Characterizations of $\mathcal{PR}$-Pseudo-Slant Warped Product Submanifold of Para-Kenmotsu Manifold with Slant Base

Abstract

In this article, we study the properties of $\mathcal{PR}$-pseudo-slant submanifold of para-Kenmotsu manifold and obtain the integrability conditions for the slant distribution and anti-invariant distribution of such submanifold. We derived the necessary and sufficient conditions for a $\mathcal{PR}$-pseudo-slant submanifold of para-Kenmotsu manifold to be a $\mathcal{PR}$-pseudo-slant warped product which are in terms of warping functions and shape operator. Some examples of $\mathcal{PR}$-pseudo-slant warped products of para-Kenmotsu manifold are also illustrated in the article.

1. Introduction

At the end of the twentieth century, B.Y. Chen initiated the study of slant submanifold as a generalization of $\mathcal{CR}$-submanifolds [1]. Later, A. Carriazo studied slant submanifolds in contact metric manifold as a special case of bi-slant submanifolds [2]. Thereafter, he studied pseudo-slant submanifolds under the name anti-slant [3]. The slant submanifold with pseudo-Riemannian metric was also initiated by B.Y. Chen et al. [4,5]. The authors of [6,7] studied slant submanifold of Kaehler and contact manifolds with respect to the pseudo-Riemannian metric. P. Alegre and A. Carriazo studied slant submanifolds in para-Hermitian manifold and provided detailed descriptions of such type of submanifolds in pseudo-Riemannian metric.

On the other hand, the study of warped product manifold is one of the most significant generalizations of Cartesian product of pseudo-Riemannian manifolds (or Riemannian manifolds). This fruitful generalization was initiated by R. L Bishop and B. O’Neill in 1969 (see [8]). The notion of warped products appeared in the physical and mathematical literature before 1969, for instance, semi-reducible space, which is used for warped product by Kruchkovich in 1957 [9]. It has been successfully utilized in general theory of relativity, black holes, and string theory. The warped product is defined as follows:

Assume that B and F are two pseudo-Riemannian manifolds with pseudo-Riemannian metric $g_B$ and $g_F$, respectively and f is a smooth function defined by $f:B⟶\left(0,1\right)$. Then, a pseudo-Riemannian manifold $M=B{\times }_{f}F$ is said to be a warped product [8,10] if it is furnished a pseudo-Riemannian warping metric g fulfilling for any tangent vector U to M as the following:

$$\begin{array}{c}\hfill g(U,U)=g({\pi }_{*}U,{\pi }_{*}U)+{\left(f\circ \pi \right)}^{2}g({\pi }_{*}^{\prime }U,{\pi }_{*}^{\prime }U),\end{array}$$

(1)

where $\pi :B\times F⟶B$ and ${\pi }^{\prime }:B\times F⟶F$ are natural projections on M, and * denotes the push-foreword map (or differential map). The smooth function f is called warping function. Moreover, the above relation is equivalent to

$$\begin{array}{c}\hfill g=g_B+f^2{g}_F.\end{array}$$

(2)

If $f:B⟶\left(0,1\right)$ is non-constant, then M is called a non-trivial (or proper) warped product, otherwise it is trivial. Now, consider any $U_1,U_2\in \mathsf{\Gamma }\left(TB\right)$ and $V_1,V_2\in \mathsf{\Gamma }\left(TF\right)$, then from the Proposition $3.1$ of [10] (page no. 49), we obtain that

$$\begin{array}{cc}\hfill & {\nabla }_{U_1}{U}_2\in \mathsf{\Gamma }\left(TB\right),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\nabla }_{U_1}{V}_1={\nabla }_{V_1}{U}_1=U_1(lnf)V_1,\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & tan\left({\nabla }_{V_1}{V}_2\right)={\nabla }_{V_1}^{{}^{\prime }}{V}_2,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & nor\left({\nabla }_{V_1}{V}_2\right)=h^F(V_1,V_2)=-\frac{g(V_1,V_2)\nabla f}{f}.\hfill \end{array}$$

(6)

where the symbols ${\nabla }^{{}^{\prime }}$ and h indicates are Levi–Civita connection on B and second fundamental form, respectively. By the consequence (3)–(6), we can conclude that for a warped product manifold $M=B{\times }_{f}F$, the submanifold F is a totally umbilical and the submanifold B is a totally geodesic in M.

In 1956, J.F. Nash derived a very useful theorem in Riemannian geometry known as Nash embedding theorem. The theorem states “every Riemannian manifold can be isometrically embedded in some Euclidean space” (see [11]). This theorem shows that any warped product of Riemannian (or pseudo-Riemannian) manifolds can be realized (or embedded) as a Riemannian (or pseudo-Riemannian) submanifold in Euclidean space. Due to this fact, B.Y. Chen asked a very interesting question in 2002. The question is “What can we conclude from an isometric immersion of an arbitrary warped product into a Euclidean space or into a space form with arbitrary codimension?” (see [10]). Thereafter, B.Y. Chen published the numerous articles on the $\mathcal{CR}$-warped products in K$\ddot{a}$hler manifold (see [12,13]). Thereafter, several authors of [14,15,16,17,18,19,20] studied pseudo-slant warped product in different ambient manifolds. In 2015, A. Ali et al. derived some useful inequalities for a pseudo-slant warped product submanifold in nearly-Kenmotsu manifold [21]. Recently, the authors of [22,23,24] studied pseudo-slant warped product submanifold of Kenmotsu manifold and derived some characterizations and inequalities.

However, in 2014, B.Y. Chen initiated a new class of warped product called $\mathcal{PR}$-warped product and found the exact solutions of the system partial differential equations associated with $\mathcal{PR}$-warped products [25]. Recently, S.K. Srivastava and A. Sharma studied $\mathcal{PR}$-semi-invariant, $\mathcal{PR}$-pseudo-slant, and $\mathcal{PR}$-semi-slant warped product of para-cosymplectic manifold in [26,27,28,29]. In the last two decades, several geometrists studied warped product submanifolds and other submanifolds in different ambient space [26,27,28,29,30,31,32,33,34,35,36,37]. Motivated by them, we analyze the geometry of $\mathcal{PR}$-pseudo-slant warped product submanifolds of para-Kenmotsu manifold which are not studied yet.

This paper is formulated as follows. The second section includes some necessary information related to para-contact and para-Kenmotsu manifold and also contains some important information about the basics of submanifolds in para-Kenmotsu manifold. Section 3 includes some useful results related to integrability of $\mathcal{PR}$-pseudo-slant submanifold in para-Kenmotsu manifold and gives examples of such submanifolds. In Section 4, we analyze the geometry of $\mathcal{PR}$-pseudo-slant warped product submanifolds in para-Kenmotsu manifold and provide some characterization results allied to shape operator and endomorphism t, and also give some examples of $\mathcal{PR}$-pseudo-slant warped product submanifold of para-Kenmotsu manifold.

2. Preliminaries

A smooth manifold ${\tilde{M}}^{2n+1}$ of dimension $(2n+1)$ furnished an almost paracontact (see [26,38,39]) structure $(\phi ,\xi ,\eta )$ which includes a $(1,1)$-type tensor field $\phi$, a vector field $\xi$, and a 1-form $\eta$ globally defined on ${\tilde{M}}^{2n+1}$ which satisfies the accompanying relation for all $U\in \mathsf{\Gamma }\left(T{M}^{2n+1}\right)$:

$$\begin{array}{c}\hfill {\phi }^{2}U=U-\eta \left(U\right)\xi ,\eta \left(\xi \right)=1.\end{array}$$

(7)

The tensor field $\phi$ induces an almost paracomplex structure $\mathcal{J}$ on a $2n$-dimensional horizontal distribution $\mathfrak{D}$ described as the kernel of 1-form $\eta$, i.e., $\mathfrak{D}=ker\left(\eta \right)$. The horizontal distribution $\mathfrak{D}$ can be expressed as an orthogonal direct sum of the two eigen distribution ${\mathfrak{D}}^{+}$ and ${\mathfrak{D}}^{-}$, the eigen distributions ${\mathfrak{D}}^{+}$ and ${\mathfrak{D}}^{-}$ having eigenvalue $+1$ and $-1$, respectively, and each has dimension n. Moreover, $\mathfrak{D}$ is invariant distribution, therefore $T{\tilde{M}}^{2n+1}$ can be expressed in the following form;

$$\begin{array}{c}\hfill T{\tilde{M}}^{2n+1}=\mathfrak{D}\oplus ⟨\xi ⟩.\end{array}$$

(8)

If ${\tilde{M}}^{2n+1}$ admits an almost paracontact structure $(\phi ,\xi ,\eta )$, then it is said to be an almost paracontact manifold [26,39]. In view of (7), we obtain

$$\begin{array}{c}\hfill \eta \circ \phi =0,\phi \circ \xi =0andrank\left(\phi \right)=2n.\end{array}$$

(9)

An almost paracontact manifold ${\tilde{M}}^{2n+1}$ is called an almost paracontact pseudo-metric manifold if it admits a pseudo-Riemannian metric of index n compatible with the triplet $(\phi ,\xi ,\eta )$ by the following relation:

$$\begin{array}{c}\hfill g(\phi U,\phi V)=\eta \left(U\right)\eta \left(V\right)-g(U,V),\end{array}$$

(10)

for all $U,V\in \mathsf{\Gamma }\left(T{\tilde{M}}^{2n+1}\right)$; $\mathsf{\Gamma }\left(T{\tilde{M}}^{2n+1}\right)$ denotes the Lie algebra on ${\tilde{M}}^{2n+1}$. The dual of the unitary structural vector field $\xi$ allied to g is $\eta$, i.e.,

$$\begin{array}{c}\hfill \eta \left(U\right)=g(U,\xi ).\end{array}$$

(11)

By the utilization of (7)–(10), we attain

$$\begin{array}{c}\hfill g(U,\phi V)+g(\phi U,V)=0.\end{array}$$

(12)

Definition 1.

An almost paracontact pseudo-metric manifold ${\tilde{M}}^{2n+1}$ is said to be a para-Kenmotsu manifold [38] if it satisfies

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_U\phi \right)V=\eta \left(V\right)\phi U+g(U,\phi V)\xi .\end{array}$$

(13)

In the relation (13), the symbol $\tilde{\nabla }$ indicates for the Levi–Civita connection with respect to g.

In (13) replacing V by $\xi$ and then applying (7), we achieve that

$$\begin{array}{c}\hfill {\tilde{\nabla }}_U\xi =-{\phi }^{2}U.\end{array}$$

(14)

Proposition 1.

On para-Kenmotsu pseudo-Riemannian manifold, the following relations holds:

$$\begin{array}{cc}\hfill & \eta \left({\tilde{\nabla }}_U\xi \right)=0,\tilde{\nabla }\eta =-\eta \otimes \eta +g,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\mathcal{L}}_{\xi }\phi =0,{\mathcal{L}}_{\xi }\eta =0,{\mathcal{L}}_{\xi }g=-2(g-\eta \otimes \eta ),\hfill \end{array}$$

(16)

where $\mathcal{L}$ denotes the Lie differentiation.

Geometry of Submanifolds

Let M be a m-dimensional paracompact and connected smooth pseudo-Riemannian manifold and ${\tilde{M}}^{2n+1}$ be a para-Kenmotsu manifold. Assume that $\psi :M⟶{\tilde{M}}^{2n+1}$ is an isometric immersion. Then $\psi \left(M\right)$ is known as an isometrically immersed submanifold of a para-Kenmotsu manifold. Let us denote that ${\psi }_{*}$ for the differential map (or push forward map) of immersion $\psi$ is characterized by ${\psi }_{*}:T_{p}M⟶T_{\psi \left(p\right)}{\tilde{M}}^{2n+1}$. Therefore, the induced pseudo-Riemannian metric $\mathfrak{g}$ on $\psi \left(M\right)$ is defined as follows: $\mathfrak{g}{(U,V)}_p=g({\psi }_{*}U,{\psi }_{*}V)$, for all $U,V\in T_{p}M$. For our convenience, we use M and p in the place of $\psi \left(M\right)$ and $\psi \left(p\right)$. Now, we denote $\mathsf{\Gamma }\left(TM\right)$ for set of all tangent vector fields on M, $\mathsf{\Gamma }\left(T{M}^{\perp }\right)$ for the set of all normal vector fields of M, ∇ for induced Levi–Civita connection on $TM$, and ${\nabla }^{\perp }$ for normal connection on the normal bundle $\mathsf{\Gamma }\left(T{M}^{\perp }\right)$. Then, Gauss and Weingarten formulas are characterized by the relation

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_{U}V=& {\nabla }_{U}V+h(U,V),\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill {\tilde{\nabla }}_U\zeta =& -A_{\zeta }U+{\nabla }_U^{\perp }\zeta ,\hfill \end{array}$$

(18)

for any $U,V\in \mathsf{\Gamma }\left(TM\right)$ and $\zeta \in \mathsf{\Gamma }\left(T{M}^{\perp }\right)$, where $A_{\zeta }$ is a shape operator and h is a second fundamental form which are allied to the normal section $\zeta$ by the following relation:

$$\begin{array}{c}\hfill g(h(U,V),\zeta )=g(A_{\zeta }U,V).\end{array}$$

(19)

The mean curvature vector H on M is described by $H=\frac{1}{m}trace\left(h\right)$. Let $p\in M$ and $\{U_1,U_2,\cdots ,U_m,U_{m+1},\cdots ,U_{2n+1}\}$ be an orthonormal basis of the $T_p{\tilde{M}}^{2n+1}$ in which $\{U_1,U_2,\cdots ,U_m\}$ are the tangent to M and $\{U_{m+1},U_{m+2},\cdots ,U_{2n+1}\}$ are normal to M. Now, we set

$$\begin{array}{c}\hfill h_{ij}^k=g(h(U_i,U_j),U_k),\end{array}$$

(20)

for $i,j\in \{1,2,\cdots ,m\}$ and $k\in \{m+1,m+2,\cdots ,2n+1\}$. The norm of h is defined by the following relation:

$$\begin{array}{c}\hfill \parallel h\parallel =\sqrt{\left(\sum _{i,j=1}^{m}g(h(U_i,U_j),h(U_i,U_j))\right)}.\end{array}$$

(21)

An isometrically immersed submanifold M of a para-Kenmotsu manifold ${\tilde{M}}^{2n+1}(\phi ,\xi ,\eta ,g)$ is said to be (see [26,39])

• Totally geodesic if h vanishes identically, i.e., $h\equiv 0$.
• Umbilical if for a normal vector field $\zeta$, shape operator $A_{\zeta }$ is proportional to identity transformation.
• Totally umbilical if M satisfies for every $U,V\in \mathsf{\Gamma }\left(TM\right)$

  $$\begin{array}{c}\hfill h(U,V)=g(U,V)H.\end{array}$$

  (22)
• Minimal if trace of h (or H) vanishes identically.
• Extrinsic sphere if M satisfies (22) and H is parallel with respect to ${\nabla }^{\perp }$.

From now on, we denote para-Kenmotsu manifold by ${\mathcal{K}}^{2n+1}$ and its pseudo-Riemannian submanifold by $\mathcal{N}$. For any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$, we substitute $tU=tan\left(\phi U\right)$ and $nU=nor\left(\phi U\right)$, where $tan$ and $nor$ are natural projections associated with the following direct sum:

$$\begin{array}{c}\hfill T_p{\mathcal{K}}^{2n+1}=T_p\mathcal{N}\oplus T_p{\mathcal{N}}^{\perp }.\end{array}$$

(23)

Thus, we can write

$$\begin{array}{c}\hfill \phi U=tU+nU.\end{array}$$

(24)

Similarly, for any $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$, we have

$$\begin{array}{c}\hfill \phi \zeta =t^{{}^{\prime }}\zeta +n^{{}^{\prime }}\zeta ,\end{array}$$

(25)

where $t^{{}^{\prime }}\zeta =tan\left(\phi \zeta \right)$ and $n^{{}^{\prime }}\zeta =nor\left(\phi \zeta \right)$. In view of (12) and (22)–(25), we attain for any $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ and $\forall {\zeta }_1,{\zeta }_2\in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$ that

$$\begin{array}{cc}\hfill & g(n^{{}^{\prime }}{\zeta }_1,{\zeta }_2)=-g({\zeta }_1,n^{{}^{\prime }}{\zeta }_2),g(tU,V)=-g(U,tV).\hfill \end{array}$$

(26)

Moreover, by the consequences of Equations (12) and (24)–(25), we have

$$\begin{array}{c}\hfill g(nU,\zeta )=-g(U,t^{{}^{\prime }}\zeta ).\end{array}$$

(27)

Further, the covariant derivative of $\phi$, t and n are characterized by, respectively,

$$\begin{array}{cc}\hfill \left({\tilde{\nabla }}_U\phi \right)V=& {\tilde{\nabla }}_U\phi V-\phi {\tilde{\nabla }}_{U}V,\hfill \end{array}$$

(28)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)V=& {\nabla }_{U}tV-t{\nabla }_{U}V,\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}n\right)V=& {\nabla }_U^{\perp }nV-n{\nabla }_{U}V,\hfill \end{array}$$

(30)

for some $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$.

Proposition 2.

Let $\mathcal{N}$ be tangent to ξ in ${\mathcal{K}}^{2n+1}$. Then, we obtain

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)V=& A_{nV}U+t^{{}^{\prime }}h(U,V)+\eta \left(V\right)tU-g(tU,V)\xi ,\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}n\right)V=& n^{{}^{\prime }}h(U,V)+\eta \left(V\right)nU-h(U,tV),\hfill \end{array}$$

(32)

for every $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$.

Proof. 

By the consequence of (17)–(18), (24), (28)–(30), we arrive at

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_U\phi \right)V+A_{nV}U=-t^{{}^{\prime }}h(U,V)+\left({\nabla }_{U}t\right)V-n^{{}^{\prime }}h(U,V)+h(U,tV)+\left({\nabla }_{U}n\right)V,\end{array}$$

for any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$. Employing (13) and (24) into the above expression, then considering tangential part and normal part of the obtained expression, we have (31) and (32), respectively. □

Proposition 3.

If ξ is normal to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$, then we acquire that

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)V=& t^{{}^{\prime }}h(U,V)+A_{nV}U,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}n\right)V=& n^{{}^{\prime }}h(U,V)+g(U,tV)\xi -h(U,tV),\hfill \end{array}$$

(34)

for all $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$.

Proof. 

Immediately, from (13), (17)–(18), (24), (28)–(30), we derive (33) and (34). □

Proposition 4.

Let $\mathcal{N}$ be tangent to ξ in ${\mathcal{K}}^{2n+1}$. Then, we receive that

$$\begin{array}{cc}\hfill \left({\nabla }_U{t}^{{}^{\prime }}\right)\zeta =& A_{n^{{}^{\prime }}\zeta }U-g(nU,\zeta )\xi -t{A}_{\zeta }U,\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill \left({\nabla }_U{n}^{{}^{\prime }}\right)\zeta =& -h(U,t^{{}^{\prime }}\zeta )-n{A}_{\zeta }U,\hfill \end{array}$$

(36)

for any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ and $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$.

Proof. 

Employing (17)–(18), (25), (29), and (30) into (28), we achieve that

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_U\phi \right)\zeta =\left({\nabla }_U{n}^{{}^{\prime }}\right)\zeta -A_{n^{{}^{\prime }}\zeta }U+t{A}_{\zeta }U+n{A}_{\zeta }U+h(U,t^{{}^{\prime }}\zeta )+\left({\nabla }_U{t}^{{}^{\prime }}\right)\zeta ,\end{array}$$

for any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$. Utilizing (13) and (24) into the above expression, we achieve (35) and (36). □

Proposition 5.

If $\mathcal{N}$ is normal to ξ in ${\mathcal{K}}^{2n+1}$, then we achieve for any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ and $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$ that

$$\begin{array}{cc}\hfill & \left({\nabla }_U{t}^{{}^{\prime }}\right)\zeta =A_{n^{{}^{\prime }}\zeta }U-t{A}_{\zeta }U+\eta \left(\zeta \right)tU,\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & \left({\nabla }_U{n}^{{}^{\prime }}\right)\zeta =-n{A}_{\zeta }U+\eta \left(\zeta \right)nU+g(U,t^{\prime }\zeta )\xi -h(U,tV).\hfill \end{array}$$

(38)

Proof. 

The process is similar to Proposition 4. □

Consider $U,\xi \in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ as two vector fields; thus, by the direct application of (14) and (17)–(18), we gain

$$\begin{array}{cc}\hfill {\nabla }_U\xi =& -{\phi }^{2}U,h(U,\xi )=0.\hfill \end{array}$$

(39)

If $\xi \in \mathsf{\Gamma }\left(T{N}^{\perp }\right)$, then by the consequence of (14) and (18), we have

$$\begin{array}{cc}\hfill A_{\xi }U=& U,{\nabla }_U^{\perp }\xi =0.\hfill \end{array}$$

(40)

In view of (39) and (40), we give the following remarks:

Remark 1.

Let ξ be tangent to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$. Then relation (39) holds on $\mathcal{N}$.

Remark 2.

Let ξ be normal to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$. Then Equation (40) holds in $\mathcal{N}$.

Proposition 6.

Let ξ be tangent to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$. Then, the endomorphism t and bundle 1-form n satisfies

$$\begin{array}{cc}\hfill & t^2+t^{{}^{\prime }}n=\mathcal{I}-\eta \otimes \xi ,\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & nt+n^{{}^{\prime }}n=0.\hfill \end{array}$$

(42)

Proof. 

Operating $\phi$ on (24), we have

$$\begin{array}{c}\hfill {\phi }^{2}U=\phi \left(tU\right)+\phi \left(nU\right).\end{array}$$

Employing (7) and (24) into the above expression, we achieve

$$\begin{array}{c}\hfill U-\eta \left(U\right)\xi =t^{2}U+ntU+t^{{}^{\prime }}nU+n^{{}^{\prime }}nU.\end{array}$$

Comparing tangential and normal parts of the above expression, we obtain (41) and (42). □

In similar way, we prove the following result:

Proposition 7.

Let ξ be normal to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$. Then, the following relations holds:

$$\begin{array}{cc}\hfill & t{t}^{{}^{\prime }}+t^{{}^{\prime }}{n}^{{}^{\prime }}=0,\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & n{t}^{{}^{\prime }}+n^{{}^{\prime 2}}=\mathcal{I}.\hfill \end{array}$$

(44)

3. $\mathcal{PR}$-Pseudo-Slant Submanifolds

Definition 2.

Let $\mathcal{N}$ be tangent to ξ in ${\mathcal{K}}^{2n+1}$. Then $\mathcal{N}$ is called a slant [40] if the quotient $\frac{g\left(tU,tU\right)}{g\left(\phi U,\phi U\right)}=\lambda \left(\theta \right)$ is constant for any non-zero spacelike or timelike vector $U\in T_p\mathcal{N}$ and for any $p\in \mathcal{N}$. The symbol θ is used for slant angle and $\lambda \left(\theta \right)$ for slant coefficient or function. In other words, if $\mathcal{N}$ is slant then λ does not depend on the vector field and point.

Remark 3.

The value of $\lambda \left(\theta \right)$ can be

(i) 

$\lambda ={cosh}^2\theta \in [1,\infty )$ for $\frac{\parallel tU\parallel }{\parallel \phi U\parallel }>1$, $tU$ is timelike or spacelike for any spacelike or timelike vector field U and $\theta >0$.

(ii) 

$\lambda \left(\theta \right)={cos}^2\theta \in [0,1]$ for $\frac{\parallel tU\parallel }{\parallel \phi U\parallel }<1$, $tU$ is timelike or spacelike for any spacelike or timelike vector field U and $0\le \theta \le 2\pi$.

(iii) 

$\lambda \left(\theta \right)=-{sinh}^2\theta \in (-\infty ,0]$ for $tU$ is timelike or spacelike for any timelike or spacelike vector field U and $\theta <0$.

Remark 4.

If $\lambda =0$, then $\mathcal{N}$ is an anti-invariant submanifold.

Remark 5.

If $\lambda =1$, then $\mathcal{N}$ is an invariant submanifold.

Example 1.

Let us consider $\tilde{M}={\mathbb{R}}^4\times {\mathbb{R}}^{+}$ together with the the usual Cartesian coordinates $(x_1,x_2,y_1,y_2,s)$. Then the structure $(\phi ,\xi ,\eta )$ over $\tilde{M}$ is defined by

$$\begin{array}{c}\hfill \phi \left(\frac{\partial }{\partial x_i}\right)=\frac{\partial }{\partial y_i},\phi \left(\frac{\partial }{\partial y_i}\right)=\frac{\partial }{\partial x_i},\phi \left(\frac{\partial }{\partial s}\right)=0,\eta =ds,\end{array}$$

(45)

where $i,j\in \{1,2\}$ and the pseudo-Riemannian metric tensor g is defined as

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_i}\right)=e^{-2s},g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_i}\right)=-e^{-2s},g\left(\frac{\partial }{\partial s},\frac{\partial }{\partial s}\right)=1,\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_k}\right)=0,g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial y_k}\right)=0,g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_k}\right)=0.\hfill \end{array}$$

(47)

Then, by simple computation, we can easily see that $\tilde{M}$ is para-Kenmotsu manifold. Suppose $M_1$, $M_2$, and $M_3$ are immersed submanifolds into $\tilde{M}$ by the immersions σ, ${\sigma }^{{}^{\prime }}$, and ${\sigma }^{{}^{″}}$ respectively, defined by

$$\begin{array}{cc}\hfill & \sigma (u,v,\alpha )=\left(u,\sqrt{3}v,\frac{3}{2}v,v,\alpha \right),\hfill \\ \hfill & \sigma (u,v,\alpha )=\left(u,\frac{1}{2}v,\sqrt{2}v,v,\alpha \right),\hfill \\ \hfill & \sigma (u,v,\alpha )=\left(u,3v,2v,v,\alpha \right).\hfill \end{array}$$

By simple computation, we conclude that $M_1$, $M_2$, and $M_3$ are slant submanifolds of type I, type II, and type III of para-Kenmotsu manifold, respectively.

Theorem 1

([40]). Let ξ be tangent to $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$. Then $\mathcal{N}$ is slant if and only if there exists a constant $\lambda \in \mathbb{R}$ such that

$$\begin{array}{c}\hfill t^2=\lambda (\mathcal{I}-\eta \otimes \xi ).\end{array}$$

(48)

In particular, λ is either ${cos}^2\theta$ or ${cosh}^2\theta$ or $-{sinh}^2\theta$.

Theorem 2

([40]). Let $\mathcal{N}$ be a slant submanifold in ${\mathcal{K}}^{2n+1}$ with $\xi \in \mathsf{\Gamma }\left(T\mathcal{N}\right)$. Then, for any $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$, we have

$$\begin{array}{cc}\hfill g(tU,tV)& =\lambda g(\phi U,\phi V),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill g(nU,nV)& =(1-\lambda )g(\phi U,\phi V).\hfill \end{array}$$

(50)

Proposition 8.

Let $\mathcal{N}$ be a slant submanifold in ${\mathcal{K}}^{2n+1}$ with slant coefficient $\lambda \left(\theta \right)$ if and only if

(i) 

$t^{\prime }nU=\left(1-\lambda \right)U$ and $ntU=-n^{\prime }nU$ for non-lightlike tangent vector field U on $\mathcal{N}$.

(ii) 

${\left(n^{\prime }\right)}^2\zeta =\lambda \zeta$ for non-lightlike normal vector field ζ.

Proof. 

Assume $\mathcal{N}$ to be slant submanifold of ${\mathcal{K}}^{2n+1}$.

(i)

Then for every $p\in \mathcal{N}$ and $U\in T\mathcal{N}$, we find

$$\begin{array}{cc}\hfill \phi U=& tU+nU,\hfill \\ \hfill {\phi }^{2}U=& \phi \left(tU+nU\right),\hfill \\ \hfill U-\eta \left(U\right)\xi =& t^{2}U+ntU+t^{\prime }nU+n^{\prime }nU.\hfill \end{array}$$

Equating tangential and normal parts and using (51), we can attain the result.

(ii)

Since, $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$, there exists $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ as $\mathcal{N}$ is slant submanifold such that $nU=\zeta$.

Now, ${\left(n^{\prime }\right)}^2\zeta =n^{\prime }{n}^{\prime }nU=-n^{\prime }ntU=n{t}^{2}U=\lambda \zeta$.

The converse can be easily derived using the same equations. □

Definition 3.

Let $\mathcal{N}$ be tangent to ξ in ${\mathcal{K}}^{2n+1}$. Then $\mathcal{N}$ is said to be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$ if its tangent bundle $T\mathcal{N}$ can orthogonally be decomposed as a direct sum of an anti-invariant distribution ${\mathfrak{D}}_{\perp }$ and a slant distribution ${\mathfrak{D}}_{\lambda }$ i.e., $T\mathcal{N}={\mathfrak{D}}_{\lambda }\oplus {\mathfrak{D}}_{\perp }\oplus ⟨\xi ⟩$, where ξ is a one-dimensional real distribution.

Let P and Q be two orthogonal projections on the slant ${\mathfrak{D}}_{\lambda }$ and anti-invariant distribution ${\mathfrak{D}}_{\perp }$, respectively. Then, for any $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$ can be expressed as follows:

$$\begin{array}{c}\hfill U=PU+QU+\eta \left(U\right)\xi .\end{array}$$

(51)

From (51), we have

$$\begin{array}{c}\hfill P^2=P,Q^2=Q,PQ=QP=0.\end{array}$$

(52)

From (24) and (51), we obtain

$$\begin{array}{c}\hfill \phi U=tPU+nPU+tQU+nQU,\end{array}$$

using the fact M is $\mathcal{PR}$-pseudo-slant, we find

$$\begin{array}{c}\hfill \phi PU=tPU+nPU+nQU,tQU=0,tPU\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\lambda }\right).\end{array}$$

(53)

This leads to the following proposition:

Proposition 9.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then the Equation (53) holds.

Theorem 3.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then the endomorphism n is parallel if and only if

$$\begin{array}{c}\hfill A_{\zeta }{V}_1=-\frac{1}{\lambda }{A}_{n^{\prime }\zeta }t{V}_1,\end{array}$$

(54)

for all $V_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\lambda }\right)$ and $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$.

Proof. 

Firstly, assume that the endomorphism n is parallel, then from (32), we obtain

$$\begin{array}{c}\hfill n^{{}^{\prime }}h(V_1,V_2)-h(V_1,t{V}_2)-\eta \left(V_2\right)n{V}_1=0.\end{array}$$

Replacing $V_2$ with $t{V}_2$ in the above equation, we obtain

$$\begin{array}{c}\hfill n^{\prime }h(V_1,t{V}_2)-h(V_1,t^2{V}_2)=0\end{array}$$

Now, using (32) in the above equation, we have $n^{\prime }h(V_1,t{V}_2)-\lambda h(V_1,V_2)=0$. Now, taking inner product with $\zeta \in \mathsf{\Gamma }\left(T{\mathcal{N}}^{\perp }\right)$ and using (19) and (26), we compute

$$\begin{array}{c}\hfill g(A_{\zeta }{V}_2,V_1)=-\frac{1}{\lambda }g(A_{n^{\prime }\zeta }t{V}_2,V_1).\end{array}$$

□

Theorem 4.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then the slant distribution ${\mathfrak{D}}_{\lambda }$ is always integrable.

Proof. 

Considering $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and $V_1,V_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\lambda }\right)$, the utilization of (10) and (17) gives $g({\nabla }_{V_1}{V}_2,W_1)=-g(\phi {\tilde{\nabla }}_{V_1}{V}_2,\phi W_1)+\eta \left({\tilde{\nabla }}_{V_1}{V}_2\right)\eta \left(W_1\right)$. By the consequences of (14), (17), (18), and (22), the above expression takes the following form:

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-g(h(V_1,t{V}_2),n{W}_1)-g({\nabla }_{V_1}^{\perp }n{V}_2,n{W}_1).\end{array}$$

In the light of Equations (36) and (40), we compute

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-g(n^{\prime }h(V_1,V_2),n{W}_1)-g(n{\nabla }_{V_1}{V}_2,n{W}_1).\end{array}$$

(55)

By interchange $V_1$ and $V_2$ into (55), we obtain

$$\begin{array}{c}\hfill g({\nabla }_{V_2}{V}_1,W_1)=-g(n^{\prime }h(V_1,V_2),n{W}_1)-g(n{\nabla }_{V_2}{V}_1,n{W}_1).\end{array}$$

(56)

In the light of (55) and (56), we achieve $g([V_1,V_2],W_1)=-g(n[V_1,V_2],n{W}_1)$, now using (50), thus, we find

$$\begin{array}{c}\hfill g([V_1,V_2],W_1)=(1-\lambda )\left(g([V_1,V_2],W_1)-\eta \left([V_1,V_2]\right)\eta \left(W_1\right)\right).\end{array}$$

(57)

By the relation (57) we conclude that ${\mathfrak{D}}_{\lambda }$ is integrable. This completes the proof. □

Remark 6.

The one-dimensional real distribution of $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$ is always integrable.

Theorem 5.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\perp }$ is integrable if and only if the shape operator satisfies

$$\begin{array}{c}\hfill A_{n{W}_1}{W}_2=A_{n{W}_2}{W}_1,\end{array}$$

(58)

$\forall W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

By the direct consequence of Equation (22), we obtain

$$\begin{array}{cc}\hfill \Phi [W_1,W_2]=t[W_1,W_2]+n[W_1,W_2]=& t{\tilde{\nabla }}_{W_1}{W}_2-t{\tilde{\nabla }}_{W_2}{W}_1+n{\tilde{\nabla }}_{W_1}{W}_2-n{\tilde{\nabla }}_{W_2}{W}_1.\hfill \end{array}$$

Since ${\mathfrak{D}}_{\perp }$ is anti-invariant distribution then $[W_1,W_2]\in \mathsf{\Gamma }\left(T{\mathfrak{D}}_{\perp }\right)$ if and only if $t{\tilde{\nabla }}_{W_1}{W}_2-t{\tilde{\nabla }}_{W_2}{W}_1=0$. By the application of (29) and (53), we observe that $-\left({\nabla }_{W_2}t\right)W_1+\left({\nabla }_{W_1}t\right)W_2=0$. In view of (31), we obtain (58). This completes the proof. □

Corollary 1.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\perp }$ is integrable if and only if the endomorphism t satisfies

$$\begin{array}{c}\hfill \left({\nabla }_{W_2}t\right)W_1=\left({\nabla }_{W_1}t\right)W_2,\end{array}$$

(59)

$\forall W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Lemma 1.

For a $\mathcal{PR}$-pseudo-slant submanifold $\mathcal{N}$ in ${\mathcal{K}}^{2n+1}$, we have

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=\frac{1}{\lambda }g(h(V_1,W_1),nt{V}_2)-g(h(V_1,t{V}_2),\phi W_1),\end{array}$$

(60)

for all $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and $V_1,V_2\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$.

Proof. 

By the consequence of (10) and (17), we have

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=\eta \left({\overline{\nabla }}_{V_1}{V}_2\right)\eta \left(W_1\right)-g(\phi {\tilde{\nabla }}_{V_1}{V}_2,\phi W_1).\end{array}$$

In view of (12) and (28), we obtain

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-g({\overline{\nabla }}_{V_1}n{V}_2,\phi W_1)-g({\tilde{\nabla }}_{V_1}t{V}_2,\phi W_1).\end{array}$$

Now using (13), (17), and (29) in the above relation,

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-g(h(V_1,t{V}_2),\phi W_1)+g({\tilde{\nabla }}_{V_1}{t}^{\prime }n{V}_2,\phi W_1)+g({\tilde{\nabla }}_{V_1}{n}^{\prime }n{V}_2,\phi W_1)\end{array}$$

The above expression reduces into the following form by the use of first part of Proposition 8 and (14):

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-g(h(V_1,t{V}_2),\phi W_1)+(1-\lambda )g({\nabla }_{V_1}{V}_2,W_1)-g({\tilde{\nabla }}_{V_1}nt{V}_2,\phi W_1).\end{array}$$

By the virtue of (18) and (19), we have (60). □

Theorem 6.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable if and only if the shape operator A satisfies

$$\begin{array}{c}\hfill g(A_{nt{V}_2}{W}_1,V_1)-g(A_{nt{V}_1}{W}_1,V_2)+g(A_{\phi W_1}t{V}_1,V_2)-g(A_{\phi W_1}{V}_1,t{V}_2)=0,\end{array}$$

(61)

$\forall W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and $V_1,V_2\in {\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$.

Proof. 

By the consequence of Lemma 1, we have

$$\begin{array}{cc}\hfill g([V_1,V_2],W_1)=& \frac{1}{\lambda }(g(h(V_1,W_1),nt{V}_2)-g(h(V_2,W_1),nt{V}_1)\hfill \\ & +g(h(t{V}_1,V_2),\phi W_1)-g(h(V_1,t{V}_2),\phi W_1))\hfill \end{array}$$

for every $V_1,V_2\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$ and $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$. In light of (19), we have

$$\begin{array}{cc}\hfill \lambda \left(g([V_1,V_2],W_1)\right)=& g(A_{nt{V}_2}{W}_1,V_1)-g(A_{nt{V}_1}{W}_1,V_2)\hfill \end{array}$$

$$\begin{array}{c}\hfill +g(A_{\phi W_1}t{V}_1,V_2)-g(A_{\phi W_1}{V}_1,t{V}_2).\end{array}$$

(62)

By the relation (62), we conclude that ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable if and only if the relation (61) holds. This completes the proof. □

Theorem 7.

Let $\mathcal{N}$ be a mixed totally geodesic $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable if and only if the shape operator A satisfies

$$\begin{array}{c}\hfill A_{n{W}_1}t{V}_1+t{A}_{n{W}_1}{V}_1=0,\end{array}$$

(63)

$\forall W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and $V_1,V_2\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$.

Proof. 

By the consequence of (10), (13), (28), and (53), we have $g([V_1,V_2],W_1)=g({\tilde{\nabla }}_{V_1}\phi W_1,\phi V_2)-g({\tilde{\nabla }}_{V_2}\phi W_1,\phi V_1)$, for every $V_1,V_2\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$ and $W_1\in \mathsf{\Gamma }\left({\mathcal{D}}_{\perp }\right)$. Now, using (17), (18), and (26) in the above expression, we have

$$\begin{array}{cc}\hfill g([V_1,V_2],W_1)=& -g(A_{n{W}_1}{V}_1,t{V}_2)+g(A_{n{W}_1}{V}_2,t{V}_1)\hfill \\ & +g({\nabla }_{V_1}^{\perp }n{W}_1,n{V}_2)-g({\nabla }_{V_2}^{\perp }n{W}_1,n{V}_1).\end{array}$$

(64)

Furthermore, by the virtue of (13), (17), (18), (26), (28), and (53), we find

$$\begin{array}{c}\hfill t{\nabla }_{V_1}{W}_1+n{\nabla }_{V_1}{W}_1+A_{n{W}_1}{V}_1={\nabla }_{V_1}^{\perp }n{W}_1-t^{{}^{\prime }}h(V_1,W_1)-n^{{}^{\prime }}h(V_1,W_1).\end{array}$$

(65)

By comparing normal components of (65), we obtain

$$\begin{array}{c}\hfill {\nabla }_{V_1}^{\perp }n{W}_1-n^{{}^{\prime }}h(V_1,W_1)=n{\nabla }_{V_1}{W}_1.\end{array}$$

(66)

Now utilizing (65) and (66) in (64), we obtain

$$\begin{array}{cc}\hfill g([V_1,V_2],W_1)=& -g(A_{n{W}_1}{V}_1,t{V}_2)+g(A_{n{W}_1}{V}_2,t{V}_1)+g\left(n{\nabla }_{V_1}{W}_1\right),n{V}_2)\hfill \\ \hfill & +g(n^{{}^{\prime }}h(V_1,W_1),n{V}_2)-g\left(n{\nabla }_{V_2}{W}_1\right),n{V}_1)-g(n^{{}^{\prime }}h(V_2,W_1),n{V}_1).\hfill \end{array}$$

By the application of (8), we have

$$\begin{array}{c}\hfill \lambda g([V_1,V_2],W_1)=g(t{A}_{n{W}_1}{V}_1,V_2)+g(A_{n{W}_1}t{V}_1,V_2).\end{array}$$

(67)

By the above expression, we conclude that ${\mathfrak{D}}_{\lambda }$ is integrable if and only if (63) holds. □

Theorem 8.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable if and only if

$$\begin{array}{cc}\hfill g(A_{n{W}_1}{V}_1,t{V}_2)-g(A_{n{W}_1}t{V}_1,V_2)+& g({\nabla }_{V_1}^{\perp }n{V}_2,n{W}_1)-g({\nabla }_{V_2}^{\perp }n{V}_1,n{W}_1)=0,\hfill \end{array}$$

(68)

for every $V_1,V_2\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$ and $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

By the consequence of (17), (18), and (22), we have

$$\begin{array}{cc}\hfill \phi [U,V]=& t{\nabla }_{V_1}{V}_2+n{\nabla }_{V_1}{V}_2-t{\nabla }_{V_2}{V}_1-n{\nabla }_{V_2}{V}_1.\hfill \end{array}$$

In light of (29), (30) and (31), we observe that

$$\begin{array}{cc}\hfill \phi [V_1,V_2]& ={\nabla }_{V_1}t{V}_2+{\nabla }_{V_1}^{\perp }n{V}_2-{\nabla }_{V_2}t{V}_1-{\nabla }_{V_2}^{\perp }n{V}_1+A_{n{V}_1}{V}_2-A_{n{V}_2}{V}_1\hfill \\ \hfill & +\eta \left(V_1\right)\phi V_2)-\eta \left(V_2\right)\phi V_1+2g(t{V}_1,V_2)\xi +h(V_1,t{V}_2)-h(t{V}_1,V_2).\hfill \end{array}$$

(69)

Now, taking the inner product in the above expression with $n{W}_1$ and using (12), where $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$;

$$\begin{array}{cc}\hfill g(\phi [V_1,V_2],n{W}_1)=& g(h(V_1,t{V}_2),n{V}_1)-g(h(t{V}_1,V_2),n{W}_1)+g({\nabla }_{V_1}^{\perp }n{V}_2,n{W}_1)\hfill \\ \hfill & -g({\nabla }_{V_2}^{\perp }n{V}_1,n{W}_1).\hfill \end{array}$$

From using (25) and (26) in the above equation, we arrive that

$$\begin{array}{cc}\hfill g(t^{{}^{\prime }}n[V_1,V_2],W_1)=& g(h(t{V}_1,V_2),n{W}_1)-g(h(V_1,t{V}_2),n{V}_1)-g({\nabla }_{V_1}^{\perp }n{V}_2,n{W}_1)\hfill \\ \hfill & +g({\nabla }_{V_2}^{\perp }n{V}_1,n{W}_1).\hfill \end{array}$$

In light of Lemma 8, we have

$$\begin{array}{cc}\hfill \left(1-\lambda \right)g([V_1,V_2],W_1)=& g(h(t{V}_1,V_2),n{W}_1)-g(h(V_1,t{V}_2),n{V}_1)-g({\nabla }_{V_1}^{\perp }n{V}_2,n{W}_1)\hfill \\ \hfill & +g({\nabla }_{V_2}^{\perp }n{V}_1,n{W}_1).\hfill \end{array}$$

(70)

Thus, Equation (70) concludes that ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable if and only if (68) holds. □

Theorem 9.

Let $\mathcal{N}$ be a pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, the distribution ${\mathfrak{D}}_{\perp }$ is integrable if and only if it A satisfies

$$\begin{array}{c}\hfill A_{n{W}_1}{W}_2=0,\end{array}$$

(71)

$\forall W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

First of all, suppose ${\mathfrak{D}}_{\perp }$ is integrable distribution, then $t{W}_2=t{W}_1=0$; this implies that ${\nabla }_{W_2}t{W}_1={\nabla }_{W_1}t{W}_2=0$. Therefore, relation (31) reduces $g(\left({\nabla }_{V_1}t\right)W_2,W_1)=g(A_{n{W}_2}{V}_1,W_1)+g(t^{\prime }h(V_1,W_2),W_1)$, for every $V_1\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus <\xi >)$; this implies that $g(A_{n{W}_2}{V}_1,W_1)=-g(t^{\prime }h(V_1,W_2),W_1)$. Now, in the light of (19) and (27), the above expression turns into $g(A_{n{W}_2}{W}_1,X)=-g(A_{n{W}_1}{W}_2,V_1)$. Thus, from (58), we obtain (71).

Conversely: suppose that $\mathcal{N}$ satisfies (71), then by utilization of (19) we have $g(t^{\prime }h(V_1,W_2),W_1)=0$. Now, employing (29) and (31) into the above expression, we achieve that $g({\nabla }_{W_2}{W}_1,V_1)=0$, which implies that ${\nabla }_{W_2}{W}_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$. This shows that ${\mathfrak{D}}_{\perp }$ is a integrable distribution. □

4. $\mathcal{PR}$-Pseudo-Slant Warped Product Submanifolds

Let $\mathcal{N}$ be tangent to $\xi$ in ${\mathcal{K}}^{2n+1}$. Then, $\mathcal{N}$ is said to be a $\mathcal{PR}$-pseudo-slant warped product if it is a warped product of type ${\mathcal{N}}_{\perp }{\times }_f{\mathcal{N}}_{\lambda }$ or ${\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$, where ${\mathcal{N}}_{\lambda }$ is slant submanifold and ${\mathcal{N}}_{\perp }$ is a anti-invariant submanifold in $\mathcal{N}$. In this paper, we only study the warped product whose base is slant, i.e., ${\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$.

Proposition 10.

Let $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ be a $\mathcal{PR}$-pseudo-slant submanifold warped product in ${\mathcal{K}}^{2n+1}$ such that $\xi \in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$. Then $\mathcal{N}$ is a $\mathcal{PR}$-product.

Proof. 

From Equation (4), we have ${\nabla }_{V_1}{W}_1={\nabla }_{W_1}{V}_1=V_1(lnf)W_1$, for $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$. Replacing by $W_1$ by $\xi$ into the above expression, we have ${\nabla }_{V_1}\xi =V_1(lnf)\xi$. With the help of (39), the above expression reduces into the given form $V_1(lnf)=0$. This completes the proof. □

Proposition 11.

There exists a non-trivial $\mathcal{PR}$-pseudo-slant submanifold warped product $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ in ${\mathcal{K}}^{2n+1}$ such that $\xi \in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$.

Proof. 

From Equation (4), we have ${\nabla }_{V_1}{W}_1={\nabla }_{W_1}{V}_1=V_1(lnf)W_1$, for $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$. Replacing by $V_1$ by $\xi$ into the above expression, we have ${\nabla }_{W_1}\xi =\xi (lnf)W_1$. In the light of (39), the above expression reduces into the following form $\xi (lnf)W_1=-W_1$. By the definition of gradient, we have

$$\begin{array}{c}\hfill \frac{\nabla f}{f}=-\xi .\end{array}$$

(72)

By the theory of differential equations we observe that Equation (72) has a solution. This shows that f is non-constant. This completes the proof.

□

Remark 7.

Let $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ be $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. Then, we have

$$\begin{array}{c}\hfill \xi (lnf)=-1.\end{array}$$

(73)

Now, we give some examples of $\mathcal{PR}$-pseudo-slant submanifold of type $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$.

Example 2.

Choose $\tilde{M}={\mathbb{R}}^8\times {\mathbb{R}}^{+}$ together with the usual Cartesian coordinates $(x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4,s)$. Then the structure $(\phi ,\xi ,\eta )$ over $\tilde{M}$ is defined by

$$\begin{array}{c}\hfill \phi \left(\frac{\partial }{\partial x_i}\right)=\frac{\partial }{\partial y_i},\phi \left(\frac{\partial }{\partial y_i}\right)=\frac{\partial }{\partial x_i},\phi \left(\frac{\partial }{\partial s}\right)=0,\eta =ds.\end{array}$$

(74)

where $i,j\in \{1,\cdots ,4\}$ and the pseudo-Riemannian metric tensor g is defined as

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_i}\right)=e^{-2s},g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_i}\right)=-e^{-2s},g\left(\frac{\partial }{\partial s},\frac{\partial }{\partial s}\right)=1,\hfill \end{array}$$

(75)

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_k}\right)=0,g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial y_k}\right)=0,g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_k}\right)=0,\hfill \end{array}$$

(76)

for all $k\in \{1,\cdots ,4\}$. Then by simple computation, we can easily see that $\tilde{M}$ is para-Kenmotsu manifold. Suppose $\mathcal{N}$ is an immersed submanifold into $\tilde{M}$ by an immersion σ which is defined by

$$\begin{array}{cc}\hfill & x_1=u,x_2=kvsinh\alpha ,x_3={\alpha }^2,x_4=0,y_1=v,\hfill \\ & y_2=kvcosh\alpha ,y_3=0,y_4={\alpha }^2-2,s=s,\hfill \end{array}$$

for $k\in \mathbb{R}$. Thus, we can easily provide the generating set for the tangent bundle of submanifold as follows:

$$\begin{array}{cc}\hfill & Z_{\alpha }=kvcosh\alpha \frac{\partial }{\partial x_2}+2\alpha \frac{\partial }{\partial x_3}+kvsinh\alpha \frac{\partial }{\partial y_2}+2\alpha \frac{\partial }{\partial y_4},\hfill \\ \hfill & Z_u=\frac{\partial }{\partial x_1},\hfill \\ \hfill & Z_v=ksinh\alpha \frac{\partial }{\partial x_2}+\frac{\partial }{\partial y_1}+kcosh\alpha \frac{\partial }{\partial y_2},\hfill \\ \hfill & Z_s=\xi .\hfill \end{array}$$

for $s\in \mathbb{R}$. The basis vector for $\phi \left(TM\right)$ is given by

$$\begin{array}{cc}\hfill & \phi Z_{\alpha }=kvsinh\alpha \frac{\partial }{\partial x_2}+2\alpha \frac{\partial }{\partial x_4}+kvcosh\alpha \frac{\partial }{\partial y_2}+2\alpha \frac{\partial }{\partial y_3},\hfill \\ \hfill & \phi Z_u=\frac{\partial }{\partial y_1},\hfill \\ \hfill & \phi Z_v=\frac{\partial }{\partial x_1}+kcosh\alpha \frac{\partial }{\partial x_2}+ksinh\alpha \frac{\partial }{\partial y_2},\hfill \\ \hfill & \phi Z_s=0.\hfill \end{array}$$

By simple calculation, we obtain that the distribution ${\mathcal{D}}_{\lambda }=span\{Z_u,Z_v\}$ is slant distribution with slant function $\lambda =\frac{1}{1+k^2}$ and the distribution ${\mathcal{D}}_{\perp }=span\left\{Z_{\alpha }\right\}$ is anti-invariant under φ. The induced metric tensor $g_{\mathcal{N}}$ on $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ is given by

$$\begin{array}{c}\hfill g_{\mathcal{N}}=d{s}^2+(d{u}^2-(1+k^2)d{v}^2)e^{-2s}+e^{-2s}{v}^{2}d{\alpha }^2.\end{array}$$

(77)

The above calculation manifests that the submanifold $\mathcal{N}$ is a form of $\mathcal{PR}$-pseudo-slant warped product of type II with warping function $f=e^{-s}v$ of para-Kenmotsu manifold.

Example 3.

Choose $\tilde{M}={\mathbb{R}}^8\times {\mathbb{R}}^{+}$ together with the usual Cartesian coordinates $(x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4,s)$. Then, the structure $(\phi ,\xi ,\eta )$ over $\tilde{M}$ is defined by

$$\begin{array}{c}\hfill \phi \left(\frac{\partial }{\partial x_i}\right)=\frac{\partial }{\partial y_i},\phi \left(\frac{\partial }{\partial y_i}\right)=\frac{\partial }{\partial x_i},\phi \left(\frac{\partial }{\partial s}\right)=0,\eta =ds.\end{array}$$

(78)

where $i,j\in \{1,\cdots ,4\}$ and the pseudo-Riemannian metric tensor g is defined as

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_i}\right)=e^{-2s},g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_i}\right)=-e^{-2s},g\left(\frac{\partial }{\partial s},\frac{\partial }{\partial s}\right)=1,\hfill \end{array}$$

(79)

$$\begin{array}{cc}\hfill & g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial x_k}\right)=0,g\left(\frac{\partial }{\partial x_i},\frac{\partial }{\partial y_k}\right)=0,g\left(\frac{\partial }{\partial y_i},\frac{\partial }{\partial y_k}\right)=0,\hfill \end{array}$$

(80)

for all $k\in \{1,\cdots ,4\}$. Then, by simple computation, we can easily see that $\tilde{M}$ is para-Kenmotsu manifold. Suppose $\mathcal{N}$ is an immersed submanifold into $\tilde{M}$ by an immersion σ which is defined by

$$\begin{array}{cc}\hfill & x_1=kusinh\alpha ,x_2=\alpha ,x_3=u,x_4=0,y_1=kucosh\alpha ,\hfill \\ & y_2=0,y_3=v,y_4=\alpha +1,s=s,\hfill \end{array}$$

for $k\in \mathbb{R}\sim \left\{1\right\}$. Thus, we can easily provide the generating set for the tangent bundle of submanifold as follows:

$$\begin{array}{cc}\hfill & Z_{\alpha }=kucosh\alpha \frac{\partial }{\partial x_1}+\frac{\partial }{\partial x_2}+kusinh\alpha \frac{\partial }{\partial y_1}+\frac{\partial }{\partial y_4},\hfill \\ \hfill & Z_u=ksinh\alpha \frac{\partial }{\partial x_1}+\frac{\partial }{\partial x_3}+kcosh\alpha \frac{\partial }{\partial y_1},\hfill \\ \hfill & Z_v=\frac{\partial }{\partial y_3},\hfill \\ \hfill & Z_s=\xi .\hfill \end{array}$$

for $s\in \mathbb{R}$. The basis vector for $\phi \left(T\mathcal{N}\right)$ is given by

$$\begin{array}{cc}\hfill & \phi Z_{\alpha }=kucosh\alpha \frac{\partial }{\partial y_1}+\frac{\partial }{\partial y_2}+kusinh\alpha \frac{\partial }{\partial x_1}+\frac{\partial }{\partial x_4},\hfill \\ \hfill & \phi Z_u=ksinh\alpha \frac{\partial }{\partial y_1}+\frac{\partial }{\partial y_3}+kcosh\alpha \frac{\partial }{\partial x_1},\hfill \\ \hfill & \phi Z_v=\frac{\partial }{\partial x_3},\hfill \\ \hfill & \phi Z_s=0.\hfill \end{array}$$

By simple calculation, we obtain that the distribution ${\mathcal{D}}_{\lambda }=span\{Z_u,Z_v\}$ is slant distribution of with slant function $\lambda =\frac{1}{1-k^2}$ and the distribution ${\mathcal{D}}_{\perp }=span\left\{Z_{\alpha }\right\}$ is anti-invariant under φ. The induced metric tensor $g_{\mathcal{N}}$ on $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ is given by

$$\begin{array}{c}\hfill g_{\mathcal{N}}=d{s}^2+e^{-2s}((1-k^2)d{u}^2-d{v}^2)+e^{-2s}{u}^{2}d{\alpha }^2.\end{array}$$

(81)

The above calculation manifests that the submanifold $\mathcal{N}$ is a form of $\mathcal{PR}$-pseudo-slant warped product of type I if $k<1$ and $\mathcal{PR}$-pseudo-slant warped product of type III if $k>1$ of para-Kenmotsu manifold with warping function $f=e^{-s}u$.

Lemma 2.

For a $\mathcal{PR}$-pseudo-slant warped product submanifold $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ in ${\mathcal{K}}^{2n+1}$, we receive for all $V_1,V_2\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1,W_2\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$ that

$$\begin{array}{cc}\hfill g(h(V_1,V_2),n{W}_1)=& g(h(V_1,W_1),n{V}_2),\hfill \end{array}$$

(82)

$$\begin{array}{cc}\hfill g(h(V_1,W_1),n{W}_2)=& g(h(V_1,W_2),n{W}_1).\hfill \end{array}$$

(83)

Proof. 

By the consequence of (17) and (28), we have

$$\begin{array}{c}\hfill g(h(V_1,V_2),n{W}_1)=g({\tilde{\nabla }}_{V_1}{V}_2,\phi W_1)-g({\tilde{\nabla }}_{V_1}{V}_2,t{W}_1).\end{array}$$

Now, applying (12) and (13) into the above expression, we achieve

$$\begin{array}{c}\hfill g(h(V_1,V_2),n{W}_1)=-g({\tilde{\nabla }}_{V_1}t{V}_2,W_1)-g({\tilde{\nabla }}_{V_1}n{W}_1,V_2)-g({\tilde{\nabla }}_{V_1}{V}_2,t{W}_1).\end{array}$$

By the utilization of (4) and (17), we obtain (82). We proceed with a similar process to prove (83). □

Lemma 3.

Let $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ be a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. Then, we obtain for all $V_1,V_2\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $U,V\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$ that

$$\begin{array}{cc}\hfill g(h(W_1,W_1),n{V}_1)=& g(h(V_1,W_1),n{W}_1)+t{V}_1(lnf)g(W_1,W_1),\hfill \end{array}$$

(84)

$$\begin{array}{cc}\hfill g(h(W_1,W_1),nt{V}_1)=& g(h(t{V}_1,W_1),nV)+\lambda \left(V_1(lnf)+\eta \left(V_1\right)\right)(W_1,W_1).\hfill \end{array}$$

(85)

Proof. 

By the consequence of (17) and (28), we have

$$\begin{array}{c}\hfill g(h(W_1,W_1),n{V}_1)=g({\tilde{\nabla }}_{W_1}{W}_1,\phi V_1)-g({\tilde{\nabla }}_{W_1}{W}_1,t{V}_1).\end{array}$$

Now, applying (12) and (13) into the above expression, we achieve

$$\begin{array}{c}\hfill g(h(W_1,W_1),n{V}_1)=-g({\tilde{\nabla }}_{W_1}\phi W_1,V_1)-g({\tilde{\nabla }}_{W_1}{W}_1,t{V}_1).\end{array}$$

By the utilization of (4), (18) and (19), we obtain (84). If we replace $V_1$ with $t{V}_1$ in (84), then we attain (85). □

Theorem 10.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, $\mathcal{N}$ is a $\mathcal{PR}$-pseudo-slant warped product submanifold if and only if

$$\begin{array}{c}\hfill A_{nt{V}_1}{W}_1-A_{\phi W_1}t{V}_1=\lambda \left(V_1\left(\mu \right)+\eta \left(V_1\right)\right)W_1,\end{array}$$

(86)

for every $V_1\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$, $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and some smooth function μ on $\mathcal{N}$ satisfies $W_2\left(\mu \right)=0$, for every $W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

Suppose that $\mathcal{N}$ is a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. Then, by the virtue of (19) and (85), we easily obtain (86) by taking $\mu =lnf$.

Conversely, suppose $\mathcal{N}$ is $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$ that satisfies (86). Then, by the application of Lemma 1 and (86), we obtain $g({\nabla }_{V_1}{V}_2,W_1)=\left(V_1\left(\mu \right)+\eta \left(V_1\right)\right)$$g(W_1,V_2)=0$. This shows that the distribution ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is totally geodesic and integrable. Now, let us denote $h^{\perp }$ as the second fundamental form of ${\mathfrak{D}}_{\perp }$. Then, by the use of (17), we have $g(h^{\perp }(W_1,W_2),V_1)=g({\tilde{\nabla }}_{W_1}{W}_2,V_1)$. In view of (10), the above expression reduces into the following form:

$$\begin{array}{c}\hfill g(h^{\perp }(W_1,W_2),V_1)=-g(\phi {\tilde{\nabla }}_{W_1}{W}_2,\phi V_1)+\eta \left(V_1\right)g({\tilde{\nabla }}_{W_1}{W}_2,\xi ).\end{array}$$

By the consequence of (13), (14), and (28), the above expression reduces into the following form:

$$\begin{array}{cc}\hfill g(h^{\perp }(W_1,W_2),V_1)=& -g({\tilde{\nabla }}_{W_1}\phi W_2,\phi V_1)+g(\left({\tilde{\nabla }}_{W_1}\phi \right)W_2,\phi V_1)+\eta \left(V_1\right)g(W_1,W_2)\hfill \\ & =-g({\tilde{\nabla }}_{W_1}\phi W_2,\phi V_1)+\eta \left(V_1\right)g(W_1,W_2).\hfill \end{array}$$

Now, using (17)–(19) and (27) in the above relation, we have

$$\begin{array}{cc}\hfill g(h^{\perp }(W_1,W_2),V_1)=& g(h(W_1,t{V}_1),\phi W_2)-g(W_2,{\tilde{\nabla }}_{W_1}{t}^{{}^{\prime }}n{V}_1)\hfill \end{array}$$

$$\begin{array}{c}\hfill -g(W_2,{\tilde{\nabla }}_{W_1}{n}^{{}^{\prime }}n{V}_1)+\eta \left(V_1\right)g(W_1,W_2).\end{array}$$

(87)

In view of (86), (87), and Lemma 8, we have

$$\begin{array}{cc}\hfill g(h^{\perp }(W_1,W_2),V_1)=& \frac{1}{\lambda }\left(g(h(W_1,t{V}_1),\phi W_2)-g(h(W_1,W_2),nt{V}_1)\right)\hfill \end{array}$$

$$\begin{array}{c}\hfill +\eta \left(V_1\right)g(W_1,W_2)=-V_1\left(\mu \right)g(W_1,W_2).\end{array}$$

(88)

By definition of gradient and (88), we have

$$\begin{array}{c}\hfill h^{\perp }(W_1,W_2)=-\nabla \left(\mu \right)g(W_1,W_2).\end{array}$$

(89)

The relation (89) shows that the distribution ${\mathfrak{D}}_{\perp }$ is totally umbilical with mean curvature $H^{\perp }=-\nabla \left(\mu \right)$, which is parallel with respect to ${\nabla }^{\perp }$. By Hiepko result and the above discussion, we conclude that the $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ is a $\mathcal{PR}$-pseudo-slant warped product submanifold of ${\mathcal{K}}^{2n+1}$. This completes the proof. □

Theorem 11.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, $\mathcal{N}$ is a mixed totally geodesic $\mathcal{PR}$-pseudo-slant warped product submanifold if and only if

$$\begin{array}{c}\hfill A_{\phi W_1}{V}_1=0,and{A}_{nt{V}_1}{W}_1=-\lambda \left(V_1\left(\mu \right)+\eta \left(V_1\right)\right)W_1,\end{array}$$

(90)

for every $V_1\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$, $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$ and some smooth function μ on $\mathcal{N}$ satisfies $W_2\left(\mu \right)=o$, for every $W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

Suppose that $\mathcal{N}$ is a mixed totally geodesic $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$, then $h(V_1,W_1)=0$, for every $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$. Therefore, by the virtue of (19) and (82), we achieve (90).

Conversely, suppose $\mathcal{N}$ is a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$ that satisfies (90). From Lemma 1 and (90), we have

$$\begin{array}{c}\hfill g({\nabla }_{V_1}{V}_2,W_1)=-\left(V_1\left(\mu \right)+\eta \left(X\right)\right)g(W_1,V_2)=0.\end{array}$$

By this expression, we easily see that the leaves of ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ are totally geodesic and integrable. Let us denote $h^{\perp }$ as the second fundamental form of ${\mathfrak{D}}_{\perp }$. Then, by the use of (17), we have $g(h^{\perp }(W_1,W_2),V_1)=g({\tilde{\nabla }}_{W_1}{W}_2,V_1)$. Now, utilizing (10), (13), (14), and (28) in the above expression, we concede that

$$\begin{array}{c}\hfill g(h^{\perp }(W_1,W_2),V_1)=-g({\tilde{\nabla }}_{W_1}\phi W_2,\phi V_1)+\eta \left(V_1\right)g(W_1,W_2).\end{array}$$

By using (17)–(19), (27), and the first part part of (90) into the above relation, we receive that

$$\begin{array}{cc}\hfill g(h^{\perp }(W_1,W_2),V_1)=& -g(W_2,{\tilde{\nabla }}_{W_1}{t}^{{}^{\prime }}n{V}_1)-g(W_2,{\tilde{\nabla }}_{W_1}{n}^{{}^{\prime }}n{V}_1)+\eta \left(V_1\right)g(W_1,W_2).\hfill \end{array}$$

(91)

In view of Lemma 8, (90) and (91), we have

$$\begin{array}{c}\hfill g(h^{\perp }(W_1,W_2),V_1)=V_1\left(\mu \right)g(W_1,W_2).\end{array}$$

(92)

By definition of gradient and (92), we have

$$\begin{array}{c}\hfill h^{\perp }(W_1,W_2)=\nabla \left(\mu \right)g(W_1,W_2).\end{array}$$

(93)

The relation (93) shows that the distribution ${\mathfrak{D}}_{\perp }$ is totally umbilical with mean curvature $H^{\perp }=\nabla \left(\mu \right)$ which is parallel with respect to ${\nabla }^{\perp }$. By Hiepko result and the above discussion, we conclude that the $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ is a mixed totally geodesic $\mathcal{PR}$-pseudo-slant warped product submanifold of ${\mathcal{K}}^{2n+1}$. □

Theorem 12.

Let $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ be a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. Then, $\mathcal{N}$ is locally a $\mathcal{PR}$-product if and only if

$$\begin{array}{c}\hfill A_{nt{V}_1}{W}_1=\lambda \eta \left(V_1\right)W_1,\end{array}$$

(94)

for every $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$.

Proof. 

By the application of Equations (10), (17), and (28), we have $g({\nabla }_{W_1}{V}_1,W_2)=-g({\tilde{\nabla }}_{W_1}\phi V_1,\phi W_2)+g(\left({\tilde{\nabla }}_{W_1}\phi \right)V_1,\phi W_2)$, for every $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$ and $W_1,W_2\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$. Now, using (10) and (27), we concede that

$$\begin{array}{c}\hfill g({\nabla }_{W_1}{V}_1,W_2)=-g({\tilde{\nabla }}_{W_1}t{V}_1,\phi W_2)-\eta \left(V_1\right)g(W_1,W_2)-g({\tilde{\nabla }}_{W_1}n{V}_1,\phi W_2).\end{array}$$

By the consequence of (12), (13), (14), (24), and (28), the above expression relation reduces into the following form:

$$\begin{array}{c}\hfill g({\nabla }_{W_1}{V}_1,W_2)=g({\tilde{\nabla }}_{W_1}{t}^2{V}_1,W_2)+g({\tilde{\nabla }}_{W_1}nt{V}_1,W_2)\\ \hfill -\eta \left(V_1\right)g(W_1,W_2)-g({\nabla }_{W_1}^{\perp }n{V}_1,\phi W_2).\end{array}$$

In light of (14), (17), (4), and Lemma 3, the above expression reduces into the following form:

$$\begin{array}{c}\hfill (1-\lambda )\left(V_1(lnf)-\eta \left(V_1\right)\right)g(W_1,W_2)=g(h(W_1,W_2),nt{V}_1)-g({\nabla }_{W_1}^{\perp }n{V}_1,\phi W_2).\end{array}$$

(95)

Interchanging $W_1$ and $W_2$ into (95), we have

$$\begin{array}{c}\hfill (1-\lambda )\left(V_1(lnf)-\eta \left(V_1\right)\right)g(W_1,W_2)=g(h(W_1,W_2),nt{V}_1)-g({\nabla }_{W_2}^{\perp }n{V}_1,\phi W_1).\end{array}$$

(96)

In view of (95) and (96), we have

$$\begin{array}{c}\hfill g({\nabla }_{W_2}^{\perp }n{V}_1,\phi W_1)=g({\nabla }_{W_1}^{\perp }n{V}_1,\phi W_2).\end{array}$$

(97)

On the other hand, by use of (13), (17), and (28), we observe that

$$\begin{array}{c}\hfill g({\nabla }_{W_1}^{\perp }n{V}_1,\phi W_2)=g(\phi {\tilde{\nabla }}_{W_1}{V}_1,\phi W_2)-\eta \left(V_1\right)g(\phi W_1,\phi W_2)\\ \hfill -g({\tilde{\nabla }}_{W_1}t{V}_1,\phi W_2).\end{array}$$

In light of (4) and (10), the above expression reduces into the following form:

$$\begin{array}{c}\hfill g({\nabla }_{W_1}^{\perp }n{V}_1,\phi W_2)=-V_1(lnf)g(W_1,W_2)+\eta \left(V_1\right)g(W_1,W_2)-g({\tilde{\nabla }}_{W_1}t{V}_1,\phi W_2).\end{array}$$

(98)

Again, interchanging $W_1$ and $W_2$ into (98), we have

$$\begin{array}{c}\hfill g({\nabla }_{W_2}^{\perp }n{V}_1,\phi W_1)=-V_1(lnf)g(W_1,W_2)+\eta \left(V_1\right)g(W_1,W_2)-g({\tilde{\nabla }}_{W_2}t{V}_1,\phi W_1).\end{array}$$

(99)

By the virtue of (98) and (99), we conclude that (97) holds if and only if

$$\begin{array}{c}\hfill g({\tilde{\nabla }}_{W_2}t{V}_1,\phi W_1)=0=-g({\tilde{\nabla }}_{W_1}t{V}_1,\phi W_2).\end{array}$$

(100)

By the utilization of (17), (24), (28), (100), and Lemma 3, we obtain

$$\begin{array}{c}\hfill \lambda \left(V_1(lnf)+\eta \left(V_1\right))g(W_1,W_2)\right)-g(h(W_1,W_2),nt{V}_1)=0.\end{array}$$

(101)

By the above relation, we can observe that f is constant if and only if the relation (94) holds. This completes the proof. □

Lemma 4.

Let $\mathcal{N}={\mathcal{N}}_{\lambda }{\times }_f{\mathcal{N}}_{\perp }$ be a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. Then, we obtain for all $U\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$, $V_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\lambda }\right)$, and $W_1\in \mathsf{\Gamma }\left(T{\mathcal{N}}_{\perp }\right)$ that

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)W_1=& -g(W_1,QU)t\nabla (lnf),\hfill \end{array}$$

(102)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)V_1=& \eta \left(U\right)A_{n{V}_1}\xi +\eta \left(V_1\right)tPU+g(PU,t{V}_1)\xi +t{V}_1(lnf)QU.\hfill \end{array}$$

(103)

$$\begin{array}{cc}\hfill \left({\nabla }_{U}t\right)t{V}_1=& \eta \left(U\right)A_{nt{V}_1}\xi +\lambda \eta \left(V_1\right)PU-\lambda \eta \left(V_1\right)g(PU,V_1)\xi \hfill \\ & +\lambda (V_1(lnf)+\eta \left(V_1\right))QU.\hfill \end{array}$$

(104)

Proof. 

By the use of (51), we have $\left({\nabla }_{U}t\right)W_1=\left({\nabla }_{PU}t\right)W_1+\left({\nabla }_{QU}t\right)W_1+\eta \left(U\right)\left({\nabla }_{\xi }t\right)W_1$. By the virtue of (4) and Definition 3, we have $\left({\nabla }_{PU}t\right)W_1=\left({\nabla }_{\xi }t\right)W_1=0$. In view of (29) and (5), we observe that $\left({\nabla }_{QU}t\right)W_1=-g(W_1,QU)t\nabla (lnf)$. By these observations, we easily concede the relation (102). By reuse of (51), we have $\left({\nabla }_{U}t\right)V_1=\left({\nabla }_{PU}t\right)V_1+\left({\nabla }_{QU}t\right)V_1+\eta \left(U\right)\left({\nabla }_{\xi }t\right)V_1$. Furthermore, by the virtue of (31), we attain $\left({\nabla }_{PU}t\right)V_1=A_{n{V}_1}PU+t^{{}^{\prime }}h(PU,V_1)+\eta \left(V_1\right)tPU-g(tPU,V_1)\xi$. Since ${\mathcal{N}}_{\lambda }$ is totally geodesic, the above expression reduces into the following form:

$$\begin{array}{c}\hfill \left({\nabla }_{PU}t\right)V_1=\eta \left(V_1\right)tPU-g(tPU,V_1)\xi .\end{array}$$

(105)

By the utilization of (4) and (51), we have

$$\begin{array}{c}\hfill \left({\nabla }_{QU}t\right)V_1=t{V}_1(lnf)QU.\end{array}$$

(106)

Similarly, we find

$$\begin{array}{c}\hfill \left({\nabla }_{\xi }t\right)V_1=A_{n{V}_1}\xi .\end{array}$$

(107)

By the application of (105)–(107), we achieve (103). If we replace $V_1$ with $t{V}_1$ in (), we easily achieve (104). □

Theorem 13.

Let $\mathcal{N}$ be a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$. Then, $\mathcal{N}$ is a $\mathcal{PR}$-pseudo-slant warped product submanifold if and only if the endomorphism t satisfies

$$\begin{array}{c}\hfill g(\left({\nabla }_{U}t\right)V,V_1)=t{V}_1\left(\mu \right)g(QU,QV)+\eta \left(V_1\right)g(PU,tPV),\end{array}$$

(108)

for every $V_1\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$, $U,V\in \mathsf{\Gamma }\left(T\mathcal{N}\right)$, and some smooth function μ on $\mathcal{N}$ satisfies $W_2\left(\mu \right)=0$, for every $W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$.

Proof. 

Suppose that M is a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$, then by (51), we obtain

$$\begin{array}{c}\hfill \left({\nabla }_{U}t\right)V=\left({\nabla }_{U}t\right)QV+\left({\nabla }_{U}t\right)PV+\eta \left(V\right)\left({\nabla }_{U}t\right)\xi .\end{array}$$

(109)

By the utilization of (14), (17), (102), and (103), we achieve that

$$\begin{array}{c}\hfill \left({\nabla }_{U}t\right)V=-\eta \left(V\right)tU-g(QU,QV)t\nabla (lnf)+\eta \left(U\right)A_{nPV}\xi \\ \hfill +\eta \left(PV\right)tPU+g(PU,tPV)\xi +tPV(lnf)QU.\end{array}$$

(110)

By taking the inner product with $V_1$ into (111), then using (39) and definition of gradient, we achieve

$$\begin{array}{c}\hfill g(\left({\nabla }_{U}t\right)V,V_1)=t{V}_1(lnf)g(QU,QV)+\eta \left(V_1\right)g(PU,tPV),\end{array}$$

(111)

By taking $\mu =lnf$ into (111) and using the fact that $\mathcal{N}$ is a warped product, we accomplished (108).

Conversely, assume that $\mathcal{N}$ is a $\mathcal{PR}$-pseudo-slant submanifold in ${\mathcal{K}}^{2n+1}$ satisfying (108). Now, replacing U with $V_2$ and V with $W_1$ in (108), we have $g(\left({\nabla }_{V_2}t\right)W_1,V_1)=0$, $V_1\in \mathsf{\Gamma }({\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩)$ and $W_1\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$. In view of (26) and (29), we have $g(h^{\lambda }(t{V}_1,V_2),W_1)=0$. This shows that ${\mathfrak{D}}_{\lambda }\oplus ⟨\xi ⟩$ is integrable and its leaves are totally geodesic in $\mathcal{N}$. Furthermore, replacing U with $W_1$ and V with $W_2$ in (108), we have $g(\left({\nabla }_{W_1}t\right)W_2,V_1)=t{V}_1\left(\mu \right)g(W_1,W_2)+\eta \left(V_1\right)g(W_1,t{V}_1)$, for every $W_1,W_2\in \mathsf{\Gamma }\left({\mathfrak{D}}_{\perp }\right)$. By (26) and orthogonality relation, we observe that

$$\begin{array}{c}\hfill g((h^{\perp }(W_1,W_2),t{V}_1)=g(t{V}_1,\nabla (lnf))g(W_1,W_2).\end{array}$$

(112)

By the relation (112), we observe that the distribution ${\mathfrak{D}}_{\perp }$ is totally umbilical with mean curvature $H^{\perp }=\nabla \left(\mu \right)$. By the application of Hiepko result [41], we can conclude that M is a $\mathcal{PR}$-pseudo-slant warped product submanifold in ${\mathcal{K}}^{2n+1}$. This completes the proof. □