Geometric inequalities of $\mathcal{PR}$-warped product submanifold in para-Kenmotsu manifold

1.   Introduction

Warped product manifold is one of the most significant generalization of Cartesian product of Riemannian manifolds (or pseudo-Riemannian manifolds). This fruitful generalization was initiated by R. L. Bishop and B. O'Neill in $1969$ (see ^[1]). But warped products viewed in the physical and mathematical literature before $1969$. For instance, semi-reducible space which is used for warped product by Kruchkovich in 1957 ^[2]. It has been successfully utilized in general theory of relativity, black holes and string theory.

Warped product geometry was taking more attention in $2002$ when Chen studied $\mathcal{CR}$-warped product in K$\ddot a$hlerian manifolds and derived several non-existence results for such warped product manifold of type $M_{\perp}\times_{f}M_{T}$, where $M_{\perp}$ and $M_{T}$ stands for anti-invariant and invariant submanifold (see, ^[3,4]). Thereafter, authors of ^[5,6] studied $\mathcal{CR}$-warped product submanifolds in Sasakian and Kenmotsu manifold and also derived some useful optimal inequalities for such warped products. After that numerous author studied the same (c.f., ^[7,8,9]). In $2018$, Siraj Uddin derived some useful optimal inequalities for semi-slant warped product submanifolds of Kenmotsu manifold ^[10].

In $2014$, B. Y. Chen explored new class of warped product termed as a $\mathcal{PR}$-warped product and find the exact solutions of the system of partial differential equations associated with $\mathcal{PR}$-warped products ^[11]. Then after, the authors of ^[12,13] studied different classes of $\mathcal{PR}$-warped products in paraCosymplectic manifold. Recently, the authors of ^[14,15] studied $\mathcal{PR}$-semi-slant and $\mathcal{PR}$-pseudo-slant warped product submanifold of para-Kenmotsu manifold. Motivated by them, we studied $\mathcal{PR}$-semi-invariant warped product submanifolds in para-Kenmotsu manifold and proved some existence and characterization results and also obtain some optimal inequalities.

This paper is formulated as follows: Section 2 includes some necessary information related to para-contact and para-Kenmotsu manifold. And also contains some important information about the submanifolds theory of para-Kenmotsu manifold. We provide basic information related to warped product submanifold and obtain some existence conditions and non-existence conditions for warped product submanifold of para-Kenmotsu manifold into the section $3$. Section $4$ includes some results related to integrability conditions of $\mathcal{PR}$-semi invariant submanifold in para-Kenmotsu manifold. In section $5$, we provide some characterization results allied to $\mathcal{PR}$-semi-invariant warped product submanifolds. We obtain optimal inequalities for $F\times_{f}B$ and $B\times_{f}F$ into the section $6$. Lastly, we give some examples of these warped products in section $7$.

2.   Preliminaries

From the literature, a smooth manifold $\widetilde{M}^{2n+1}$ of dimension $(2n+1)$ furnished an almost paracontact structure $(\varphi, \xi, \eta)$ which includes a $(1, 1)$-type tensor field $\varphi$, a vector field $\xi$ and a $1$-form $\eta$ globally defined on $\widetilde{M}^{2n+1}$ which satisfies the accompanying relation for all $U\in\Gamma(TM^{2n+1})$ ^[12,16]:

The tensor field $\varphi$ 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. $\mathfrak{D} = ker(\eta)$. 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 $T\widetilde{M}^{2n+1}$ can be expressed in the following form;

If $\widetilde{M}^{2n+1}$ admits an almost paracontact structure $(\varphi, \xi, \eta)$ then it is said to be an almost paracontact manifold ^[12,16]. In view of (2.1), we obtain

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

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

By the utilization of (2.1)–(2.4), we attain

Definition 2.1. An almost paracontact manifold $\widetilde{M}^{2n+1}$ is said to be a para-Kenmotsu manifold ^[17] if it fulfills one additional condition

In the relation (2.7); symbol $\widetilde{\nabla}$ indicates Levi-Civita connection with respect to above defined metric $g$.

Replace $V$ by $\xi$ into (2.7) then applying (2.1), we achieve that

Proposition 2.2. On para-Kenmotsu pseudo Riemannian manifold the following relations hold

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

Geometry of submanifolds

Let $M$ be a paracompact and connected smooth pseudo-Riemannian manifold of dimension $m$ and $\widetilde{M}^{2n+1}$ be a para-Kenmotsu manifold. Let $\psi:M\longrightarrow \widetilde{M}^{2n+1}$ be an isometric immersion. Then $\psi \left(M\right)$ is known as an isometrically immersed submanifold of a para-Kenmotsu manifold. Let us denote $\psi_{*}$ for the differential map (or push forward map) of immersion $\psi$ is characterized by $\psi_{*}:T_{p}M\longrightarrow T_{\psi(p)}\widetilde{M}^{2n+1}$. Therefore, the induced pseudo-Riemannian metric $\mathfrak{g}$ on $\psi(M)$ 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 $p$ on the place of $\psi(M)$ and $\psi(p)$. Now, denoting $\Gamma({TM})$ for set of all vector fields on $M$, $\Gamma(TM^{\perp})$ for the set of all normal vector fields of $M$, $\nabla$ for induced Levi-Civita connection on $TM$ and $\nabla^{\perp}$ for normal connection on the normal bundle $\Gamma(TM^{\perp})$. Then the Gauss and Weingarten formulas are characterized by the following relations:

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

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

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

An isometrically immersed pseudo-Riemannian submanifold $M$ in a para-Kenmotsu manifold $\widetilde{M}^{2n+1}$ is said to be

● totally geodesic if $h$ is 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 for any tangent vectors $U$ and $V$, $M$ satisfies:

● minimal if trace of $h$ (or $H$) vanishes identically;

● extrinsic sphere if $M$ satisfies (2.16) and $H$ is parallel with respect to $\nabla^{\perp}$ (for more details see, ^[12,16]).

From now and in all we will denote the para-Kenmotsu manifold by $\mathcal{K}^{2n+1}$ and its pseudo-Riemannian submanifold by $\mathcal{N}$. Let $tan:T_{p}\mathcal{K}^{2n+1}\longrightarrow T_{p}\mathcal{N}$ and $nor:T_{p}\mathcal{K}^{2n+1} \longrightarrow T_{p} \mathcal{N}^{\perp}$ be two endomorphism. Then for any $U\in\Gamma(T\mathcal{N})$,

we can write

where $tU = tan(\varphi U)$ and $nU = nor (\varphi U)$. Similarly, for any $\zeta\in\Gamma(T\mathcal{N}^{\perp})$, we have

where $t^{'}\zeta = tan(\varphi \zeta)$ and $n^{'}\zeta = nor (\varphi \zeta)$. In view of (2.6) and (2.16)–(2.18), we attain for any $U, V\in\Gamma(T\mathcal{N})$ and $\forall \zeta_{1}, \zeta_{2}\in\Gamma(T\mathcal{N}^{\perp})$ that

Moreover, by the consequences of (2.6), (2.17) and (2.18), we have

Moreover, the covariant derivative of $\varphi$, $n$ and $t$ are charcterized by

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

Proposition 2.3. If $\mathcal{N}$ is tangent to $\xi$ in a $\mathcal{K}^{2n+1}$. Thus, we have

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

Proof. By the consequence of (2.11), (2.12), (2.17) and (2.21)–(2.23), we have

for any $U\in\Gamma(T\mathcal{N})$. Employing (2.7) and (2.17) into above expression then considering tangential part and normal part of obtained expression, we have (2.24) and (2.25), respectively.

Proposition 2.4. Let $\mathcal{N}$ is normal to $\xi$ in a $\mathcal{K}^{2n+1}$. Then, we have

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

Proof. Immediately, form (2.7), (2.11), (2.12), (2.17), (2.21) and (2.22), we derive (2.26) and (2.27).

Proposition 2.5. If $\xi$ is tangent to $\mathcal{N}$ in $\mathcal{K}^{2n+1}$. Thus, we obtain

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

Proof. Employing (2.11), (2.12), (2.18), (2.22) and (2.23) into (2.21), we achieve that

for any $U\in\Gamma(TM)$. Utilizing (2.7) and (2.17) into above expression, we achieve (2.28) and (2.29).

Proposition 2.6. If $\mathcal{N}$ is normal to $\xi$ in $\mathcal{K}^{2n+1}$. Then we achieve for any $U\in\Gamma(T\mathcal{N})$ and $\zeta\in\Gamma(T\mathcal{N}^{\perp})$ that

Proof. From (2.7), (2.11), (2.12), (2.17), (2.18) and (2.21), we achieve

Now, employing (2.22) and (2.23) into (2.32) then comparing tangential and normal parts, we have (2.30) and (2.31).

Let us consider $U, \xi \in\Gamma(T\mathcal{N})$, thus by the direct application of (2.8) and (2.11), we have

If $\xi\in\Gamma(T\mathcal{N}^{\perp})$, then by the consequence of (2.8) and (2.12), we have

From above we conclude the following remarks

Remark 2.7. Let $M$ is tangent to $\xi$ in $\mathcal{K}^{2n+1}$, then the relation (2.33) holds on $\mathcal{N}$.

Remark 2.8. Let $M$ is normal to $\xi$ in $\mathcal{K}^{2n+1}$, then the Eq (2.34) holds in $\mathcal{N}$.

Lemma 2.9. If $\mathcal{N}$ is tangent to $\xi$ in $\mathcal{K}^{2n+1}$, then the endomorphism $t$ and bundle $1$-form $n$ satisfies:

Proof. Operating $\varphi$ on (2.17), we have

Employing (2.1) and (2.17) into above expression, we achieve

Comparing tangential and normal part of above expression, we get (2.35) and (2.36).

In similar way we prove the following result:

Lemma 2.10. If $\xi$ is normal to $\mathcal{N}$ in $\mathcal{K}^{2n+1}$. Then, we obtain

3.   Warped product submanifold

Let $(B, g_{B})$ and $(F, g_{F})$ be two pseudo-Riemannian manifolds with pseudo-Riemannian metric $g_{B}$ and $g_{F}$, respectively and $f:B\longrightarrow \left(0, \infty\right)$ is a $C^{\infty}$-function. Then, $\mathcal{N} = B\times_{f} F$ is called warped product manifold ^[1,12] if the pseudo-Riemannian product manifold $B\times F$ furnished a pseudo-Riemannian warping metric $g$ fulfill the accompanying condition:

where $f$ is called warping function. If $f$ is constant on $B$ then $M$ is called trivial warped product manifold. For a warped product $\mathcal{N} = B\times_{f} F$, $F$ is called a fiber and $B$ is called a base. The leaves $B\times\lbrace p \rbrace = \sigma_{2}^{-1}(q)$ and $\lbrace p \rbrace\times F = \sigma_{1}^{-1}(p)$, for $(p, q)\in \mathcal{N}$ are pseudo-Riemannian manifolds. Now, recall the following proposition.

Proposition 3.1. For all $V_{1}, V_{2}\in\Gamma(TF)$ and $U_{1}, U_{2}\in \Gamma(TB)$, we obtain on $\mathcal{N} = B\times_{f} F$ that

where $grad(f)$ indicates the gradient of positive function $f$ characterized by $g(\nabla f, U_{1}) = U_{1}(f)$.

Remark 3.2. If $\mathcal{N} = B\times_{f}F$ is a warped product manifold, then the pseudo-Riemannian submanifold $F$ is totally umbilical and the pseudo-Riemannian submanifold $B$ is totally geodesic in $\mathcal{N}$.

In $1956$, J. F. Nash derived a very useful theorem in Riemannian geometry which is known as Nash embedding theorem. The theorem states "every Riemannian manifold can be isometrically embedded in some Euclidean spaces" (see, ^[18]). This theorem shows that any warped product of Riemannian (or pseudo-Riemannian) manifold can be realized (or embedded) as a Riemannian (or pseudo-Riemannian) submanifolds 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, ^[19]). Thereafter several geometers studied warped product submanifold into a different ambient manifolds. After that the warped product was become very active and populor research area among the geometers. Due to this fact, we have studied the warped product submanifold in para-Kenmotsu manifold which is not studied yet. Now, we prove some results related to existence of warped product submanifolds in para-Kenmotsu manifold.

Proposition 3.3. There does not exists a non-trivial warped product submanifold $\mathcal{N} = B\times_{f}F$ in $\mathcal{K}^{2n+1}$ such that $\xi\in\Gamma(TF)$.

Proof. If there exist a non-trivial warped product $\mathcal{N} = B\times_{f}F$, then by the consequence of (3.3), we attain $\nabla_{U_{1}}V_{1} = U_{1}(\ln f)V_{1}$ for all $U_{1}\in\Gamma(TB)$ and $V_{1}\in\Gamma(TF)$. Now taking $V_{1} = \xi$ then applying (2.1) and (2.33) into above expression, we obtain $U_{1} (\ln f)g(\xi, \xi) = -g(U_{1}, \xi) = 0$. This shows that $f$ is constant function, contradiction.

Proposition 3.4. There exists a non-trivial pseudo-Riemannian warped product submanifold $\mathcal{N} = B\times_{f}F$ in $\mathcal{K}^{2n+1}$ such that $\xi\in\Gamma(TB)$.

Proof. If there exists a non-trivial warped product submanifold then by the utilization of relation (3.3), we obtain $\xi(\ln f)\vert V_{1} \vert^{2} = -\vert V_{1} \vert^{2}$ for any space-like or time-like vector field $V_{1}\in\Gamma(TF)$. Now using Proposition 3.3 into above expression, we have $\nabla f = -\xi$, which is first order partial differential equation, in this case $f$ is not constant.

Above Propositions shows that the warped product submanifold in $\mathcal{K}^{2n+1}$ exists if $\xi$ is tangent to the first factor. Thus by the direct consequence of (2.8) and (3.3), we have

4.   $\mathcal{PR}$-semi invariant submanifolds

Definition 4.1. Let $\mathcal{N}$ be tangent to $\xi$ in $\mathcal{K}^{2n+1}$. Then $\mathcal{N}$ is said to be a $\mathcal{PR}$-semi-invariant ^[12] if there exists a $\varphi$-invariant distribution $\mathfrak{D}_{T}$ and a $\varphi$-anti-invariant distribution $\mathfrak{D}_{\perp}$ satisfying

Let $P$ and $Q$ are two orthogonal projections on $\mathfrak{D}_{T}$ and $\mathfrak{D}_{\perp}$, respectively. Thus for $U\in\Gamma(T\mathcal{N})$ can be expressed as follows:

From (4.2), we have

By the application of (2.17) and (4.2), we achieve that,

using the fact $\mathcal{N}$ is $\mathcal{PR}$-semi invariant, we acquire that

In the light of (2.18), we attain that $\varphi(tU+nU) = U-\eta(U)\xi$. Reuse of (2.18) and (4.4) gives $t^{2} = U-\eta(U)\xi$. In view of last expression and (2.19), we conclude that the paracontact structure admits on $\mathfrak{D}_{T}\oplus\langle\xi\rangle$. Therefore, the dimension of the distribution $\mathfrak{D}_{T}$ must be even. Despite if we denotes $\nu$ for $\varphi$-invariant subspace of $T\mathcal{N}^{\perp}$, thus the normal bundle $T\mathcal{N}^{\perp}$ can be expressed as follows:

Theorem 4.2. Let $\psi:\mathcal{N}\longrightarrow\mathcal{K}^{2n+1}$ be an isometric immersion. Then necessary and sufficient condition for $\mathcal{N}$ to be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ is $n\circ t = 0$.

Proof. Firstly, suppose that $\mathcal{N}$ is immersed as a $\mathcal{PR}$-semi invariant submanifold in para-Kenmotsu manifold $\mathcal{K}^{2n+1}$, then by application of (2.17), (4.2) and (4.4), we achieve

Thus the relation (4.6) shows that the $n\circ t = 0$.

Conversely, let $M$ be a pseudo-Riemannian submanifold tangent to $\xi$ in a para-Kenmotsu manifold $\mathcal{K}^{2n+1}$ satisfying $n\circ t = 0$. Therefore, by the consequences of (2.6) and (2.35), we attain that $g(U, tt^{'}\zeta) = 0$, $\forall U\in\Gamma(T\mathcal{N})$ and $\zeta\in\Gamma(T\mathcal{N}^{\perp})$. In view of (2.37), we have $t^{'}n^{'} = 0$. From (2.35) and (2.38), we find

If we put $t^{2} = P$ and $\mathcal{I}-t^{2} = Q$, we have (4.4). This implies that $P$ and $Q$ are orthogonal projections on distributions $D_{T}$ and $D_{\perp}$, respectively. Further, from (4.7) we achieve $t = tP$, $n = nQ$, $nP = 0$ and $tQ = 0$. This shows that $\mathfrak{D}_{T}$ and $\mathfrak{D}_{\perp}$ are invariant and anti-invariant, respectively. Hence completes the proof of theorem.

Lemma 4.3. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then we obtain

for every $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$ and $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$.

Proof. By the consequence (2.7), (2.11) and (2.21), we obtain $g(h(U_{1}, \varphi U_{2}), \varphi \zeta) = g(\varphi\widetilde{\nabla}_{U_{1}}U_{2}, \varphi \zeta)$. By direct use of (2.4) into above expression gives Eq (4.8). Similarly, we obtain (4.9) and (4.10).

Lemma 4.4. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then we have

for any $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$ and $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$.

Proof. By the direct consequence of (2.4), (2.11)–(2.13), (2.22) and (4.4), we achieve (4.11) and (4.12).

Theorem 4.5. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then, the invariant distribution $\mathfrak{D}_{T}$ is integrable if and only if $h$ fulfills

for any $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$.

Proof. By the consequence of (2.11), we obtain for any $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$ that;

Form (2.7) and (2.21), above equation reduces into the following form

By the application of Eqs (2.11), (2.17) and (4.4) above expression reduces into the following form

Interchange the role of $U_{1}$ and $U_{2}$ into the above expression, we obtain

In view of (4.14) and (4.15), we get the result.

Theorem 4.6. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then the distribution $\mathfrak{D}_{\perp}$ is integrable if shape operator satisfying:

for any $V_{1}, V_{2}\in\Gamma(\mathfrak{D}_{\perp})$.

Proof. Let us consider any $U_{1}\in\Gamma(\mathfrak{D}_{T})$, then by the consequence of Eq (2.11), we enlist

Now employing Eqs (2.7) and (2.21) into above relation then we attain

Applying (2.12) and (4.4) into above expression, we achieve

Interchange the role of $U_{1}$ and $U_{2}$ into (4.17), we have

From (4.17) and (4.18), we get (4.16). This completes the proof.

Lemma 4.7. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then, we conclude that

for any $U_{1}\in\Gamma(\mathfrak{D}_{T})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$.

Proof. By the direct consequence of (2.24), (2.25), (4.3) and (4.4).

5.   $\mathcal{PR}$-semi invariant warped product

A $\mathcal{PR}$-semi-invariant warped product submanifold was studied in ^[12]. In this section, we study $\mathcal{PR}$-semi-invariant warped products into a para-Kenmotsu manifold. It is defined as $B\times_{f}F$ or $F\times_{f}B$, where $B$ is a $\varphi$-anti-invariant submanifold of $\mathcal{N}$ and $F$ be a $\varphi$-invariant submanifold of $\mathcal{N}$ which are the integral manifolds of anti-invariant distributions $\mathfrak{D}_{\perp}$ and invariant distributions $\mathfrak{D}_{T}$, respectively (see, ^[12]). If $f$ is a constant function, then $\mathcal{N}$ is called $\mathcal{PR}$-semi-invariant product (or trivial or proper warped product).

Proposition 5.1. There does not exists a $\mathcal{PR}$-semi-invariant non-trivial warped product submanifold of the form $\mathcal{N} = F\times_{f}B$ in $\mathcal{K}^{2n+1}$ such that $\xi\in\Gamma(TB)$.

Proof. If there exists a proper warped product then $\nabla_{U_{1}}\xi = U_{1} (\ln f)\xi$, for all tangent vector field $U_{1}$ in $\Gamma(TF)$. In view of (2.1) and (2.33), we have $U_{1} (\ln f) = 0$. This means $f$ can't be non-constant, contradiction.

Proposition 5.2. There does not exists a $\mathcal{PR}$-semi-invariant non-trivial warped product submanifold of the form $\mathcal{N} = B\times_{f}F$ in $\mathcal{K}^{2n+1}$ such that $\xi\in\Gamma(TF)$.

Proof. If there exists a proper warped product then $\nabla_{U_{1}}\xi = U_{1}(\ln f)\xi$, for all $U_{1}\in\Gamma(TF)$. In view of (2.1) and (2.33), we have $U_{1} (\ln f) = 0$. This relation shows that the function $f$ is constant, contradiction.

Proposition 5.3. There does not exists a $\mathcal{PR}$-semi-invariant non-trivial warped product submanifold of the form $\mathcal{N} = B\times_{f}F$ in $\mathcal{K}^{2n+1}$ such that $\xi\in\Gamma(T\mathcal{N}^{\perp})$.

Proof. Let $\mathcal{N}$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product with $\xi\in\Gamma(TM^{\perp})$. Then by the consequence of (2.4), (2.7), (2.11), (2.21) and (3.3), we obtain $\forall U_{1}\in\Gamma(TF)$ and $V_{1}\in\Gamma(TB)$ that;

Now, we replace $V_{1}$ by $\varphi V_{1}$ into above relation and applying Eq (2.1), we obtain

since $V_{1}$ is not a lightlike vector, therefore, $f$ is constant on $F$, contradiction. Hence complete the proof.

5.1.   $\mathcal{PR}$-semi-invariant warped product of the form $\mathit{\boldsymbol{B}}\times_{\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{F}}$

In this section, we analyze the geometry of $\mathcal{PR}$-semi-invariant warped product submanifold of type $B\times_{f}F$. By the virtue of Proposition 5.2 implies for above mention warped product $\xi$ is tangent to $B$. We derive some important results for $\mathcal{PR}$-semi-invariant warped product submanifold of type $B\times_{f}F$:

Lemma 5.4. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{K}^{2n+1}$. Then, we achieve for all $U_{1}, U_{2}\in\Gamma(TF)$ and $V_{1}, V_{2}\in\Gamma(TB)$ that:

Proof. From the Eqs (2.4), (2.11), (2.13) and (4.4), we obtain $g(A_{nV_{1}}U_{1}, U_{2}) = -g(\widetilde{\nabla}_{U_{1}}U_{2}, \varphi V_{1})$. Now applying (2.6) and (2.21) into last expression, we arrive that

In view of (2.7), above equation reduces into the following form

By virtue of (2.11), (2.17), (5.4) and property of Riemannian connection gives

In light of (2.19) and (3.3), we get (5.1). Employing (2.7), (2.11), (2.12), (2.17), (2.18) and (4.4) into (2.21), we achieve that

Beacause $\mathcal{N}$ is warped product then $B$ is totally geodesic then considering the tangential part of obtained expression

Using (3.2) and (4.4), we arrive at Eq (5.2). Using (2.7), (2.11), (2.17), (2.18) and (4.4) into (2.21), then we find

Now using (3.3) into above expression then after taking normal part of above expression, we have

Interchange the role of $U_{1}$ and $U_{2}$ into above relation gives (5.3).

Theorem 5.5. Let $\mathcal{N} = B\times_{f}F$ be a $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{K}^{2n+1}$. Then the both distributions $\mathfrak{D}_{\perp}$ and $\mathfrak{D}_{T}$ are integrable.

Proof. By direct consequence of (4.4), Theorem 4.5, Theorem 4.6 and Lemma 5.4, we easily achieve the result.

Lemma 5.6. If $\mathcal{N} = B\times_{f}F$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold in $\mathcal{K}^{2n+1}$. Then, we have

for all $V_{1}, V_{2}, V_{1}^{\prime}\in\Gamma(TB)$ and $U_{1}, U_{2}\in\Gamma(TF)$.

Proof. By the application of Eqs (2.6), (2.11), (2.17) and (4.4), we have

Now utilizing (2.7) and (2.21) into above expression, we achieve

By the consequence of (2.11), (2.17), (4.4) and (5.9), the above relation reduces into the following form

By the virtue of (2.19), (3.3) and (5.10), we achieve (5.6). By the consequence of (2.6), (2.11), (2.17) and (4.4), we obtain $g(h(U_{1}, V_{1}), nV_{2}) = -g(\varphi \widetilde{\nabla}_{U_{1}}V_{1}, V_{2})$. In light of (2.7) and (2.21), we get (5.7). Proceed similar process for accomplish the Eq (5.8).

Lemma 5.7. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{K}^{2n+1}$. Then for all $U_{1}, U_{2}\in\Gamma(TF)$ and $V_{1}, V_{2}\in\Gamma(TB)$, we obtain

Proof. By the direct use of (3.3), (3.4) and (4.4), we obtain (5.11)–(5.14).

Now, we prove some results related to characterization of $\mathcal{PR}$-semi invariant warped product submanifold:

Theorem 5.8. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then $\mathcal{N}$ is form a $\mathcal{PR}$-semi-invariant warped product submanifold if and only if the shape operator $A$ satisfies:

for any $C^{\infty}$-function $\mu$ over $\mathcal{N}$ which satisfying $U_{2}(\mu) = 0$, $\forall U_{2}\in\Gamma(\mathfrak{D}_{T})$.

Proof. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial warped product submanifold in $\mathcal{K}^{2n+1}$.

Thus, by the direct application of Eqs (2.17), (4.4) and (5.1), we attain for any $V_{1}\in\Gamma(TB)$ and $U_{1}\in\Gamma(TF)$ that

If we take $\mu = \ln f$ in above expression and use fact that $\mathcal{N}$ is warped product, then we accomplished (5.15).

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.15). Thus, by the utilization of (2.7), (2.11) and (2.16), we obtain for all $U_{1}\in\Gamma(\mathfrak{D}_{T})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp}\oplus\langle\xi\rangle)$ that

In light of (2.12) and (5.15), the above equation taking into the form

Above discussion demonstrate that the distribution $\mathfrak{D}_{\perp}\oplus\langle\xi\rangle$ is form totally geodesic foliation. Furthermore, by the consequence of (2.4), (2.11) and (2.12), we obtain $\forall U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp}\oplus\langle\xi\rangle)$ that

Now employing (2.7), (2.8), (2.21) and (4.4) into above relation:

In view of (2.4), (2.12) and (5.15) above relation reduces into the following form

Above calculation shows that the distribution $\mathfrak{D}_{\perp}$ defines spherical foliation. After using Hiepko Theorem ^[20], we can conclude that $B\times_{f}F$ is a warped product.

Theorem 5.9. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then $\mathcal{N}$ is form a non-trivial $\mathcal{PR}$-semi-invariant warped product $\mathcal{N} = B\times_{f}F$ if and only if the endomorphism $t$ satisfies the following condition:

for any $C^{\infty}$-function $\mu$ over $\mathcal{N}$ fulfilling $U_{2}(\mu) = 0$, $\forall U_{2}\in\Gamma(\mathfrak{D}_{T})$.

Proof. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial warped product submanifold in $\mathcal{K}^{2n+1}$,

then by the application of (4.2), we attain that

We obtain by the utilization of (2.24) and Lemma 5.7 that

Then, by taking $\mu = \ln f$ and using the fact that $\mathcal{N}$ is warped product, we accomplished (5.16).

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.16). If we interchange $U$ with $V_{1}$ and $V$ with $V_{2}$ into the Eq (5.16), then we obtain for any $V_{1}, V_{2}\in\Gamma(\mathfrak{D}_{\perp}\oplus\langle\xi\rangle$ and $U_{1}\in\Gamma(\mathcal{D}_{T})$ that;

On the other hand, by the use Eqs (2.23), (3.4) and (5.16), we compute

By above relation, we accomplish that

Above calculation proves that the $\mathfrak{D}_{T}$ defines spherical foliation i.e., the distribution is totally umbilical and has a parallel mean curvature. And the distribution $\mathfrak{D}_{T}$ always integrable. The relation (5.17) shows that the the distribution $\mathfrak{D}_{\perp}\oplus\langle\xi\rangle$ is defines totally geodesic foliation. After using Hiepko Theorem ^[20], we conclude that $\mathcal{N}$ is form a $\mathcal{PR}$-semi-invariant warped product of type $B\times_{f}F$, where $F$ is leaf of the distribution $\mathfrak{D}_{T}$ and $B$ is leaf of the distribution $\mathfrak{D}_{\perp}$.

Theorem 5.10. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$. Then $\mathcal{N}$ is form a non-trivial $\mathcal{PR}$-semi-invariant warped product $\mathcal{N} = B\times_{f}F$ if and only if the endomorphism $n$ satisfies:

for all $U, V\in\Gamma(T\mathcal{N})$ and a smooth function $\mu$ over $\mathcal{N}$ satisfying $U_{2}(\mu) = 0$, $\forall U_{2}\in\Gamma(\mathfrak{D}_{T})$ and $\xi(\mu) = -1$.

Proof. Let $\mathcal{N} = F\times_{f}B$ be a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{K}^{2n+1}$ such that $F$ is a $\varphi$-invariant submanifold and $B$ is a $\varphi$-anti-invariant submanifold, then from (4.2) we have

By the utilization of (2.24) and (3.4), we attain

taking $\mu = \ln f$ in above equation to obtain (5.19).

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.19). Then, if we replace $U$ by $U_{1}$ and $V$ by $U_{2}$ into the relation (5.19), we acquire that

for $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$. Now applying Eq (2.22) into above expression, we obtain

Above discussion show that the distribution $\mathfrak{D}_{T}$ is integrable. By the consequence of (2.4), (2.17) and (4.4), we have

In view of (2.8), (2.22) and (2.25), we attain

Now employing (2.4) and (2.17) into above relation, we acquire that

This proves that the $\mathfrak{D}_{T}$ defines spherical foliation. Despite that, if we replace $U$ by $V_{1}$ and $V$ by $U_{1}$ into the Eq (5.19), then we get

for $U_{1}\in\mathfrak{D}_{T}$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp}\oplus\langle\xi\rangle)$. Thus by the consequence of (2.4), (2.8) and (2.17), we obtain

for any $U_{1}\in\mathfrak{D}_{T}$ and $V_{1}, V_{2}\in\Gamma(\mathfrak{D}_{\perp}\oplus\langle\xi\rangle)$. Now utilizing (2.22) and (5.22) into above expression, we arrive $g(\nabla_{V_{1}}V_{2}, U_{1}) = 0.$ Above calculation shows that the distribution $\mathfrak{D}_{\perp}\oplus\langle\xi\rangle$ defines totally geodesic foliation. Therefore, by an application of Hiepko Theorem ^[20], we can conclude that $\mathcal{N}$ is a $\mathcal{PR}$-semi invariant warped product of type $B\times_{f}F$, where $B$ is leaf of the distribution $\mathfrak{D}_{\perp}\oplus\langle\xi\rangle$ and $F$ is leaf of the distribution $\mathfrak{D}_{T}$. Hence completes the proof.

5.2.   $\mathcal{PR}$-semi-invariant warped product submanifold of the form $\mathit{\boldsymbol{F}}\times_{\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{B}}$

In this section, we analyze the geometry of $\mathcal{PR}$-semi-invariant warped product submanifold of type $F\times_{f}B$. We derive some important results for such warped products:

Lemma 5.11. Let $\mathcal{N} = F\times_{f}B$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold in $\mathcal{K}^{2n+1}$, thus we obtain for any $U_{1}\in\Gamma(TF)$ and $V_{1}, V_{2}\in\Gamma(TB)$ that

Proof. In view of (2.6), (2.7) and (2.11), we obtain

Now, we just use use (3.3) into above expression to obtain (5.23). We achieve (5.24) if we replace $U_{1}$ by $\varphi U_{1}$ into (5.23).

Lemma 5.12. For a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold $\mathcal{N} = F\times_{f}B$ in $\mathcal{K}^{2n+1}$, we obtain

for all $V_{1}\in\Gamma(TB)$, $V\in\Gamma(T\mathcal{N})$ and $U_{1}\in\Gamma(TF)$.

Proof. In view of (2.23) and (4.2), we obtain

Now using (2.24), (3.3) and (4.4) into above expression, we obtain (5.25). For the proof of (5.26), we consider

In above expression we just utilize (2.23), (2.24) and (3.4) to achieve (5.26). Similarly, by the use of (2.23), (2.25), (3.3), (3.4), (4.2) and (4.4), we easily achieve (5.27) and (5.28).

Theorem 13. Let $\psi:\mathcal{N}\longrightarrow\mathcal{K}^{2n+1}$ be an isometric immersion and $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold. Then $\mathcal{N}$ is a $\mathcal{PR}$-semi-invariant warped product submanifold $\mathcal{N} = F\times_{f}B$ if and only if $A$ satisfies:

where $\mu$ is a smooth function on $\mathcal{N}$ satisfying $V_{2}(\mu) = 0$, for all $V_{2}\in\Gamma(\mathfrak{D}_{\perp})$.

Proof. Let $\mathcal{N} = F\times_{f}B$ be a non-trivial warped product submanifold $\mathcal{K}^{2n+1}$, then, in view of (2.6), (2.7), (2.11) and (4.4), we have

for any $V_{1}\in\Gamma(TB)$ and $U_{1}\in\Gamma(TF)$. By using the fact that $\mathcal{N}$ is a warped product and taking $\mu = \ln f$, then we acquire that $V_{2}(\mu) = 0$, for $V_{2}\in\Gamma(TB)$.

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.30). Therefore, in light of Eqs (2.6) and (2.11), we attain that for all $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T}\oplus\langle\xi\rangle)$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$ that;

In view of (2.12) and (5.30) above expression reduces into the following form

Above expression shows that the anti-invariant distribution $\mathfrak{D}_{\perp}$ defines totally geodesic foliation. Further, by the consequence of (2.6), (2.7), (2.12), (2.21) and (4.4)), we achieve

for all $U_{1}\in\Gamma(\mathfrak{D}_{T}\oplus\langle\xi\rangle)$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$. At this moment we applying (5.30) into above relation, then we accomplish that

Above discussion proves that the $\mathfrak{D}_{T}$ defines spherical foliation. By using Hiepko Theorem ^[20], we can conclude that $\mathcal{N}$ is a $\mathcal{PR}$-semi-invariant warped product of type $F\times_{f}B$, where $F$ is leaf of the distribution $\mathfrak{D}_{T}\oplus\langle\xi\rangle$ and $B$ is leaf of the distribution $\mathfrak{D}_{\perp}$. This finishes the proof of theorem.

Theorem 5.14. Let $\psi:\mathcal{N}\longrightarrow\mathcal{K}^{2n+1}$ be an isometric immersion. Then necessary and sufficient condition for a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product $\mathcal{N} = F\times_{f}B$ if and only if endomorphism $t$ satisfies

for every $U, V\in\Gamma(T\mathcal{N})$, where $\mu$ is smooth function on $\mathcal{N}$ satisfying $V_{1}(\mu) = 0$, $\forall V_{1}\in\Gamma(\mathfrak{D}_{\perp})$.

Proof. Let $\mathcal{N} = F\times_{f}B$ be a $m$-dimensional non-trivial $\mathcal{PR}$-semi-invariant warped product such that $B$ is a $\varphi$-anti-invariant submanifold and $F$ is a $\varphi$-invariant submanifold, then we attain by the use of (4.2) that

By the utilization of (2.24), (5.25) and (5.26), we have

taking $\mu = \ln f$ in above equation to obtain (5.32).

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.32). Then, with the help of (4.4) and (5.32), we obtain

for any $U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T}\oplus\langle \xi\rangle)$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$. The Eq (5.33) implies that $D_{T}\oplus\langle\xi\rangle$ defines totally geodesic foliation. However, if we substitute $V_{1}$ at the place $U$ and $U_{2}$ at the place $V$ into the Eq (5.32), then we get

for any $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$ and $U_{2}\in\Gamma(\mathfrak{D}_{T}\oplus\langle \xi\rangle)$. Now taking inner product with $V_{2}\in\Gamma(\mathfrak{D}_{\perp})$ into (5.34), we achieve that

Above calculation demonstrate that the distribution $\mathfrak{D}_{\perp}$ defines spherical foliation. After using the Hiepko theorem ^[20], we can conclude that $\mathcal{N}$ is form a $\mathcal{PR}$-semi-invariant warped product of type $F\times_{f}B$, where $F$ is leaf of $\mathfrak{D}_{T}$ and $B$ is leaf of $\mathfrak{D}_{\perp}$. This accomplished the proof of theorem.

Theorem 5.15. Let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold $\mathcal{K}^{2n+1}$. Then $\mathcal{N}$ is a non-trivial $\mathcal{PR}$-semi-invariant warped product of the form $F\times_{f}B$ if and only if

for any $U, V\in\Gamma(T\mathcal{N})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$, where $\mu$ is smooth function on $M$ satisfying $V_{1}(\mu) = 0$.

Proof. Let $F\times_{f}B$ be a proper $\mathcal{PR}$-semi invariant warped product submanifold of dimension $m$ in $\mathcal{K}^{2n+1}$ such that $B$ is a $\varphi$-anti-invariant submanifold and $F$ is a $\varphi$-invariant submanifold, subsequently we obtain from (4.2) that

By the utilization of (2.24), (5.27) and (5.28), we have

taking $\mu = \ln f$ in above equation to obtain (5.35).

Conversely, let $\mathcal{N}$ be a $\mathcal{PR}$-semi-invariant submanifold in $\mathcal{K}^{2n+1}$ satisfying (5.32). Replace $U$ with $U_{1}$ and $V$ with $U_{2}$ in the relation (5.35), we have

$U_{1}, U_{2}\in\Gamma(\mathfrak{D}_{T})$. In light of (2.4), (2.8) and (2.17), we attain

for any $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$. Now utilize (2.22) and (5.36) into above expression, we achieve

The relation (5.37) implies that the distribution $\mathfrak{D}_{T}\oplus\langle\xi\rangle$ defines totally geodesic foliation. Moreover, if we replace $U$ by $V_{1}$ and $V$ by $U_{2}$ into the Eq (5.32), then we get

for any $U_{2}\in\Gamma(\mathfrak{D}_{T})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$. By the utilization of (2.4), (2.17) and (4.4) into above expression, we obtain

for any $U_{2}\in\Gamma(\mathfrak{D}_{T})$ and $V_{1}\in\Gamma(\mathfrak{D}_{\perp})$. Now employing (2.4), (2.17), (2.22) and (5.35) into above relation

Above calculation demonstrate that the distribution $\mathfrak{D}_{\perp}$ defines spherical foliation. After using Hiepko Theorem ^[20], we can deduce that $\mathcal{N}$ is a $\mathcal{PR}$-semi invariant warped product of type $F\times_{f}B$, where $B$ is leaf of the distribution $\mathfrak{D}_{\perp}$ and $F$ is leaf of the distribution $\mathfrak{D}_{T}$. This proves the the result.

6.   Optimal inequalities

6.1.   Inequalities $\mathcal{PR}$-semi invariant warped product submanifold of type $\mathit{\boldsymbol{B}}\times_{\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{F}}$

In this section, we derive some inequalities for warped product submanifold of type $\mathcal{N} = B\times_{f}F$ into para-Kenmotsu manifold. We assume the dimension of $B$ is $s+1$ and the dimension of $F$ is $2r$. Now, we construct a frame field for $F\times_{f}B$ as follows:

● An orthonormal basis $\lbrace U_{i}, U_{i}^{*} = \varphi U_{i}\rbrace$ for $\mathfrak{D}_{T}$, where $i = 1, 2, ...r$. Further, one can suppose $\epsilon_{i} = g(U_{i}, U_{i}) = 1$ and $\epsilon^{*}_{i} = g(U^{*}_{i}, U^{*}_{i}) = -1$.

● An orthonormal basis $\lbrace V_{a}, \xi\rbrace$ for $\mathfrak{D}_{\perp}$, where $i = 1, 2, \cdot, s$ and $\epsilon_{a} = g(V_{a}, V_{a})$.

● An orthonormal basis $\lbrace V^{*}_{a} = \varphi V_{a}, \zeta_{b}, \zeta^{*}_{b} = \varphi\zeta_{b}\rbrace$ for $TM^{\perp}$, $b = 1, 2, ...t$. Moreover, one can assume $\epsilon_{\alpha} = g(\zeta_{\alpha}, \zeta_{\alpha})$ and $\epsilon^{*}_{\alpha} = g(\zeta^{*}_{\alpha}, \zeta^{*}_{\alpha})$.

Theorem 6.1. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{N}$ such that $B$ is a time-like submanifold. Then, $h$ satisfies the following relation:

where $\Vert h_{\nu}^{\mathfrak{D}_{T}}\Vert^{2} = g(h_{\nu}(\mathfrak{D}_{T}, \mathfrak{D}_{T}), h_{\nu}(\mathfrak{D}_{T}, \mathfrak{D}_{T}))$. Moreover, if equality holds then $M$ is mixed totally geodesic and if $h(\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp})\perp\nu$, then we have

Proof. If $\mathfrak{D}_{\perp}$ is a time-like distribution then $\epsilon_{a} = -1$. Now the square norm of $h$ is given by:

First, we consider

Now using the fact that $\mathfrak{D}_{T}$ is $\varphi$-invariant distribution, then we have

With the help of (6.5) and Lemma 4.3, the Eq (6.4) reduces into the following form

Now using (4.8) and the intergrability condition into above expression, we compute

Now, applying (5.6), (5.12) and (5.25) into above relation

Above expression reduces into the following form

By adding and subtracting same quantity into above expression

By the definition of gradient above relation reduces into the following form

Now consider

By the utilization of $h(V_{a}, V_{b}) = h^{c}_{ab}nV_{c}+h^{\alpha}_{ab}\zeta_{\alpha}+h^{\alpha^{*}}_{ab}\zeta_{\alpha^{*}}$ into above relation, we achieve,

Last term is vanishes since $(h^{\alpha}_{ab})^{2} = (h^{\alpha^{*}}_{ab})^{2}$. Therefore, above expression reduces into following form,

Lastly, consider

Now using

These expressions employing into (6.11), we have

Since $\mathfrak{D}_{\perp}$ is totally geodesic then $h_{i^{*}a}^{c} = h_{ia}^{c}$. So, above relation become

By the virtue of Lemma 4.3 and Lemma 4.4, we obtain

By utilization of (6.9), (6.10) and (6.13), we obtain (6.1). So, equality holds if and only if $\Vert h(\mathfrak{D}_{T}, \mathfrak{D}_{\perp})\Vert = 0$, this follows $(2)$. The statement $(3)$ directly follows from (6.1) and (6.9).

Theorem 6.2. Let $\mathcal{N} = B\times_{f}F$ be a non-trivial $\mathcal{PR}$-semi invariant warped product submanifold in $\mathcal{K}^{2n+1}$ such that $B$ is a space-like submanifold then $h$ satisfying the following relation

Moreover, if equality holds then $\mathcal{N}$ is mixed totally geodesic and if $h(\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp})\perp\nu$, then we receive that

6.2.   Inequalities $\mathcal{PR}$-semi invariant warped product submanifold of type $\mathit{\boldsymbol{F}}\times_{\mathit{\boldsymbol{f}}}\mathit{\boldsymbol{B}}$

In this section, we derive some inequalities for warped product submanifold of type $F\times_{f}B$ into para-Kenmotsu manifold. We assume the dimension of $B$ is $s$ and the dimension of $F$ is $2r+1$. Now, we construct a frame field for $F\times_{f}B$ as follows:

● An orthonormal basis $\lbrace U_{i}, X^{*} = \varphi U_{i}, \xi\rbrace$ for $\mathfrak{D}_{T}$, where $i = 1, 2, ...r$. Further, one can suppose $\epsilon_{i} = g(U_{i}, U_{i}) = 1$ and $\epsilon^{*}_{i} = g(U^{*}_{i}, U^{*}_{i}) = -1$.

● An orthonormal basis $\lbrace V_{a}\rbrace$ for $\mathfrak{D}_{\perp}$, where $i = 1, 2, \cdot, s$ and $\epsilon_{a} = g(V_{a}, V_{a})$.

● An orthonormal basis $\lbrace Z^{*}_{a} = \varphi V_{a}, \zeta_{b}, \zeta^{*}_{b} = \varphi\zeta_{b}\rbrace$ for $TM^{\perp}$, $b = 1, 2, ...t$. Moreover, one can assume $\epsilon_{\alpha} = g(\zeta_{\alpha}, \zeta_{\alpha})$ and $\epsilon^{*}_{\alpha} = g(\zeta^{*}_{\alpha}, \zeta^{*}_{\alpha})$.

Theorem 6.3. Let $\mathcal{N} = F\times_{f}B$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold in $\mathcal{K}^{2n+1}$ such that $\xi$ is tangent to $B$ and $B$ is a time-like submanifold with $\nabla^{\perp}(\varphi F)\subset\varphi(F)$. Then, $h$ satisfies

where $\nabla^{F} \ln f$ is gradient of $F$. Moreover, equality holds if $\Vert h(\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp})\Vert = 0$ and if $M$ is mixed totally geodesic submanifold then the inequality (6.16) taking the following form

Proof. Now the square norm of $h$ is given by:

First, we consider

Now using the fact that $\mathfrak{D}_{T}$ is $\varphi$-invariant distribution, then we have

With the help of (6.20) and Lemma 4.3, the Eq (6.19) reduces into following form

Now using (4.8) and the intergrability condition into above expression, we calculate

Now, consider

By the utilization of $h(V_{a}, V_{b}) = h^{c}_{ab}nV_{c}+h^{\alpha}_{ab}\zeta_{\alpha}+h^{\alpha^{*}}_{ab}\zeta_{\alpha^{*}}$ into (6.22), we achieve,

Last term is vanishes, since $(h^{\alpha}_{ab})^{2} = (h^{\alpha^{*}}_{ab})^{2}$. Therefore, above expression reduces into following form,

Lastly, taking

In this case $h(U_{i}, V_{a}) = h^{b}_{ia}nV_{b}+h_{ia}^{\alpha}\zeta_{\alpha}+h_{ia}^{\alpha^{*}}\zeta_{\alpha^{*}}$ and $h(U_{i^{*}}, V_{a}) = h^{b}_{i^{*}a}nZ_{b}+h_{i^{*}a}^{\alpha}\zeta_{\alpha}+h_{i^{*}a}^{\alpha^{*}}\zeta_{\alpha^{*}}$. Last expressions employing into (6.24) then after applying (5.6), we obtain

Now employing Lemma 5.11 into above expression, then we find

By utilization of (6.21), (6.23) and (6.25), we obtain (6.16).

Theorem 6.4. Let $\mathcal{N} = F\times_{f}B$ be a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold in $\mathcal{K}^{2n+1}$ such that $\xi$ is tangent to $B$ and $B$ is a space-like submanifold with $\nabla^{\perp}(\varphi F)\subset\varphi(F)$. Then the following results holds:

(1) $h$ is fulfills

where $\Vert h_{\nu}^{\mathfrak{D}_{T}}\Vert^{2} = g(h_{\nu}(\mathfrak{D}_{T}, \mathfrak{D}_{T}), h_{\nu}(\mathfrak{D}_{T}, \mathfrak{D}_{T}))$.

(2) If $\Vert h(\mathfrak{D}_{\perp}, \mathfrak{D}_{\perp})\Vert = 0$, then equality holds in (6.26).

(3) If $\mathcal{N}$ is mixed totally geodesic submanifold then the inequality (6.26) taking the following form

7.   Examples

Example 7.1. Choose $\widetilde{M} = \mathbb{R}^{4}\times \mathbb{R^{+}}$ together the the usual Cartesian coordinates $(x_{1}, x_{2}, x_{3}, x_{4}, s)$. Then the structure $(\varphi, \xi, \eta)$ over $\widetilde{M}$ is defined by

where $e_{i} = \frac{\partial}{\partial x_{i}}$, for $i\in\lbrace 1, 2, 3, 4, 5\rbrace$ and $e_{6} = \frac{\partial}{\partial s}$, and the pseudo-Riemannian metric $g$ is defined as

Then by the simple computation, we can easily see that $\widetilde{M}$ is para-Kenmotsu manifold. Suppose $M$ be an immersed submanifold into $\widetilde{M}$ by an immersion $\sigma$ which is given by

So, we can easily provide the generating set for the tangent bundle of submanifold as follows:

The space $\varphi(TM)$ with respect to $\varphi$ is spanned by the following vectors

Clearly, the vectors $\varphi Z_{1}$ is orthogonal to the tangent bundle $TM$. Therefore, the distribution $span\lbrace Z_{1}\rbrace$ is anti-invariant under $\varphi$ and the distribution $span\lbrace Z_{2}\rbrace$ is invariant under $\varphi$. The induced metric $g_{M}$ on $M$ is given by:

Above discussion demonstrate that $M$ is a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold of type $M_{T}\times_{f}M_{\perp}$ with warping function $f = \alpha e^{-s}$.

Example 7.2. Let us consider $\widetilde{M} = \mathbb{R}^{10}\times \mathbb{R^{+}}$ together the the usual Cartesian coordinates $(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7}, x_{8}, x_{9}, x_{10}, s)$. Then the structure $(\varphi, \xi, \eta)$ over $\widetilde{M}$ is defined by

where $e_{i} = \frac{\partial}{\partial x_{i}}$, for $i\in\lbrace 1, 3, ...., 9\rbrace$ and $i\in\lbrace 2, 4, ...., 10\rbrace$, and the pseudo-Riemannian metric tensor $g$ is defined as

Then by simple computation, we can easily see that $\widetilde{M}$ is para-Kenmotsu manifold. Suppose $M$ be an immersed submanifold into $\widetilde{M}$ by an immersion $\sigma$ which is defined by

So, we can easily provide the generating set for the tangent bundle of submanifold as follows:

The space $\varphi (TM)$ is spanned by the following vectors

Clearly, the vectors $\varphi Z_{\theta}$ and $\varphi Z_{\phi}$ is orthogonal to $TM$. Therefore, the distribution $\mathcal{D}_{\perp} = span\lbrace Z_{\theta}, Z_{\phi} \rbrace$ is anti-invariant under $\varphi$ and the distribution $\mathcal{D}_{T} = span\lbrace Z_{\alpha}, Z_{\beta} \rbrace$ is invariant under $\varphi$. The induced metric tensor $g_{M}$ on $M = M_{T}\times_{f}M_{\perp}$ is given by:

Above calculation manifest that $M$ is a form a non-trivial $\mathcal{PR}$-semi-invariant warped product submanifold of $\widetilde{M}$ such that the warping function $f = (\beta^{2}-\alpha^{2})^{\frac{1}{2}}e^{-s}$.

The authors confirm that the data supporting the findings of this study are available within the article (and/or) its supplementary materials.

8.   Conclusions

In this paper, we obtained existence and non-existence conditions for warped product submanifold of para-Kenmotsu manifold, derived results related to integrability conditions of $\mathcal{PR}$-semi invariant submanifold in para-Kenmotsu manifold. We established characterization results allied to $\mathcal{PR}$-semi-invariant warped product submanifolds. Furthermore, optimal inequalities for $F\times_{f}B$ and $B\times_{f}F$ are obtained and examples of these warped products are illustrated.

Acknowledgments

The authors would like to express their gratitude to the Deanship of Scientific Research at King Khalid University, Saudi Arabia for providing a funding research group under the research grant R. G. P. 2/130/43. Also, The authors express their gratitude to Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2022R27), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia. Anuj Kumar: Supported by CSIR, Human Resource Development Group, India through SRF [09/1196(0001)/2018-EMR-I].

Conflict of interest

The authors declare no conflicts of interest.