a semi-slant $\xi ^\perp$-Riemannian submersion such that $D_1=span(\frac{\partial }{\partial x_{1}},\frac{\partial }{\partial x_{2}}, \frac{\partial }{\partial y_{1}},\frac{\partial }{\partial y_{2}})$ and $D_2=span(-\cos \alpha \frac{\partial }{\partial x_{3}}-\sin \alpha \frac{\partial }{\partial x_{4}},\frac{\partial }{\partial y_{3}})$ with semi-slant angle $\theta =\alpha .$

Example 3.23

Let F be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} F: &{} \mathbb {R}^{13} &{} \longrightarrow &{} \mathbb {R}^{7} \\ &{} (x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},y_{1},y_{2},y_{3},y_{4},y_{5},y_{6},z) &{} &{} (\frac{x_1-x_2}{\sqrt{2}},\frac{y_1-y_2}{\sqrt{2}},\frac{x_3+x_4}{\sqrt{2}},\frac{y_3+y_4}{\sqrt{2}}, \frac{x_5-x_6}{\sqrt{2}},y_{5},z). \end{array} \end{aligned}$$

Then the submersion F is a semi-slant $\xi ^\perp$-Riemannian submersion such that $D_1=span(\frac{\partial }{\partial x_{1}}+\frac{\partial }{\partial x_{2}}, \frac{\partial }{\partial y_{1}}+\frac{\partial }{\partial y_{2}}, \frac{\partial }{\partial x_{3}}-\frac{\partial }{\partial x_{2}}, \frac{\partial }{\partial y_{3}}-\frac{\partial }{\partial y_{4}})$ and $D_2=span(\frac{\partial }{\partial x_{5}}+\frac{\partial }{\partial x_{6}}, \frac{\partial }{\partial y_{6}})$ with semi-slant angle $\alpha =\frac{\pi }{4}.$

4 Hemi-Slant $\xi ^{\perp }$-Riemannian Submersions

Very recently Ramazan Sarıand Mehmet Akif Akyol [36] also introduced and studied hemi-slant $\xi ^\perp$-submersions and obtained interesting results. In this Sect. 4, our aim is to discuss briefly some results of this paper.

Definition 4.1

Let $(M,\varphi ,\xi ,\eta , g_{M})$ be a Sasakian manifold and $(N, g_{N})$ be a Riemannian manifold. Suppose that there exists a Riemannian submersion $\phi :M\longrightarrow N$ such that $\xi$ is normal to $ker\phi _{*}$. Then $\phi$ is called a hemi-slant $\xi ^{\perp }$-Riemannian submersion if the vertical distribution $ker\phi _{*}$ of $\phi$ admits two orthogonal complementary distributions $\mathcal {D}_{\perp }$ and $\mathcal {D}_{\theta }$ such that $\mathcal {D}_{\perp }$ is anti-invariant and $\mathcal {D}_{\theta }$ is slant, i.e., we have

$$\begin{aligned} \ker \phi _{*}=\mathcal {D}_{\perp }\oplus \mathcal {D}_{\theta }. \end{aligned}$$

In this case, the angle $\theta$ is called the slant angle of the hemi-slant $\xi ^{\perp }$-Riemannian submersion.

If $\theta \ne 0,\frac{\pi }{2}$ then we say that the submersion is proper hemi-slant $\xi ^{\perp }$-Riemannian submersion. Now, we are going to give some proper examples in order to guarantee the existence of hemi-slant $\xi ^{\perp }$-Riemannian submersions in Sasakian manifolds and demonstrate that the method presented in this paper is effective. Note that, $(\mathbb {R}^{2n+1},\varphi , \eta , \xi , g_{\mathbb {R}^{2n+1}})$ will denote the manifold $\mathbb {R}^{2n+1}$ with its usual contact structure given by

$$\begin{aligned} \eta =\frac{1}{2}(dz-\sum _{i=1}^{n}y^{i}dx^{i}),\,\,\,\xi =2\frac{\partial }{\partial z}, \end{aligned}$$

$$\begin{aligned} g=\eta \otimes \eta +\frac{1}{4}\sum _{i=1}^{n}(dx^{i}\otimes dx^{i}+dy^{i}\otimes dy^{i}), \end{aligned}$$

$$\begin{aligned} \varphi (\sum _{i=1}^{n}(X_{i}\partial x^{i}+Y_{i}\partial y^{i})+Z\partial z)= \sum _{i=1}^{n}(Y_{i}\partial x^{i}-X_{i}\partial y^{i}) \end{aligned}$$

where $(x_{1},..,x_{n},y_{1},...,y_{n},z)$ denotes the Cartesian coordinates on $\mathbb {R}^{2n+1}.$

Example 4.2

Every anti-invariant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold onto a Riemannian manifold is a hemi-slant $\xi ^{\perp }$-Riemannian submersion with $\mathcal {D}_{\theta }=\{0\}$.

Example 4.3

Every slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold onto a Riemannian manifold is a hemi-slant $\xi ^{\perp }$-Riemannian submersion with $\mathcal {D}_{\perp }=\{0\}$.

Example 4.4

Let $\phi$ be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} \phi : &{} (\mathbb {R}^{9},g_{\mathbb {R}^{9}}) &{} \rightarrow &{} (\mathbb {R}^{5},g_{\mathbb {R}^{5}}) \\ &{} (x_{1},x_{2},x_{3},x_{4},y_{1},y_{2},y_{3},y_{4},z) &{} &{} (\frac{x_{1}+y_{2} }{\sqrt{2}},\frac{x_{2}+y_{1}}{\sqrt{2}},\sin \gamma x_{3}-\cos \gamma x_{4},y_{4},z) \end{array} \end{aligned}$$

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

$$\begin{aligned} \ker \phi _{*}=Sp\{&V_{1}=-\partial x_{1}+\partial y_{2},V_{2}=-\partial x_{2}+ \partial y_{1},V_{3}=-\cos \gamma \partial x_{3}-\sin \gamma \partial x_{4},\\&V_{4}=\partial y_{3}\} \end{aligned}$$

and

$$\begin{aligned} {(\ker \phi _{*})}^{\perp }=Sp\{&W_{1}=\partial x_{1}+\partial y_{2},W_{2}=\partial x_{2}+\partial y_{1},W_{3}=\sin \gamma \partial x_{3}-\cos \gamma \partial x_{4},\\&W_{4}=\partial y_{4},W_{5}=\partial z\} \end{aligned}$$

hence we have $\varphi V_{1}=W_{2},\varphi V_{2}=W_{1}.$ Thus, it follows that $\mathcal {D}_{\perp }=sp\{V_{1},V_{2}\}$ and $\mathcal {D}_{\theta }=sp\{V_{3},V_{4}\}$ is a slant distribution with slant angle $\theta =\gamma .$ Thus, $\phi$ is a slant $\xi ^\perp$-submersion. Also by direct computations, we have

$$\begin{aligned} g_{\mathbb {R}^{9}}(W_{i},W_{i})=g_{\mathbb {R}^{5}}(\varphi W_{i},\varphi W_{i}),\ \ \ i=1,...,5 \end{aligned}$$

which show that $\phi$ is a slant $\xi ^\perp$-Riemannian submersion.

Example 4.5

Let F be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} F : &{} (\mathbb {R}^{9},g_{\mathbb {R}^{9}}) &{} \longrightarrow &{} (\mathbb {R}^{5},g_{\mathbb {R}^{5}}) \\ &{} (x_{1},...,y_{1},...,z) &{} &{} (\frac{x_{1}+y_{2} }{\sqrt{2}},\frac{x_{2}+y_{1}}{\sqrt{2}},\frac{x_{3}+x_{4}}{\sqrt{2}},\frac{ y_{3}+y_{4}}{\sqrt{2}},z). \end{array} \end{aligned}$$

The submersion F is hemi-slant $\xi ^{\perp }$-Riemannian submersion such that $\mathcal {D}_{\perp }=span\{\partial x_{1}- \partial y_{2},\partial x_{2}-\partial y_{1}\}$ and $\mathcal {D}_{\theta }=span\{\partial x_{3}+\partial x_{4},\partial y_{3}+ \partial y_{4}\}$ with hemi-slant angle $\theta =0.$

Example 4.6

Let $\pi$ be a submersion defined by

$$\begin{aligned} \begin{array}{cccc} \pi : &{} (\mathbb {R}^{7},g_{\mathbb {R}^{7}}) &{} \longrightarrow &{} (\mathbb {R}^{4},g_{\mathbb {R}^{4}}) \\ &{} (x_{1},...,y_{1},...,z) &{} &{} (\frac{x_{1}+x_{2} }{\sqrt{2}},\sin \gamma x_{3}-\cos \gamma y_{4},\cos \beta x_{4}-\sin \beta y_{3},z). \end{array} \end{aligned}$$

The submersion $\pi$ is a hemi-slant $\xi ^{\perp }$-Riemannian submersion such that $\mathcal {D}_{\perp }=span\{\partial x_{1}- \partial x_{2}\}$ and $\mathcal {D}_{\theta }=span\{\cos \gamma \partial x_{3}-\sin \gamma \partial y_{4},\sin \beta \partial x_{4}-\cos \beta \partial y_{3}\}$ with hemi-slant angle $\theta =\alpha +\beta$.

Let $\phi$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M,\varphi ,\xi ,\eta ,g_{M})$ onto a Riemannian manifold $(N,g_{N})$. Then, for $U\in \Gamma (ker\phi _{*})$, we put

$$\begin{aligned} U=\mathcal {P}U+\mathcal {Q}U \end{aligned}$$

where $\mathcal {P}U\in \Gamma (\mathcal {D}_{\perp })$ and $\mathcal {Q}U\in \Gamma (\mathcal {D}_{\theta })$. For $Z\in \Gamma (TM)$, we have

$$\begin{aligned} Z=\mathcal {V}Z+\mathcal {H}Z \end{aligned}$$

where $\mathcal {V}Z\in \Gamma (ker\phi _{*})$ and $\mathcal {H}Z\in \Gamma (ker\phi _{*})^{\perp }$.

We denote the complementary distribution to $\varphi \mathcal {D}_{\perp }$ in $(ker\phi _{*})^{\perp }$ by $\mu$. Then we have

$$\begin{aligned} (ker\phi _{*})^{\perp }=\varphi \mathcal {D}_{\perp }\oplus \mu , \end{aligned}$$

where $\varphi (\mu )\subset \mu .$ Hence $\mu$ contains $\xi .$ For $V\in \Gamma (ker\phi _{*})$, we write

$$\begin{aligned} \varphi V=\rho V+\omega V \end{aligned}$$

(4.1)

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

$$\begin{aligned} \varphi X=\mathcal {B}X+\mathcal {C}X, \end{aligned}$$

(4.2)

where $\mathcal {B}X$ and $\mathcal {C}X$ are vertical (resp. horizontal) components of $\varphi X$, respectively. Then the horizontal distribution $(ker\phi _{*})^{\perp }$ is decomposed as

$$\begin{aligned} (ker\phi _{*})^{\perp }=\varphi \mathcal {D}_{\perp }\oplus \mu , \end{aligned}$$

here $\mu$ is the orthogonal complementary distribution of $\mathcal {D}_{\perp }$ and it is both invariant distribution of $(ker\phi _{*})^{\perp }$ with respect to $\varphi$ and contains $\xi .$ Then by using (2.3), (2.4), (4.1) and (4.2), we get

$$\begin{aligned} (\nabla _{V}^{^{M}}\rho )W=\mathcal {B}T_{V}W-T_{V}\omega W \end{aligned}$$

(4.3)

$$\begin{aligned} (\nabla _{V}^{^{M}}\omega )W=\mathcal {C}T_{V}W-T_{V}\rho W \end{aligned}$$

(4.4)

for $V,W\in \Gamma (ker\phi _{*})$, where

$$\begin{aligned} (\nabla _{V}^{^{M}}\rho )W=\hat{\nabla }_{V}\rho W-\rho \hat{\nabla }_{V}W \end{aligned}$$

and

$$\begin{aligned} (\nabla _{V}^{^{M}}\omega )W=\mathcal {H}\nabla _{V}^{^{M}}\omega W-\omega \hat{\nabla }_{V}W. \end{aligned}$$

Lemma 4.7

Let $\phi :M \rightarrow N$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_{M})$ onto a Riemannian manifold $(N,g_{N}).$ Then we have

$$\begin{aligned} \rho ^{2}W=\cos ^{2}\theta W,\text { }W\in \Gamma (\mathcal {D}_{\theta }), \end{aligned}$$

(4.5)

where $\theta$ denotes the hemi-slant angle of $ker\phi _{*}.$

Lemma 4.8

Let $\phi :M \rightarrow N$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M,\varphi , \eta , \xi ,g_{M})$ onto a Riemannian manifold $(N,g_{N}).$ Then we have

$$\begin{aligned} g_{M}(\rho U,\rho V)=\cos ^{2}\theta g_M(U,V) \end{aligned}$$

(4.6)

$$\begin{aligned} g_M(\omega U,\omega V)=\sin ^{2}\theta g_M(U,V) \end{aligned}$$

(4.7)

for all $U,V\in \Gamma (ker\phi _{*}).$

4.1 Integrable and Parallel Distributions

Theorem 4.9

Let $\phi$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_{M})$ onto Riemannian manifold $(N,g_{N})$ with a hemi-slant angle $\theta$. Then the distribution $\mathcal {D}_{\perp }$ is integrable if and only if we have

$$\begin{aligned} g_M(\mathcal {T}_{U}\varphi V-\mathcal {T}_{V}\varphi U,\rho Z)=g_N((\nabla \phi _{*})(V,\varphi U)-(\nabla \phi _{*})(U,\varphi V),\phi _{*}(\omega Z)) \end{aligned}$$

for any $U,V\in \Gamma (\mathcal {D}_{\perp })$ and $Z\in \Gamma (\mathcal {D}_{\theta }).$

Proof

For $U,V\in \Gamma (TM)$, by using (2.9) and (2.10), we have

$$\begin{aligned} g_M(\nabla ^{M}_{U}V,Z)=g_M(\nabla ^{M}_{U}\varphi V,\varphi Z). \end{aligned}$$

(4.8)

For $U,V\in \Gamma (\mathcal {D}_{\perp }),$ $Z\in \Gamma (\mathcal {D}_{\theta }),$ using (2.9 ) and (4.8), we have

$$\begin{aligned} g_M([U,V],Z)=g_M(\nabla ^{M}_{U}\varphi V,\varphi Z)-g_M(\nabla ^{M}_{V}\varphi U,\varphi Z). \end{aligned}$$

On the other hand, we get

$$\begin{aligned} g_M([U,V],Z)=g_M(\mathcal {T}_{U}\varphi V-\mathcal {T}_{V}\varphi U,\rho Z)+g_M( \mathcal {H}(\nabla ^{M}_{U}\varphi V)-\mathcal {H}(\nabla ^{M}_{V}\varphi U),wZ). \end{aligned}$$

Or,

$$\begin{aligned} g_M([U,V],Z)&=g_M(\mathcal {T}_{U}\varphi V-\mathcal {T}_{V}\varphi U,\rho Z)\\&+g_N(\phi _{*}(\nabla ^{M}_{U}\varphi V)-\phi _{*}(\nabla ^{M}_{V}\varphi U),\phi _{*}(\omega Z)) \end{aligned}$$

which proves assertion.    $\square$

Theorem 4.10

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then the distribution $\mathcal {D}_{\theta }$ is integrable if and only if we have

$$\begin{aligned} g_N((\nabla \phi _{*})(Z,\omega W)-(\nabla \phi _{*})(W,\omega Z),\varphi U)= g_M(\mathcal {T}_{Z}\omega \rho W-\mathcal {T}_{W}w\rho Z,U) \end{aligned}$$

for any $Z,W\in \Gamma (\mathcal {D}_{\theta })$ and $U\in \Gamma (\mathcal {D}_{\perp }).$

Proof

For $Z,W\in \Gamma (\mathcal {D}_{\theta })$ and $U\in \Gamma (\mathcal {D}_{\perp }),$ using (2.9) and (4.8) we have

$$\begin{aligned} g_M([Z,W],U)= & {} g_M(\nabla ^{M}_{Z}\varphi W,\varphi U)-g_M(\nabla ^{M}_{W}\varphi Z,\varphi U). \end{aligned}$$

Therefore, by using (4.1), we get

$$\begin{aligned} g_M([Z,W],U)= & {} -g_M(\nabla ^{M}_{Z}\rho ^{2}W,U)-g_M(\nabla ^{M}_{Z}\omega \rho W,U)\\&+g_M(\nabla ^{M}_{Z}\omega W,\varphi U) +g_M(\nabla ^{M}_{W}\rho ^{2}Z,U)\\&+g_M(\nabla ^{M}_{W}\omega \rho Z,U)-g_M(\nabla ^{M} _{W}\omega Z,\varphi U). \end{aligned}$$

Now, by virtue of (3.16), we obtain

$$\begin{aligned} g_M([Z,W],U)= & {} \cos ^{2}\theta g_M([Z,W],U)-g_M(\nabla ^{M}_{Z}\omega \rho W,U)\\&+g_M(\nabla ^{M}_{Z}\omega W,\varphi U) +g_M(\nabla ^{M}_{W}\omega \rho Z,U)\\&-g_M(\nabla ^{M}_{W}\omega Z,\varphi U). \end{aligned}$$

Then we have

$$\begin{aligned} \sin ^{2}\theta g_M([Z,W],U)&=g_M(\nabla ^{M}_{W}\omega \rho Z-\nabla ^{M}_{Z}\omega \rho W,U)\\&+g_M(\nabla ^{M}_{Z}\omega W-\nabla ^{M}_{W}\omega Z,\varphi U). \end{aligned}$$

On the other hand, we have

$$\begin{aligned} \sin ^{2}\theta g_M([Z,W],U)= & {} g_M(\mathcal {T}_{W}\omega \rho Z-\mathcal {T}_{Z}\omega \rho W,U)\\&+g_M(H (\nabla ^{M}_{Z}\omega W)-\mathcal {H}(\nabla ^{M}_{W}\omega Z),\varphi U) \\= & {} g_M(\mathcal {T}_{W}\omega \rho Z-\mathcal {T}_{Z}\omega \rho W,U)\\&+g_N(\phi _{*}(\nabla ^{M} _{Z}\omega W)-\phi _{*}(\nabla ^{M}_{W}\omega Z),\varphi U) \end{aligned}$$

which proves assertion.    $\square$

Theorem 4.11

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then the distribution $D_{\perp }$ is parallel if and only if

$$\begin{aligned} g_M(\phi _{*}(\nabla _{U}V),\phi _{*}(\omega \rho Z))=g_M(\varphi \nabla _{U}V,\omega Z) \end{aligned}$$

and

$$\begin{aligned} g_M(\hat{\nabla }_{U}\rho V +\mathcal {T}_{U}\omega V,BX)=-g_M(\mathcal {T}_{U}\rho V+ \mathcal {H}(\nabla _{U}\omega V),CX) \end{aligned}$$

for any $U,V\in \Gamma (D_{\perp }),$ $Z\in \Gamma (D_{\theta }), X\in \Gamma ((\ker \phi _{*})^{\perp })$.

Proof

For $U,V\in \Gamma (D_{\perp }),$ $Z\in \Gamma (D_{\theta })$ using (2.9), we get

$$\begin{aligned} g_M(\nabla _{U}V,Z)= & {} g_M(\varphi \nabla _{U}V,\varphi Z)+\eta (\nabla _{U}V)\eta (Z) \\= & {} g_M(\varphi \nabla _{U}V,\varphi Z). \end{aligned}$$

Or,

$$\begin{aligned} g_M(\nabla _{U}V,Z)=-g_M(\nabla _{U}V,\rho ^{2}Z+\omega \rho Z+\varphi \omega Z). \end{aligned}$$

Then one obtains

$$\begin{aligned} \sin ^{2}\theta g_M(\nabla _{U}V,Z)=-g_M(\mathcal {H}(\nabla _{U}V),\omega \rho Z)+g_M(\varphi \nabla _{U}V,\omega Z). \end{aligned}$$

By property of $\phi$, we get

$$\begin{aligned} \sin ^{2}\theta g_M(\nabla _{U}V,Z) =-g_N(\phi _{*}(\nabla _{U}V),\phi _{*}(\omega \rho Z))+g_M(\varphi \nabla _{U}V,\omega Z). \end{aligned}$$

On the other hand, for $U,V\in \Gamma (D_{\perp }),X\in \Gamma ((\ker \phi _{*})^{\perp })$, we have

$$\begin{aligned} g_M(\nabla _{U}V,X)=g_M(\nabla _{U}\varphi V,\varphi X). \end{aligned}$$

Now, by virtue of (2.3) and (4.1), we obtain

$$\begin{aligned} g_M(\nabla _{U}V,X)&=g_M(\mathcal {T}_{U}\rho V,CX)+g_M(\hat{\nabla }\rho V,BX)\\&+g_M(\mathcal {T}_{U}\omega V,BX)+g_M(\mathcal {H}(\nabla _{U}\omega V),CX) \end{aligned}$$

which completes the proof.    $\square$

Theorem 4.12

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then the distribution $D_{\theta }$ is parallel if and only if

$$\begin{aligned} g_N(\phi _{*}(\omega W),(\nabla \phi _{*})(Z,\varphi U))=g_M(\rho W,\mathcal {T} _{Z}\varphi U) \end{aligned}$$

and

$$\begin{aligned}&g_N((\nabla \phi _{*})(\nabla _{Z}\omega \rho W),\phi _{*}(X))-g_N((\nabla \phi _{*})(\nabla _{Z}\omega W),\phi _{*}(CX))\\&=-g_M(\mathcal {T}_{Z}\omega W,BX)+ g_M(\omega W,Z) \eta (X). \end{aligned}$$

for all $Z,W\in \Gamma (D_{\theta }),$ $U\in \Gamma (D_{\perp }), X\in \Gamma ((\ker \phi _{*})^{\perp })$.

Theorem 4.13

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then $D_{\perp }$ defines a totally geodesic foliation on M if and only if

$$\begin{aligned} g_N((\nabla \phi _{*})(U,\varphi V),\phi _{*}(\omega Z))=-g_M(\mathcal {T} _{U}V,\omega \rho Z) \end{aligned}$$

and

$$\begin{aligned} g_M(\mathcal {T}_{U}\varphi V,BX)=g_N((\nabla \phi _{*})(U,\varphi V),\phi _{*}(CX)) \end{aligned}$$

for any $U,V\in \Gamma (D_{\perp }),Z\in \Gamma (D_{\theta }), X\in \Gamma ((\ker \phi _{*})^{\perp })$.

Proof

For $U,V\in \Gamma (D_{\perp }),Z\in \Gamma (D_{\theta }),$ from (2.9), (2.3), (2.4), (4.1) to (4.5), we have

$$\begin{aligned} g_M(\nabla _{U}V,Z)=\cos ^{2}\theta g_M(\nabla _{U}V,Z)-g_M(\mathcal {T}_{U}V,\omega \rho Z)+g_M(\mathcal {H }(\nabla _{U}\varphi V),wZ). \end{aligned}$$

Or,

$$\begin{aligned} \sin ^{2}\theta g_M(\nabla _{U}V,Z)=-g_M(\mathcal {T}_{U}V,w\rho Z)-g_N(\phi _{*}(\nabla _{U}\varphi V),\phi _{*}(\omega Z)). \end{aligned}$$

On the other hand, for $X\in \Gamma ((\ker \phi _{*})^{\perp })$, we have

$$\begin{aligned} g_M(\nabla _{U}V,X) =g_M(\mathcal {T}_{U}\varphi V,BX)+g_M(\mathcal {H}(\nabla _{U}\varphi V),CX). \end{aligned}$$

Or,

$$\begin{aligned} g_M(\nabla _{U}V,X)=g_M(\mathcal {T}_{U}\varphi V,BX)-g_N(\phi _{*}(\nabla _{U}\varphi V),\phi _{*}(CX)). \end{aligned}$$

This completes the proof.    $\square$

Theorem 4.14

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M,\varphi ,\eta ,\xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then $D_{\theta }$ defines a totally geodesic foliation on M if and only if

$$\begin{aligned} g_N((\nabla \phi _{*})(Z,\omega W),\phi _{*}(\varphi U))=-g_M(\mathcal {T}_{Z}\omega \rho W,U) \end{aligned}$$

and

$$\begin{aligned} g_N((\nabla \phi _{*})(Z,\omega \rho W),\phi _{*}(X))+g_N((\nabla \phi _{*})(Z,\omega W),\phi _{*}(CX))=g_M(\mathcal {T}_{Z}\omega W,BX) \end{aligned}$$

for any $Z,W\in \Gamma (D_{\theta }),U\in \Gamma (D_{\perp }),X\in \Gamma ((\ker \phi _{*})^{\perp })$.

4.2 Hemi-Slant $\xi ^{\perp }$-Riemannian Submersions on Sasakian Space Forms

A plane section in the tangent space $T_pM$ at $p\in M$ is called a $\varphi$-section if it is spanned by a vector X orthogonal to $\xi$ and $\varphi X$. The sectional curvature of $\varphi$-section is called $\varphi$-sectional curvature. A Sasakian manifold with constant $\varphi$-sectional curvature c is a Sasakian space form. The Riemannian curvature tensor of a Sasakian space form is given by

$$\begin{aligned} R^{M}(X,Y,Z,W)&=\frac{c+3}{4}\{g_M(Y,Z)g_M(X,W)-g_M(X,Z)g_M(Y,W)\} \nonumber \\&+\frac{c-1}{4}\{g_M(Y,W)\eta (X)\eta (Z)-g_M(X,W)\eta (Y)\eta (Z)\nonumber \\&+g_M(X,Z)\eta (Y)\eta (W)-g_M(Y,Z)\eta (X)\eta (W) \nonumber \\&+g_M(\varphi Y,Z)g_M(\varphi X,W)-g_M(\varphi X,Z)g_M(\varphi Y,W) \nonumber \\&-2g_M(\varphi X,Y)g_M(\varphi Z,W)\} \end{aligned}$$

(4.9)

for any $X, Y, Z, W\in \Gamma (TM)$ [39].

Theorem 4.15

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi , g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then we have

$$\begin{aligned} \widehat{R}(U,V,W,S)&=\frac{c+3}{4}\{g_M(V,S)g_M(U,W)-g_M(U,S)g_M(V,W)\} \\&+g_M(\mathcal {T}_{V}W,\mathcal {T}_{U}S)-g_N(\mathcal {T}_{U}W,\mathcal {T}_{V}S) \nonumber \end{aligned}$$

(4.10)

and

$$\begin{aligned} \widehat{K}(U,V)=\frac{c+3}{4}\{g_M(U,V)^{2}-1\}+g_M(\mathcal {T}_{V}U,\mathcal {T}_{U}V)-g_M(\mathcal {T}_{U}U,\mathcal {T}_{V}V) \end{aligned}$$

(4.11)

for all $U,V,S,W\in \Gamma (\mathcal {D}^{\perp }).$

Proof

For any $U,V,S,W\in \Gamma (\mathcal {D}_{\perp })$ by using (4.9), $\varphi U\in \Gamma ((\ker \phi _{*})^{\perp })$ and $\eta (U)=0,$ then we have

$$\begin{aligned} R^{M}(U,V,S,W)=\frac{c+3}{4}\{g_M(V,S)g_M(U,W)-g_M(U,S)g_M(V,W)\}. \end{aligned}$$

(4.12)

Hence, we have

$$\begin{aligned} \widehat{R}(U,V,W,S)&=\frac{c+3}{4}\{g_M(V,S)g_M(U,W)-g_M(U,S)g_M(V,W)\} \\&+g_M(\mathcal {T}_{V}W,\mathcal {T}_{U}S)-g_M(\mathcal {T}_{U}W,\mathcal {T}_{V}S) \end{aligned}$$

which completes the proof.    $\square$

Corollary 4.16

Let $\phi$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M^{m}, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$ and $m\ge 3$. If $\mathcal {D}_{\perp }$ is totally geodesic, then M is flat if and only if $c=-3.$

Theorem 4.17

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. If $\mathcal {D}_{\perp }$ is totally geodesic, then

$$\begin{aligned} \widehat{\tau }_{\perp }=\frac{c+3}{2}q(1-2q) \end{aligned}$$

where $\widehat{\tau }_{\perp }$ is the scaler curvature.

Proof

We have

$$\begin{aligned} \widehat{S}_{\perp }(U,V)=\underset{i=1}{\overset{2q}{\sum }}\widehat{R} (E_{i},U,V,E_{i}) \end{aligned}$$

where $\{E_1,...,E_{2q}\}$ is ortonormal basis on $\Gamma (\mathcal {D}_\perp )$ and $U,V\in \Gamma (\mathcal {D}_{\perp }).$ Thus, one obtains

$$\begin{aligned} \widehat{S}_{\perp }(U,V)=\underset{i=1}{\overset{2q}{\sum }}\{\frac{c+3}{4} \{g_M(U,E_{i}) g_M(E_{i},V)- g_M(E_{i},E_{i})g_M(U,V)\}\}. \end{aligned}$$

Or,

$$\begin{aligned} \widehat{S}_{\perp }(U,V)=\frac{c+3}{4}(1-2q)g_M(U,V). \end{aligned}$$

(4.13)

By taking $U=V=E_{k}, k=1,...,2q$, we get the result.    $\square$

Corollary 4.18

Let $\phi$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. If $\mathcal {D}_{\perp }$ is totally geodesic distribution, then $\mathcal {D}_{\perp }$ is Einstein.

Theorem 4.19

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. Then we have

$$\begin{aligned} \widehat{R}(K,L,P,W)&=\frac{c+3}{4}\{g_M(L,P)g_M(K,W)-g_M(K,P)g_M(L,W)\} \nonumber \\&+\frac{c-1}{4}\{g_M( \varphi L,P)g_M(\varphi K,W) \nonumber \\&-g_M(\varphi K,P)g_M(\varphi L,W)-2g_M(\varphi K,L)g_M(\varphi P,W)\} \nonumber \\&+g_M(\mathcal {T}_{L}P,\mathcal {T}_{K}W)-g_M(\mathcal {T}_{K}P,\mathcal {T}_{L}W) \end{aligned}$$

(4.14)

and

$$\begin{aligned} \widehat{K}(K,L)&=\frac{c+3}{4}\{g_M(L,K)g_M(K,L)-g_M(K,K)g_M(L,L)\} \nonumber \\&-3\frac{c-1}{4}g_M(\varphi K,L)+g_M(T_{L}K,T_{K}L)-g_M(\mathcal {T}_{K}K,\mathcal {T}_{L}L) \end{aligned}$$

(4.15)

for all $K, L, P, N\in \Gamma (\mathcal {D}_{\theta }).$

Theorem 4.20

Let $\phi$ be a hemi-slant $\xi ^{\perp }$ Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. If $\mathcal {D}_{\theta }$ is totally geodesic, then we have

$$\begin{aligned} \widehat{k}_{\theta }=p\frac{(c+3)(2p-1)+3(c-1)\cos ^{2}\theta }{2}. \end{aligned}$$

Proof

For any $K,L\in \Gamma (\mathcal {D}_{\theta }),$ using (4.14), we derive

$$\begin{aligned} \widehat{S}_{\theta }(K,L)=\frac{c+3}{4}(2p-1)g_M(K,L)+3\frac{c-1}{4}\cos ^{2}\theta g_M(K,L) \end{aligned}$$

(4.16)

where $\{E_1,...,E_{2p}\}$ is orthonormal basis on $\Gamma (\mathcal {D}_{\theta })$. From the above equation, we obtain the proof.    $\square$

Corollary 4.21

Let $\phi$ be a hemi-slant $\xi ^{\perp }$-Riemannian submersion from a Sasakian manifold $(M, \varphi , \eta , \xi ,g_M)$ onto Riemannian manifold $(N,g_N)$ with a hemi-slant angle $\theta$. If $\mathcal {D}_{\theta }$ is totally geodesic distribution, then $\mathcal {D}_{\theta }$ is Einstein.

5 Quasi Hemi-slant Submanifolds of Cosymplectic Manifolds

In this Sect. 5, we will finally discuss some results of quasi hemi-slant submanifolds introduced and studied by Rajendra Prasad et al. [38]. First, we have

Definition 5.1

A submanifold M of an almost contact metric manifold $\overline{M}$ is called a quasi hemi-slant submanifold if there exist distributions D, $D^{\theta }$ and $D^{\perp }$ such that (i) TM admits the orthogonal direct decomposition as

$$\begin{aligned} TM=D\oplus D^{\theta }\oplus D^{\perp }\oplus <\xi > . \end{aligned}$$

(ii) The distribution D is $\phi$ invariant, i.e., $\phi D=D$.(iii) For any nonzero vector field $X\in (D^{\theta })_{p},$ $p\in M,$ the angle $\theta$ between JX and $(D^{\theta })_{p}$ is constant and independent of the choice of point p and X in $(D^{\theta })_{p}.$

(iv) The distribution $D^{\perp }$ is $\phi$ anti-invariant, i.e., $\phi D^{\perp }\subseteq T^{\perp }M$.

In this case, we call $\theta$ the quasi hemi-slant angle of M. Suppose the dimension of distributions D, $D^{\theta }$ and $D^{\perp }$ are $n_{1},n_{2}$ and $n_{3}$, respectively. Then we can easily see the following particular cases:

(i) If $n_{1}=0$, then M is a hemi-slant submanifold.

(ii) If $n_{2}=0$; then M is a semi-invariant submanifold.

(iii) If $n_{3}=0$, then M is a semi-slant submanifold.

We say that a quasi hemi-slant submanifold M is proper if $D\ne \{0\}$, $D^{\perp }\ne \{0\}$ and $\theta \ne 0,\frac{\pi }{2}$.

This means that the notion of quasi hemi-slant submanifold is a generalization of invariant, anti-invariant, semi-invariant, slant, hemi-slant, semi-slant submanifolds. Let M be a quasi hemi-slant submanifold of an almost contact metric manifold $\overline{M}$. We denote the projections of $X\in \Gamma (TM)$ on the distributions D, $D^{\theta }$ and $D^{\perp }$ by P, Q and R, respectively. Then we can write for any $X\in \Gamma (TM)$

$$\begin{aligned} X=PX+QX+RX+\eta \left( X\right) \xi . \end{aligned}$$

(5.1)

Now we put

$$\begin{aligned} \phi X=TX+NX, \end{aligned}$$

(5.2)

where TX and NX are tangential and normal components of $\phi X$ on M. Using (5.1) and (5.2), we obtain

$$\begin{aligned} \phi X=TPX+NPX+TQX+NQX+TRX+NRX. \end{aligned}$$

Since $\phi D=D$ and $\phi D^{\perp }\subseteq T^{\perp }M$, we have $NPX=0$ and $TRX=0$. Therefore, we get

$$\begin{aligned} \phi X=TPX+TQX+NQX+NRX. \end{aligned}$$

(5.3)

Then for any $X\in \Gamma (TM),$ it is easy to see that

$$\begin{aligned} TX=TPX+TQX \end{aligned}$$

and

$$\begin{aligned} NX=NQX+NRX. \end{aligned}$$

For any $V \in \Gamma (T^{\perp }M)$, we can put

$$\begin{aligned} \phi V = tV+nV \end{aligned}$$

where tV and nV are the tangential and normal componenets of $\phi V$ on M, respectively.

An almost contact metric manifold is called a cosymplectic manifold if $(\widehat{\nabla }_{X}\phi )Y=0,$ $\widehat{\nabla } _{X}\xi =0\,\,$ $\forall \,\, X,\, Y\in \Gamma (T\widehat{M}),$ where $\widehat{\nabla }$ represents the Levi-Civita connection of $(\widehat{M}, g).$

The covariant derivative of $\phi$ is defined as

$$\begin{aligned} (\widehat{\nabla }_{X}\phi )Y=\widehat{\nabla }_{X}\phi Y-\phi \widehat{\nabla }_{X}Y. \end{aligned}$$

If $\widehat{M}$ is a cosymplectic manifold, then we have

$$\begin{aligned} \phi \widehat{\nabla }_{X}Y=\widehat{\nabla }_{X}\phi Y. \end{aligned}$$

Let M be a Riemannian manifold isometrically immersed in $\widehat{M}$ and the induced Riemannian metric on M is denoted by the same symbol g throughout this paper. Let A and h denote the shape operator and second fundamental form, respectively, of submanifolds of M into $\widehat{M}.$ The Gauss and Weingarten formulas are given by

$$\begin{aligned} \widehat{\nabla }_{X}Y=\nabla _{X}Y+h(X,Y) \end{aligned}$$

and

$$\begin{aligned} \widehat{\nabla }_{X}V=-A_{V}X+\nabla _{X}^{\perp }V \end{aligned}$$

for any vector fields $X,\, Y\in \Gamma (T{M})$ and V on $\Gamma (T^{\perp }M),$ where $\nabla$ is the induced connection on M and $\nabla ^{\perp }$ represents the connection on the normal bundle $T^{\perp }M$ of M and $A_{V}$ is the shape operator of M with respect to normal vector V $\in$ $\Gamma (T^{\perp }M).$ Moreover, $A_{V}$ and the second fundamental form h :  $TM\otimes TM\longrightarrow T^{\perp }M$ of M into $\widehat{M}$ are related by

$$\begin{aligned} g(h(X,Y),V)=g(A_{V}X,Y), \end{aligned}$$

for any vector fields $X,Y\in \Gamma (TM)$ and V on $\Gamma (T^{\perp }M).$

5.1 Integrability of Distributions

Theorem 5.2

Let M be a proper quasi hemi-slant submanifold of a cosymplectic manifold $\overline{M}.$ Then the invariant distribution D is integrable if and only if

$$\begin{aligned} g(\nabla _{X}TY-\nabla _{Y}TX,TQZ)=g(h(Y,TX)-h(X,TY),NQZ+NRZ) \end{aligned}$$

for any X,  $Y\in \Gamma (D)$ and $Z\in \Gamma (D^{\theta }\oplus D^{\perp }).$

Proof

For a cosymplectic manifold, we have

$$\begin{aligned} \overline{\nabla }_{X} \xi =0 \,\,\, \forall \,\,\, X \in \Gamma (D). \end{aligned}$$

(5.4)

If $Y \in \Gamma (D)$, then $g(Y, \xi )=0$. Thus, one gets

$$\begin{aligned} g(\overline{\nabla }_{X} Y, \xi )+g(Y, \overline{\nabla }_{X} \xi )=0. \end{aligned}$$

(5.5)

Now, $g([X, Y], \xi )=g(\overline{\nabla }_{X} Y, \xi )-g(\overline{\nabla }_{Y} X, \xi )=0$.

Also, we have

$$\begin{aligned}&g([X,Y],Z) =g(\overline{\nabla }_{X}\phi Y,\phi Z)-g(\overline{\nabla } _{Y}\phi X,\phi Z)=g(\nabla _{X}TY \nonumber \\&\,\,\,\,-\nabla _{Y}TX,TQZ)+g(h(X,TY)-h(Y,TX),NQZ+NRZ) \end{aligned}$$

which completes the proof.    $\square$

Similarly, we have

Theorem 5.3

Let M be a proper quasi hemi-slant submanifold of a cosymplectic manifold $(\overline{M},g,\phi ).$ Then the slant distribution $D^{\theta }$ is integrable if and only if

$$\begin{aligned}&g(A_{NW}Z-A_{NZ}W,TPX) =g(A_{NTW}Z-A_{NTZ}W,X) \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+g(\nabla _{Z}^{\perp }NW-\nabla _{W}^{\perp }NZ,NRX) \nonumber \end{aligned}$$

for any $Z,\, W$ $\in \Gamma (D^{\theta })$ and X $\in \Gamma (D\oplus D^{\perp }).$

Theorem 5.4

Let M be a quasi hemi-slant submanifold of a cosymplectic manifold $\overline{M}.$ Then the anti-invariant distribution $D^{\perp }$ is integrable if and only if

$$\begin{aligned} g(T([Z, W]), TX)=g(\nabla _{W}^{\perp }NZ-\nabla _{Z}^{\perp }NW, NQX) \end{aligned}$$

for any Z, W $\in \Gamma (D^{\perp })$ and X $\in \Gamma (D\oplus D^{\theta }).$

5.2 Totally Geodesic Foliations

Theorem 5.5

Let M be a proper quasi hemi-slant submanifold of a cosymplectic manifold $\overline{M}.$ Then M is totally geodesic if and only if

$$\begin{aligned}&g(h(X,PY)+\cos ^{2}\theta h(X,QY),U) =g(\nabla _{X}^{\perp }NTQY,U) \nonumber \\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,+g(A_{NQY}X+A_{NRY}X,tU)-g(\nabla _{X}^{\perp }NY,nU) \end{aligned}$$

for any $X,Y\in \Gamma (TM)$ and $U\in \Gamma (T^{\perp }M)$.

Proof

For any $X,\, Y\in \Gamma (TM),$ $\ U\in \Gamma \left( T^{\perp }M\right)$, we have

$$\begin{aligned} g(\overline{\nabla }_{X}Y,U)= & {} g(\overline{\nabla }_{X}PY,U)+g(\overline{\nabla }_{X}QY,U)+g(\overline{\nabla }_{X}RY,U) \\= & {} g(\overline{\nabla }_{X}\phi PY,\phi U)+g(\overline{\nabla }_{X}TQY,\phi U)+g(\overline{\nabla }_{X}NQY,\phi U) \\&+g(\overline{\nabla }_{X}\phi RY,\phi U). \end{aligned}$$

$$\begin{aligned} g(\overline{\nabla }_{X}Y,U)= & {} g(h(X,PY)+\cos ^{2}\theta h(X,QY),U)-g(\nabla _{X}^{\perp }NTQY,U) \nonumber \\&-g(A_{NQY}X+A_{NRY}X,tU)+g(\nabla _{X}^{\perp }NY,nU) \end{aligned}$$

which completes the proof.    $\square$

Similarly, we have

Theorem 5.6

Let M be a proper quasi hemi-slant submanifold of a cosymplectic manifold $\overline{M}.$ Then anti-invariant distribution $D^{\perp }$ defines totally geodesic foliation if and only if

$$\begin{aligned} g(A_{\phi Y}X, TPZ+tQZ) =g(\nabla _{X}^{\perp }{\phi Y}, nQZ),\,\,\,\, g(A_{\phi Y}X, tV) =g(\nabla _{X}^{\perp }\phi Y, nV) \end{aligned}$$

for any X,  $Y\in \Gamma (D^{\perp })$, $Z\in \Gamma (D\oplus D^{\theta })$ and $V\in \Gamma \left( T^{\perp }M\right) .$

Theorem 5.7

Let M be a proper quasi hemi-slant submanifold of a cosymplectic manifold $\overline{M}.$ Then the slant distribution $D^{\theta }$ defines a totally geodesic foliation on M if and only if

$$\begin{aligned}&g(\nabla _{X}^{\perp }NY,NRZ) =g(A_{NY}X,TPZ)-g(A_{NTY}X,Z),\text { and } \nonumber \\&g(A_{NY}X,tV) =g(\nabla _{X}^{\perp }NY,nV)-g(\nabla _{X}^{\perp }NTY,V) \end{aligned}$$

for any $X,Y\in \Gamma (D^{\theta })$, $Z\in \Gamma (D\oplus D^{\perp })$ and $V\in \Gamma \left( T^{\perp }M\right) .$

5.3 Examples

Now we discuss few examples from [38]

Example 5.8

Let us consider a 15-dimensional differentiable manifold

$$\begin{aligned} \overline{M} =\{(x_{i},y_{i,}z)=(x_{1},x_{2},...,x_{7},y_{1},y_{2},...,y_{7},z)\in \mathrm {R}^{15}\}. \end{aligned}$$

And choose the vector fields

$$\begin{aligned} E_{i}=\frac{\partial }{\partial y_{i}},\, \, E_{7+i}=\frac{\partial }{\partial x_{i}}, \, \, E_{15}=\xi =\frac{\partial }{\partial z},\text { for } i=1,\, 2,\, ...,7. \end{aligned}$$

Let g be a Riemannian metric defined by

$$\begin{aligned} g=(dx_{1})^{2}+(dx_{2})^{2}+\cdots +(dx_{7})^{2}+(dy_{1})^{2}+(dy_{2})^{2}+\cdots +(dy_{7})^{2}+(dz)^{2}. \end{aligned}$$

We define $\left( 1,1\right)$-tensor field $\phi$ as

$$\begin{aligned} \phi \left( \frac{\partial }{\partial x_{i}}\right) =\frac{\partial }{\partial y_{i}}, \, \, \phi \left( \frac{\partial }{\partial y_{j}} \right) =-\frac{\partial }{\partial x_{j}},\, \ \ \phi \left( \frac{\partial }{\partial z}\right) =0 \ \ \forall \ i,j=1,2,...,7. \end{aligned}$$

Thus, $(\overline{M},\phi ,\xi ,\eta ,g)$ is an almost contact metric manifold. Also, we can easily show that $(\overline{M},\phi ,\xi ,\eta ,g)$ is a cosymplectic manifold of dimension 15.

Let M be a submanifold of $\overline{M}$ defined by

$$\begin{aligned} f\left( u,v,w,r,s,t,q\right) =\left( u,w,0,\frac{s}{\sqrt{2}},0,\frac{t}{\sqrt{2}},0,v,r\cos \theta ,r\sin \theta ,0,\frac{s}{\sqrt{2}},0,\frac{t}{\sqrt{2}},q\right) , \end{aligned}$$

where $0< \theta < \frac{\pi }{2}$. Now the tangent bundle of M is spanned by the set $\{Z_{1},Z_{2},Z_{3},Z_{4},Z_{5},Z_{6},Z_{7}\}$, where

$$\begin{aligned} Z_{1}=\frac{\partial }{\partial x_{1}},\, \ Z_{2}=\frac{\partial }{ \partial y_{1}}, \ Z_{3}=\frac{\partial }{\partial x_{2}}, \end{aligned}$$

$$\begin{aligned} Z_{4}=\cos \theta \frac{\partial }{\partial y_{2}}+\sin \theta \frac{ \partial }{\partial y_{3}},{ \ }Z_{5}=\frac{1}{\sqrt{2}}\left( \frac{ \partial }{\partial x_{4}}+\frac{\partial }{\partial y_{5}}\right) , \end{aligned}$$

$$\begin{aligned} Z_{6}=\frac{1}{\sqrt{2}}\left( \frac{\partial }{\partial x_{6}}+\frac{ \partial }{\partial y_{7}}\right) ,{ \ }Z_{7}=\frac{\partial }{\partial z}. \end{aligned}$$

Thus, we have

$$\begin{aligned} \phi Z_{1}=\frac{\partial }{\partial y_{1}},{ \ }\phi Z_{2}=-\frac{ \partial }{\partial x_{1}},{ \ }\phi Z_{3}=\frac{\partial }{\partial y_{2}}, \end{aligned}$$

$$\begin{aligned} \phi Z_{4}=-\left( \cos \theta \frac{\partial }{\partial x_{2}}+\sin \theta \frac{\partial }{\partial x_{3}}\right) ,{ \ }\phi Z_{5}=\frac{1}{\sqrt{ 2}}\left( \frac{\partial }{\partial y_{4}}-\frac{\partial }{\partial x_{5}} \right) , \end{aligned}$$

$$\begin{aligned} \phi Z_{6}=\frac{1}{\sqrt{2}}\left( \frac{\partial }{\partial y_{6}}-\frac{ \partial }{\partial x_{7}}\right) ,{ \ }\phi Z_{7}=0. \end{aligned}$$

Now, let the distributions $D=span\{Z_{1},Z_{2}\},\ \ D^{\theta }=span\{Z_{3},Z_{4}\},\ \ D^{\perp }=span\{Z_{5},Z_{6}\}.$ And D is invariant, $D^{\theta }$ is slant with slant angle $\theta$ and$\ D^{\perp }$ is anti-invariant.