Geometric Inequalities for Skew CR-Warped Product Submanifolds in Locally Conformal Almost Cosymplectic Manifolds

Abstract

In this paper, we investigate contact skew CR-warped product submanifolds of locally conformal almost cosymplectic manifolds, a framework that simultaneously generalizes warped product pseudo-slant, semi-slant, and contact CR-submanifolds. We first establish a necessary and sufficient characterization theorem showing that a proper contact skew CR-submanifold with integrable slant distribution admits a local warped product structure if and only if certain shape operator conditions involving the slant angle and the warping function are satisfied. Subsequently, we derive sharp geometric inequalities for the squared norm of the second fundamental form in terms of the warping function, the slant angle, and the conformal factor of the ambient manifold. The equality cases are completely characterized and lead to strong rigidity results, namely that the base manifold is totally geodesic while the slant fiber is totally umbilical in the ambient space. Several applications are presented, showing that our results recover and extend a number of known inequalities and classification theorems for warped product submanifolds in cosymplectic, Kenmotsu, and Sasakian geometries as special cases.

1. Introduction and Motivations

The theory of submanifolds plays a fundamental role in differential geometry, particularly in understanding how intrinsic and extrinsic geometric properties interact under additional geometric structures. Among the most studied classes are CR-submanifolds, slant submanifolds, and their various generalizations in almost Hermitian and almost contact metric manifolds. A unified framework that encompasses these classes is provided by the notion of skew CR-submanifolds, originally introduced by G. S. Ronsse [1] in the setting of Kähler manifolds. This concept naturally generalizes invariant, anti-invariant, semi-slant, and pseudo-slant submanifolds, thereby offering a flexible geometric structure for studying mixed distributions. Their study was extended to almost contact metric manifolds by Tripathi [2], a locally product Riemannian manifold by Tastan [3] and later to other geometric settings by several authors (see [4]). Parallel to this development, warped product manifolds, introduced by Bishop and O’Neill [5], have played a fundamental role in geometry and mathematical physics. The systematic study of warped product submanifolds in complex and contact manifolds was initiated by Chen [6,7], who established sharp inequalities involving the second fundamental form. Subsequent works investigated warped product submanifolds in Kähler, cosymplectic, Kenmotsu, and nearly cosymplectic manifolds (see [8,9,10,11,12]).

The interaction between skew CR-geometry and warped products was first explored by Şahin [13] in Kähler manifolds and later extended to cosymplectic manifolds by Khursheed et al. [14]. More recently, contact skew CR-warped product submanifolds were studied in Sasakian manifolds by Uddin et al. [15]. However, the case of locally conformal almost cosymplectic manifolds, introduced by Olszak [16], remains largely unexplored. This class naturally generalizes cosymplectic and Kenmotsu manifolds [17,18].

Motivated by these observations, we study contact skew CR-warped product submanifolds of the form $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ locally conformal almost cosymplectic manifolds $\tilde{\mathrm{Q}}$ such that ${\mathrm{Q}}_1={\mathrm{Q}}_T\times {\mathrm{Q}}_{\perp }$. We establish a characterization theorem for the existence of such warped product structures and derive sharp inequalities for the squared norm of the second fundamental form in terms of the warping function and the slant angle. The equality cases yield rigidity results showing that the base manifold is totally geodesic and the slant fiber is totally umbilical. Our results recover and extend several known inequalities for warped product pseudo-slant, semi-slant, and skew CR-submanifolds as special cases [4,14,19,20,21,22].

2. Definitions and Basic Results

An almost contact structure $(\phi ,\xi ,\eta ,g)$ endowed with $(2m+1)$-dimensional manifold $\tilde{\mathrm{Q}}$ satisfies the following necessary compatibility conditions

$$\begin{array}{c}\hfill {\phi }^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1,\phi \left(\xi \right)=0,\eta \circ \varphi =0.\end{array}$$

(1)

$$\begin{array}{c}\hfill g(\phi E_1,\phi E_2)=g(E_1,E_2)-\eta \left(E_1\right)\eta \left(E_2\right),\eta \left(E_1\right)=g(E_1,\xi ),\end{array}$$

(2)

for any $E_1,E_2$ tangent to $\tilde{\mathrm{Q}}.$ Moreover, $\phi$, $\xi ,$ and $\eta$ stand for $(1,1)$-tensor, a vector field, a dual 1-form. A contact manifold $\tilde{\mathrm{Q}}$ together with (1) and (2) meets the following conditions according [16]

$$\begin{array}{cc}\hfill & \left({\tilde{\nabla }}_{E_1}\phi \right)E_2=\lambda \left\{g(\phi E_1,E_2)\xi -\eta \left(E_2\right)\phi E_1\right\}\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill & {\tilde{\nabla }}_{E_1}\xi =\lambda \left\{E_1-\eta \left(E_1\right)\xi \right\}.\hfill \end{array}$$

(4)

The manifold $\tilde{Q}$ is said to be a locally conformal almost cosymplectic manifold, where $\tilde{\nabla }$ denotes the Levi–Civita connection. Moreover, $\Gamma \left(T\tilde{\mathrm{Q}}\right)$ stands for the Lie algebra of vector fields.

Remark 1. 

A locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$ is generalized to a cosymplectic manifold and Kenmotsu manifold $\lambda =0$ and $\lambda =1$, respectively (see [17,18]).

The Weingarten and Gauss formulas, where ∇(${\nabla }^{\perp }$) is induced connections on the tangent bundle $T\mathrm{Q}$ (normal bundle $T^{\perp }\mathrm{Q}$) of submanifold $\mathrm{Q}\subseteq \tilde{\mathrm{Q}}$, are given by

$$\begin{array}{c}\hfill \left(i\right){\tilde{\nabla }}_{E_1}{E}_2=-A_{E_2}{E}_1+{\nabla }_{E_1}^{\perp }{E}_2,\left(ii\right){\tilde{\nabla }}_{E_1}{E}_2={\nabla }_{E_1}{E}_2+\mathfrak{B}(E_1,E_2),\end{array}$$

(5)

for each $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right)$ and $E_2\in \Gamma \left(T^{\perp }\mathrm{Q}\right)$, where $\mathfrak{B}$ and $A_{E_2}$ are the second fundamental form and shape operator. They are related as follows:

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,E_2),E_2)=g(A_{E_2}{E}_1,E_2).\end{array}$$

(6)

For any $E_1\in \Gamma \left(T\mathrm{Q}\right)$, from (4) and (5) (i), we may write

$$\begin{array}{c}\hfill \left(i\right){\nabla }_{E_1}\xi =\lambda \left\{E_1-\eta \left(E_1\right)\xi \right\},\left(ii\right)\mathfrak{B}(E_1,\xi )=0.\end{array}$$

(7)

If $E_1\in \Gamma \left(T\mathrm{Q}\right)$ and $E_2\in \Gamma \left(T^{\perp }\mathrm{Q}\right)$,

$$\begin{array}{c}\hfill \left(i\right)\phi E_1=T{E}_1+F{E}_1,\left(ii\right)\phi E_2=t{E}_2+f{E}_2\end{array}$$

(8)

where $T{E}_1\left(t{E}_2\right)$ and $F{E}_1\left(f{E}_2\right)$ stand for tangential and normal components of $\phi E_1\left(\phi E_2\right)$, respectively. $(1,1)$-tensor T follows that

$$\begin{array}{c}\hfill g(T{E}_1,E_2)=-g(E_1,T{E}_2),\end{array}$$

(9)

for every $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right)$. It should be noted that throughout the manuscript, we will use $\mathrm{Q}$ as a submanifold of locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$, that is, $\mathrm{Q}\subseteq \tilde{\mathrm{Q}}.$ Now, we have the following classification of submanifold theory:

(a)

A submanifold $\mathrm{Q}$ is said to be minimal in $\tilde{\mathrm{Q}}$ if its mean curvature vector $\mathrm{H}$ vanishes.

(b)

If the second fundamental form $\mathfrak{B}$ is vanished everywhere on $\mathrm{Q}$, then $\mathrm{Q}$ is totally geodesic in $\tilde{\mathrm{Q}}.$

(c)

A totally umbilical submanifold $\mathrm{Q}$ in $\tilde{\mathrm{Q}}$ implies $\mathfrak{B}(E_1,E_2)=g(E_1,E_2)\mathrm{H}$, for any $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right).$

(d)

If the tangent bundle $T_p\mathrm{Q}$ of $\mathrm{Q}$ is satisfied $\phi \left(T_p\mathrm{Q}\right)\subseteq T_p\mathrm{Q}$ for $\xi \in \mathrm{Q}$, then $\mathrm{Q}$ is an invariant submanifold. An anti-invariant submanifold is satisfied $\phi \left(T_p\mathrm{Q}\right)\subset T_p^{\perp }\mathrm{Q}$ for the normal bundle $T_p^{\perp }\mathrm{Q}$ of $\mathrm{Q}$ at point p.

(e)

If the angle $\gamma \left(E_1\right)$ between $\phi E_1$ and $T_p\mathrm{Q}$ is constant for any vector $0\ne E_1\in \mathrm{Q}$, then $\mathrm{Q}$ is a slant submanifold of $\tilde{\mathrm{Q}}$ along a slant angle $\gamma \left(E_1\right)$ (see [23,24,25] for more detail). Moreover, $\mathrm{Q}$ is a slant if and only if it satisfies the following:

$$\begin{array}{c}\hfill T^2=\nu (-I+\eta \otimes \xi ),\end{array}$$

(10)

for $\nu \in [0,1]$ and $\nu ={cos}^2\gamma$ [24].

(f)

If the tangent bundle $T\mathrm{Q}$ is satisfied $T\mathrm{Q}=\mathfrak{D}\oplus {\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle$ for invariant distribution $\mathfrak{D}$, anti-invariant ${\mathfrak{D}}^{\perp }$. Moreover, $\langle \xi \rangle$ is the 1-dimensional distribution spanned by the structure vector field $\xi$ [24].

Motivated by (10), we have some useful relations,

$$\begin{array}{cc}\hfill g(T{E}_1,T{E}_2)=& {cos}^2\gamma \left(g(E_1,E_2)-\eta \left(E_1\right)\eta \left(E_2\right)\right),\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill g(F{E}_1,F{E}_2)=& {sin}^2\gamma \left(g(E_1,E_2)-\eta \left(E_1\right)\eta \left(E_2\right)\right),\hfill \end{array}$$

(12)

for any $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right)$. In [10], the following results were proved:

$$\begin{array}{c}\hfill \left(a\right)tF{E}_1={sin}^2\gamma (-E_1+\eta \left(E_1\right)\xi ),\left(b\right)fF{E}_1=-FT{E}_1,\end{array}$$

(13)

for any $E_1\in \Gamma \left(T\mathrm{Q}\right)$. There are some other classifications as follows:

(i)

A submanifold $\mathrm{Q}$ stand for semi-slant submanifold if the tangent bundle $T\mathrm{Q}$ is satisfied $T\mathrm{Q}=\mathfrak{D}\oplus {\mathfrak{D}}^{\gamma }\oplus \langle \xi \rangle$ for $\phi$-invariant $\mathfrak{D}$ and proper slant ${\mathfrak{D}}^{\gamma }$ with slant angle $\gamma .$

(ii)

A submanifold $\mathrm{Q}$ stand for pseudo-slant submanifold if tangent bundle $\mathrm{Q}$ $T\mathrm{Q}$ is satisfied $T\mathrm{Q}={\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}^{\gamma }\oplus \langle \xi \rangle$ for anti-invariant ${\mathfrak{D}}^{\perp }$ and slant distribution ${\mathfrak{D}}^{\gamma }$ with angle $\gamma .$

Now, if tangent space $T\mathrm{Q}$ has orthonormal basis $\{{\alpha }_1,{\alpha }_2\dots {\alpha }_n\}$ and the normal bundle $T^{\perp }\mathrm{Q}$ has the orthonormal basis ${\alpha }_r\in \{{\alpha }_{n+1},{\alpha }_{n+2},\dots {\alpha }_m\}$, then we define

$$\begin{array}{c}\hfill {\mathfrak{B}}_{cd}^r=g(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\alpha }_r){,\left|\right|\mathfrak{B}\left|\right|}^2=\sum _{c,d=1}^{n}g(\mathfrak{B}({\alpha }_c,{\alpha }_d),\mathfrak{B}({\alpha }_c,{\alpha }_d)).\end{array}$$

(14)

Furthermore, consider the gradient $\nabla \psi$ of the function $\psi$ on $\mathrm{Q}$; then, for any $E_1\in \Gamma \left(T\mathrm{Q}\right)$, we have the following:

$$\begin{array}{c}\hfill {\left|\right|\nabla \psi \left|\right|}^2=\sum _{c=1}^n{\left({\alpha }_c\left(\psi \right)\right)}^2,g(\nabla \psi ,E_1)=E_1\left(\psi \right).\end{array}$$

(15)

A covariant derivative of $\phi$ is defined as

$$\begin{array}{c}\hfill \left({\tilde{\nabla }}_{E_1}\phi \right)E_2={\tilde{\nabla }}_{E_1}\phi E_2-\phi {\tilde{\nabla }}_{E_1}{E}_2\end{array}$$

(16)

for any $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right).$

3. Contact Skew CR-Submanifolds

As an extension of CR-submanifolds and slant submanifolds of almost Hermitian manifolds, G. S. Ronsse (cf. [1]) established the idea of skew CR-submanifolds of Kaehler manifolds. To generalize the above classes of submanifolds, Tripathi [2] extended this study to contact skew CR-submanifolds by engaging the property of $T^2=\mathcal{Q}$ on the submanifold for a bounded linear operator T that is symmetric. It is obvious from (3) and (8) that

$$\begin{array}{c}\hfill g(T{E}_1,E_2)=-g(E_1,T{E}_2)\end{array}$$

for any $E_1,E_2\in \Gamma \left(T\mathrm{Q}\right).$ This shows that $\mathcal{Q}$ is a symmetric operator, i.e.,

$$\begin{array}{c}\hfill g(\mathcal{Q}{E}_1,E_2)=g(E_1,\mathcal{Q}{E}_2).\end{array}$$

Its eigenvalues are real and diagonalizable. Its eigenvalues are also bounded by 0 and $-1$. For every point $p\in Q$, we can write

$$\begin{array}{c}\hfill {\mathfrak{D}}_p^{\Lambda }=Ker{\{T^2+{\Lambda }^2\left(p\right)I\}}_p\end{array}$$

where $-{\Lambda }^2\left(p\right)$ is an eigenvalue of $T^2$ such that $\Lambda \left(p\right)\in [0,1]$ and I is an identity operator. It is reminded that ${\mathfrak{D}}_p^0=KerT$ and ${\mathfrak{D}}_p^1=KerF$, where ${\mathfrak{D}}_p^0$ and ${\mathfrak{D}}_p^1$ are the maximal anti-invariant and invariant subspaces under $\phi$ of tangent space a $T_p\mathrm{Q}$, respectively. We now use $\mathfrak{D}\oplus <\xi >$ and ${\mathfrak{D}}^{\perp }$ to represent the distributions ${\mathfrak{D}}_p^1$ and ${\mathfrak{D}}_p^0$, respectively. If $-{\Lambda }_1^2\left(p\right),\dots -{\Lambda }_k^2\left(p\right)$ are eigenvalues of symmetric and diagonalizable operator $T^2$ at $p\in \mathrm{Q}$. Then a T-invariant subspace of $T_p\mathrm{Q}$ if $T_p\mathrm{Q}$ can be decomposed as a direct sum of mutually orthogonal eigenspaces, i.e.,

$$\begin{array}{c}\hfill T_p\mathrm{Q}={\mathfrak{D}}_p^{{\Lambda }_1}\oplus \dots \oplus {\mathfrak{D}}_p^{{\Lambda }_k},\end{array}$$

for each ${\mathfrak{D}}_p^{{\Lambda }_c}$, $1\le c\le k$. Furthermore, if ${\Lambda }_c\ne 0$, then we find that ${\mathfrak{D}}_p^{{\Lambda }_c}$ is even dimensional. If there exist the functions ${\Lambda }_c$ defined $\mathrm{Q}$ for each $1\le c\le k$ and k is an integer with eigenvalues 0 and 1 such that

$$\begin{array}{c}\hfill T_p\mathrm{Q}=\mathfrak{D}\oplus {\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}_p^{{\Lambda }_1}\oplus \dots \oplus {\mathfrak{D}}_p^{{\Lambda }_k}\oplus <\xi {>}_p\end{array}$$

with each $-{\lambda }_c^2\left(p\right)$ are distinct eigenvalues of $T^2$ for the point $p\in \mathrm{Q}$ and the dimensions of ${\mathfrak{D}}_p$, ${\mathfrak{D}}_p^{\perp }$ and ${\mathfrak{D}}^{{\lambda }_c}$ are independent of $p\in \mathrm{Q}$. Then, $\mathrm{Q}$ is called the generic submanifold of $\tilde{\mathrm{Q}}$. Moreover, $\mathrm{Q}$ is called a skew CR-submanifold if each ${\lambda }_c$ is constant on $\mathrm{Q}$. Special cases of a generic submanifold include invariant, anti-invariant, and slant submanifolds.

Definition 1 

([15]). A submanifold $\mathrm{Q}$ stands for a contact skew CR-submanifold in $\tilde{\mathrm{Q}}$ if the tangent bundle $T\mathrm{Q}$ is satisfied $T\mathrm{Q}=\mathfrak{D}\oplus {\mathfrak{D}}^{\perp }\oplus {\mathfrak{D}}^{\gamma }\oplus <\xi >$ that include invariant, anti-invariant and slant distributions $\mathfrak{D}$, ${\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\gamma }$, respectively. Moreover, $<\xi >$ is a 1-dimensional distribution spanned by ξ.

Motivated by the above definition the normal bundle $T^{\perp }\mathrm{Q}$ is satisfied $T^{\perp }\mathrm{Q}=\phi {\mathfrak{D}}^{\perp }\oplus F{\mathfrak{D}}^{\gamma }\oplus \nu$ for an invariant subspace $\nu$ under $\phi .$ Assuming that $d_1$, $d_2$ and $d_3$ are the dimensions of the distributions $\mathfrak{D}$, ${\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\gamma }$, respectively. Consequently, $\mathrm{Q}$ is invariant if $d_2=0$ and $d_3=0$. $\mathrm{Q}$ is an anti-invariant submanifold of $\tilde{\mathrm{Q}}$ if $d_1=0$, and $d_3=0$. If $d_1=0$ and $d_2=0$, then $\mathrm{Q}$ is a proper slant submanifold. If neither of $d_i=0,i=1,2,3$ is zero then $\mathrm{Q}$ proper contact skew CR-submanifold.

Now we obtain some initial results.

Proposition 1. 

Let $\mathrm{Q}$ be a proper contact skew CR-submanifold of order one in a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$. Then, we have

$$\begin{array}{c}\hfill g({\nabla }_{E_1}{E}_2,V_1)={csc}^2\gamma \left(g(\mathfrak{B}(E_1,\phi E_2),F{V}_1)-g(\mathfrak{B}(E_1,E_2),FT{V}_1)\right)\end{array}$$

(17)

$$\begin{array}{c}\hfill g({\nabla }_{G_1}{G}_2,V_1)={sec}^2\gamma \left(g(\mathfrak{B}(G_1,G_2),FT{V}_1)-g(\mathfrak{B}(G_1,T{V}_1),\phi G_2)\right),\end{array}$$

(18)

for any $E_1,E_2\in \Gamma \left(\mathfrak{D}\right)$, $G_1,G_2\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \xi )$ and $V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$.

Proof. 

On using (5) (i) and (8) (ii), we have

$$\begin{array}{c}\hfill g({\nabla }_{E_1}{E}_2,V_1)=g(\phi {\tilde{\nabla }}_{E_1}{E}_2,\phi V_1)=g(\phi {\tilde{\nabla }}_{E_1}{E}_2,T{V}_1)+g(\phi {\tilde{\nabla }}_{E_1}{E}_2,F{V}_1),\end{array}$$

for $E_1,E_2\in \Gamma \left(\mathfrak{D}\right)$ and $V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$. Using (3) and (16), we derive

$$\begin{array}{c}\hfill g({\nabla }_{E_1}{E}_2,V_1)=-g({\tilde{\nabla }}_{E_1}{E}_2,T^2{V}_1)-g({\tilde{\nabla }}_{E_1}{E}_2,FT{V}_1)+g({\tilde{\nabla }}_{E_1}\phi E_2,F{V}_1).\end{array}$$

Thus, (10) and relation (5) (ii) give (17). For next result

$$\begin{array}{c}\hfill g({\nabla }_{G_1}{G}_2,V_1)=g(\phi {\tilde{\nabla }}_{G_1}{G}_2,\phi V_1)=g(\phi {\tilde{\nabla }}_{G_1}{G}_2,T{V}_1)-g({\tilde{\nabla }}_{G_1}{G}_2,\phi F{V}_1),\end{array}$$

for any $G_1,G_2\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \xi )$. Implementation of (16) and (8) (ii), we obtain

$$\begin{array}{cc}\hfill g({\nabla }_{G_1}{G}_2,V_1)=& g({\tilde{\nabla }}_{G_1}\phi G_2,T{V}_1)-g(\left({\tilde{\nabla }}_{G_1}\phi \right)G_2,T{V}_1)-g({\tilde{\nabla }}_{G_1}{G}_2,tF{V}_1)\hfill \\ \hfill & -g({\tilde{\nabla }}_{G_1}{G}_2,fF{V}_1).\hfill \end{array}$$

Utilizing (3), (5) (ii), (8) (ii) and (13), we get

$$\begin{array}{c}\hfill {cos}^2\gamma g({\nabla }_{G_1}{G}_2,V_1)=g(\mathfrak{B}(G_1,G_2),FT{V}_1)-g(\mathfrak{B}(G_1,T{V}_1),\phi G_2),\end{array}$$

which gives our assertion. Hence, the proof is complete. □

Proposition 2. 

On a contact skew CR-submanifold $\mathrm{Q}$ of a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$, we have

$$\begin{array}{c}\hfill g({\nabla }_{G_1}{E}_1,V_1)={csc}^2\gamma \left(g(\mathfrak{B}(G_1,\phi E_1),F{V}_1)-g(\mathfrak{B}(E_1,G_1),FT{V}_1)\right),\end{array}$$

(19)

$$\begin{array}{c}\hfill g({\nabla }_{E_1}{G}_1,V_1)={sec}^2\gamma \left(g(\mathfrak{B}(E_1,G_1),FT{V}_1)-g(\mathfrak{B}(E_1,T{V}_1),\phi G_1)\right),\end{array}$$

(20)

for any $E_1\in \Gamma \left(\mathfrak{D}\right)$, $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \xi )$ and $V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$.

Proof. 

As ambient space $\tilde{\mathrm{Q}}$ is a locally conformal almost cosymplectic manifold and $\mathrm{Q}$ is a contact skew CR-submanifold, then taking account of (10), we obtain

$$\begin{array}{cc}\hfill g({\nabla }_{G_1}{E}_1,V_1)=& g({\tilde{\nabla }}_{G_1}\phi E_1,F{V}_1)-g({\tilde{\nabla }}_{G_1}{E}_1,T^2{V}_1)-g({\tilde{\nabla }}_{G_1}{E}_1,FT{V}_1)\hfill \\ \hfill =& {cos}^2\gamma g({\nabla }_{G_1}{E}_1,V_1)+g(\mathfrak{B}(G_1,\phi E_1),F{V}_1)-g(\mathfrak{B}(G_1,E_1),FT{V}_1).\hfill \end{array}$$

Thus, the first part follows from the above relation. The second part can be followed similarly to the first. □

4. Contact Skew CR-Warped Product Submanifolds

The general warped product of type $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_2$ is defined in [5] such that f be a warping function defined on base ${\mathrm{Q}}_1$ that is a positive differentiable function. The following formula is defined for vector fields $V_1$ and $G_1$ on ${\mathrm{Q}}_1$ and ${\mathrm{Q}}_2,$ respectively, in Lemma 7.3 [5]

$$\begin{array}{c}\hfill {\nabla }_{V_1}{G}_1={\nabla }_{G_1}{V}_1=V_1(lnf)G_1,\end{array}$$

(21)

where ∇ denotes the Levi–Civita connection on base $\mathrm{Q}$. It is defined that the base ${\mathrm{Q}}_1$ is totally geodesic in $\mathrm{Q}$, and the fiber ${\mathrm{Q}}_2$ is totally umbilical $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_2$. Motivated by these concepts, skew CR-warped product submanifolds in the Kaehler manifold are discussed [13] and the contact version is classified in [15]. In our study, we consider the contact skew CR-warped product $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ in a locally conformal almost cosymplectic manifold $\tilde{Q}$ that includes invariant, anti-invariant, and slant submanifolds $\mathfrak{D}$, ${\mathrm{Q}}_{\perp }$, and ${\mathrm{Q}}^{\gamma }$, respectively, such that ${\mathrm{Q}}_1={\mathrm{Q}}_T\times {\mathrm{Q}}_{\perp }$ is a CR-product. For simplicity throughout this section, the tangent spaces of ${\mathrm{Q}}_T,{\mathrm{Q}}_{\perp }$ and ${\mathrm{Q}}_{\gamma }$ are represented by $\mathfrak{D},{\mathfrak{D}}^{\perp }$ and ${\mathfrak{D}}^{\gamma }$, respectively. Also, ${\mathfrak{D}}_1$ is the tangent space of ${\mathrm{Q}}_1$.

It is clear that when $\xi$ is tangent to ${\mathrm{Q}}_1$ due to the non-trivial contact skew CR-warped product, then we have

$$\begin{array}{c}\hfill {\nabla }_{V_1}\xi =\lambda V_1,\end{array}$$

for any $V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$. Using (21) and then taking the inner product with $V_1$, we get

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

For non-trivial contact skew CR-warped product in $\tilde{\mathrm{Q}}$, we have a lemma;

Lemma 1. 

On a contact skew CR-warped product submanifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ of locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$, the following relation holds

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,V_1),F{V}_2)=-\phi E_1(lnf)g(V_1,V_2)+\left(U(lnf)-\lambda \eta \left(E_1\right)\right)g(T{V}_1,V_2),\end{array}$$

(22)

for any $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$ and $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ such that $\xi \in {\mathrm{Q}}_1={\mathrm{Q}}_T\times {\mathrm{Q}}_{\perp }$.

Proof. 

Based on (3), (5) (i) and (8) (i), we have

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,V_1),F{V}_2)=g({\tilde{\nabla }}_{V_1}{E}_1,\phi V_2)-g({\tilde{\nabla }}_{V_1}{E}_1,T{V}_2),\end{array}$$

for any $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$ and $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$. Then, from (2), (16), and (21), we obtain

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,V_1),F{V}_2)=g(\left({\tilde{\nabla }}_{V_1}\phi \right)E_1,V_2)-g({\tilde{\nabla }}_{V_1}\phi E_1,V_2)-E_1(lnf)g(V_1,T{V}_2).\end{array}$$

Again using (3) and (21), we get the desired result, which completely proves the lemma. □

Interchanging $V_1$ by $T{V}_1$ in (22), we get

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,T{V}_1),F{V}_2)=-\phi E_1(lnf)g(T{V}_1,V_2)-{cos}^2\gamma \left(E_1(lnf)-\lambda \eta \left(E_1\right)\right)g(V_1,V_2).\end{array}$$

(23)

Interchanging $E_1$ by $\phi E_1,$ for any $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$ in (22), we derive

$$\begin{array}{c}\hfill g(\mathfrak{B}(\phi E_1,V_1),F{V}_2)=\phi E_1(lnf)g(T{V}_1,V_2)+\left(E_1(lnf)-\lambda \eta \left(E_1\right)\right)g(V_1,V_2).\end{array}$$

(24)

Also, if we interchange $V_2$ by $T{V}_2$ in (22), then we obtain

$$\begin{array}{c}\hfill g(\mathfrak{B}(E_1,V_1),FT{V}_2)=\phi E_1(lnf)g(T{V}_1,V_2)+{cos}^2\gamma \left(E_1(lnf)-\lambda \eta \left(E_1\right)\right)g(V_1,V_2).\end{array}$$

(25)

From (23), (24) and (25), we deduce that

$$\begin{array}{c}\hfill g(A_{F{V}_2}\phi E_1-A_{FT{V}_2}{E}_1,V_1)={sin}^2\gamma \left(E_1(lnf)-\lambda \eta \left(E_1\right)\right)g(V_1,V_2).\end{array}$$

(26)

Lemma 2. 

On a contact skew CR-warped product submanifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ of locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$, then we have

$$\begin{array}{cc}\hfill g(\mathfrak{B}(V_1,T{V}_2),\phi G_1)& -g(\mathfrak{B}(V_1,G_1),FT{V}_2)\hfill \\ \hfill & ={cos}^2\gamma \left(\lambda \eta \left(G_1\right)-G_1(lnf)\right)g(V_1,V_2),\hfill \end{array}$$

(27)

for any $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$ and $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$.

Proof. 

As result $T{V}_2$ and $\phi G_1$ are normal, we have

$$\begin{array}{c}\hfill g(\mathfrak{B}(V_1,T{V}_2),\phi G_1)=g({\tilde{\nabla }}_{V_1}T{V}_2,\phi G_1)=-g({\tilde{\nabla }}_{V_1}\phi G_1,T{V}_2),\end{array}$$

for $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus <\xi >)$. Thus (16) gives

$$\begin{array}{c}\hfill g(\mathfrak{B}(V_1,T{V}_2),\phi G_1)=g({\tilde{\nabla }}_{V_1}{G}_1,T^2{V}_2)+g({\tilde{\nabla }}_{V_1}{G}_1,FT{V}_2)-g(\left({\tilde{\nabla }}_{V_1}\phi \right)G_1,T{V}_2).\end{array}$$

Then, using (3), (5) (i), and (10), we derive

$$\begin{array}{cc}\hfill g(\mathfrak{B}(V_1,T{V}_2),\phi G_1)=& -{cos}^2\gamma g({\nabla }_{V_1}{G}_1,V_2)+g(\mathfrak{B}(V_1,G_1),FT{V}_2)\hfill \\ \hfill & +\lambda \eta \left(G_1\right)g(T{V}_1,T{V}_2).\hfill \end{array}$$

Therefore, applying (11) and (21) in the preceding equation gives the required result. □

Now, we give our first main result as follows:

Theorem 1. 

Every proper contact skew CR-submanifold $\mathrm{Q}$ of a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$ with integrable slant distribution ${\mathfrak{D}}^{\gamma },$ is a locally contact skew CR-warped product submanifold if and only if

(i)

$A_{F{V}_1}\phi E_1-A_{FT{V}_1}{E}_1={sin}^2\gamma \left(E_1\left(\mu \right)-\lambda \eta \left(E_1\right)\right)V_1,$

for some smooth functions μ on $\mathrm{Q}$ such that $V_2\left(\mu \right)=0$, for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$.

(ii)

$A_{\phi G_1}T{V}_1-A_{FT{V}_1}{G}_1={cos}^2\gamma \left(\lambda \eta \left(G_1\right)-G_1\left(\mu \right)\right)V_1,$

for some smooth functions μ on $\mathrm{Q}$ such that $V_2\left(\mu \right)=0$, for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$.

Proof. 

The first part of the theorem can easily get from (26) and (27) due to $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ being a contact skew CR-warped product submanifold of a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$. Conversely, assume that $\mathrm{Q}$ is a proper contact skew CR-submanifold of $\tilde{\mathrm{Q}}$. Then Proposition 1, gives

$$\begin{array}{cc}\hfill & g({\nabla }_{E_2}{E}_1,V_1)={csc}^2\gamma g(A_{F{V}_1}\phi E_2-A_{FT{V}_1}{E}_2,E_1),\hfill \\ \hfill & g({\nabla }_{G_1}{G}_2,V_1)=-{sec}^2\gamma g(A_{\phi G_2}T{V}_1-A_{FT{V}_1}{G}_2,G_1).\hfill \end{array}$$

As a consequence of the theorem’s hypothesis, i.e., the conditions (i) and (ii) and the orthogonality of vector fields, we deduce that $g({\nabla }_{E_1}{E}_2,V_1)=0$ and $g({\nabla }_{G_1}{G}_2,V_1)=0$, which mean that ${\nabla }_{E_1}{E}_2$ and ${\nabla }_{G_1}{G}_2$ lie in ${\mathfrak{D}}_1$ for any $E_1,E_2\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$ and $G_1,G_2\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$. Notice that ${\mathfrak{D}}_1=\mathfrak{D}\oplus {\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle$. In a similar way from Proposition 2 and the given conditions (i) and (ii), we get ${\nabla }_{E_1}{G}_1$ and ${\nabla }_{G_1}{E}_1$ also belong to ${\mathfrak{D}}_1$. Therefore, we conclude that the leaves of ${\mathfrak{D}}_1$ are totally geodesic in $\mathrm{Q}$ which is a leaf of ${\mathrm{Q}}_1={\mathrm{Q}}_T\times {\mathrm{Q}}_{\perp }$. If ${\mathfrak{B}}^{\gamma }$ is the second fundamental form of ${\mathrm{Q}}_{\gamma }$ into $\mathrm{Q}$ and ${\mathrm{Q}}_{\gamma }$ is a leaf of ${\mathfrak{D}}^{\gamma }$ due to ${\mathfrak{D}}^{\gamma }$ being integrable, then from (5) (i) and (3), we derive

$$\begin{array}{c}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E)=g({\nabla }_{V_1}{V}_2,E)=g({\tilde{\nabla }}_{V_1}{V}_2,E),\end{array}$$

(28)

for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $E=(E_1,G_1)\in \Gamma \left({\mathfrak{D}}_1\right)$ such that $E_1\in \Gamma \left(\mathfrak{D}\right)$ and $G_1\in \Gamma \left({\mathfrak{D}}^{\perp }\right)$. Now, we calculate $g({\mathfrak{B}}^{\gamma }(V_1,V_2),E)$ separately as follows:

$$\begin{array}{c}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E_1)=g({\tilde{\nabla }}_{V_1}{V}_2,E_1)=-g({\tilde{\nabla }}_{V_1}{E}_1,V_2)=-g(\phi {\tilde{\nabla }}_{V_1}{E}_1,\phi V_2)\end{array}$$

for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $\xi$ along to $(\mathfrak{D}$ i.e., $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle ).$ Using (16), (8) (i) and (3), we are able to derive

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E_1)=& -g({\tilde{\nabla }}_{V_1}\phi E_1,T{V}_2)+g(\left({\tilde{\nabla }}_{V_1}\phi \right)E_1,T{V}_2)\hfill \\ \hfill & -g({\tilde{\nabla }}_{V_1}\phi E_1,F{V}_2)+g(\left({\tilde{\nabla }}_{V_1}\phi \right)E_1,F{V}_2)\hfill \\ \hfill =& -g(\phi {\tilde{\nabla }}_X\phi E_1,\phi T{V}_2)-\lambda \eta \left(E_1\right)g(T{V}_1,T{V}_2)\hfill \\ \hfill & -g(\mathfrak{B}(V_1,\phi E_1),F{V}_2)-\lambda \eta \left(E_1\right)g(F{V}_1,F{V}_2)\hfill \\ \hfill =& -g({\tilde{\nabla }}_{V_1}{\phi }^2{E}_1,T^2{V}_2)-\lambda {cos}^2\gamma \eta \left(E_1\right)g(V_1,V_2)\hfill \\ \hfill & -g(A_{F{V}_2}\phi E_1,V_1)-g({\tilde{\nabla }}_X{\phi }^2{E}_1,FT{V}_2)\hfill \\ \hfill & -\lambda {sin}^2\gamma \eta \left(E_1\right)g(V_1,V_2).\hfill \end{array}$$

Using (1) and (10), the following equation is derived

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E_1)=& -{cos}^2\gamma g({\tilde{\nabla }}_{V_1}{E}_1,V_2)+{cos}^2\gamma \eta \left(E_1\right)g({\tilde{\nabla }}_{V_1}\xi ,V_2)\hfill \\ \hfill & -\lambda \eta \left(E_1\right)g(V_1,V_2)-g(A_{F{V}_2}\phi E_1,V_1)\hfill \\ \hfill & +g({\tilde{\nabla }}_{V_1}{E}_1,FT{V}_2)-\eta \left(E_1\right)g({\tilde{\nabla }}_{V_1}\xi ,FT{V}_2).\hfill \end{array}$$

In view of (5) (i) and (4), the following result can be drawn:

$$\begin{array}{c}\hfill {sin}^2\gamma g({\mathfrak{B}}^{\gamma }(V_1,V_2),E_1)=-\lambda {sin}^2\gamma \eta \left(E_1\right)g(V_1,V_2)-g(A_{F{V}_2}\phi E_1-A_{FT{V}_2}{E}_1,V_1).\end{array}$$

Applying the first assumption of the theorem on the right-hand side of the preceding equation,

$$\begin{array}{c}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E_1)=-E_1\left(\mu \right)g(V_1,V_2)=-g(V_1,V_2)g({\nabla }^{\lambda }\left(\mu \right),E_1),\end{array}$$

(29)

where ${\nabla }^{\lambda }\left(\mu \right)$ is the gradient of $\mu$ along ${\mathrm{Q}}_T$. Similarly, if $\xi$ is along to ${\mathfrak{D}}^{\perp }$, then we have

$$\begin{array}{c}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),G_1)=g({\tilde{\nabla }}_{V_1}{V}_2,G_1)=g(\phi {\tilde{\nabla }}_{V_1}{V}_2,\phi G_1)+\lambda \eta \left(G_1\right)g({\tilde{\nabla }}_{V_1}{V}_2,\xi )\end{array}$$

for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$. It is simplified by using (16) as follows:

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),G_1)& =g({\tilde{\nabla }}_{V_1}\phi V_2,\phi G_1)-g(\left({\tilde{\nabla }}_{V_1}\phi \right)V_2,\phi G_1)-\lambda \eta \left(G_1\right)g(V_1,V_2)\hfill \\ \hfill & =g({\tilde{\nabla }}_{V_1}T{V}_2,\phi G_1)+g({\tilde{\nabla }}_{V_1}F{V}_2,\phi G_1)-\lambda \eta \left(G_1\right)g(V_1,V_2)\hfill \\ \hfill & =g(\mathfrak{B}(V_1,T{V}_2),\phi G_1)-g({\tilde{\nabla }}_{V_1}tF{V}_2,G_1)-g({\tilde{\nabla }}_{V_1}fF{V}_2,G_1)\hfill \\ \hfill & +g(\left({\tilde{\nabla }}_{V_1}\phi \right)F{V}_2,G_1)-\lambda \eta \left(G_1\right)g(V_1,V_2).\hfill \end{array}$$

According to (3) and (13), the following conclusions are reached:

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),G_1)& =g(A_{\phi G_1}T{V}_2,V_1)+{sin}^2\gamma g({\tilde{\nabla }}_{V_1}{V}_2,G_1)+g({\tilde{\nabla }}_{V_1}FT{V}_2,G_1)\hfill \\ \hfill & +\lambda \eta \left(G_1\right)g(F{V}_1,F{V}_2)-\lambda \eta \left(G_1\right)g(V_1,V_2).\hfill \end{array}$$

Then, from (5) (ii) and (12), we get

$$\begin{array}{cc}\hfill {cos}^2\gamma g({\mathfrak{B}}^{\gamma }(V_1,V_2),G_1)& =g(A_{\phi G_1}T{V}_2-A_{FTY}{G}_1,V_1)\hfill \\ \hfill & +{sin}^2\gamma \eta \left(G_1\right)g(V_1,V_2)-\lambda \eta \left(G_1\right)g(V_1,V_2).\hfill \end{array}$$

The implementation of the second assumption condition (ii) of the theorem, we arrive at

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),G_1)& =-G_1\left(\mu \right)g(V_1,V_2)=-g(V_1,V_2)g({\overrightarrow{\nabla }}^{\perp }\left(\mu \right),G_1)\hfill \end{array}$$

(30)

where ${\overrightarrow{\nabla }}^{\perp }\left(\mu \right)$ is the gradient of $\mu$ along ${\mathrm{Q}}_{\perp }$. Then from (28), (29) and (30), we conclude that

$$\begin{array}{cc}\hfill g({\mathfrak{B}}^{\gamma }(V_1,V_2),E)& =-g(V_1,V_2)g({\nabla }^T\left(\mu \right)+{\overrightarrow{\nabla }}^{\perp }\left(\mu \right),E),\hfill \end{array}$$

which is equivalent to

$$\begin{array}{c}\hfill {\mathfrak{B}}^{\gamma }(V_1,V_2)=-\left({\nabla }^T\left(\mu \right)+{\overrightarrow{\nabla }}^{\perp }\left(\mu \right)\right)g(V_1,V_2).\end{array}$$

(31)

Consequently, ${\mathrm{Q}}_{\gamma }$ is a totally umbilical submanifold of $\mathrm{Q}$ such that the mean curvature vector $H^{\gamma }=\left({\nabla }^T\left(\mu \right)+{\overrightarrow{\nabla }}^{\perp }\left(\mu \right)\right)$. Furthermore, for each $V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ the hypothesis $V_2\mu =0$, shows that the leaves of ${\mathfrak{D}}^{\gamma }$ are extrinsic spheres in $\mathrm{Q}$. It is sufficient to involve the Hiepko’s Theorem [26], $\mathrm{Q}$ has become a locally warped product manifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$, where f is the warping function. This proves the theorem completely. □

5. Inequalities for Contact Skew CR-Warped Product Submanifolds

This section provides an inequality between the length of the second fundamental form ${\parallel \mathfrak{B}\parallel }^2$ and the length of the warping function ${\parallel \nabla lnf\parallel }^2$ for a contact skew CR-warped product submanifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ in $\tilde{\mathrm{Q}}$ that include the slant angle also. In order to be able to use them later, we first provide the following lemmas.

Lemma 3. 

On a contact skew CR-warped product submanifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ in a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$, we have

(i)

$g(\mathfrak{B}(V_1,E_1),\phi G_1)=0$;

(ii)

$g(\mathfrak{B}(E_1,G_1),F{V}_1)=0$;

• for any $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle ),V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$.

Proof. 

For any $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle ),V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$, we have

$$\begin{array}{c}\hfill g(\mathfrak{B}(V_1,E_1),\phi G_1)=g({\tilde{\nabla }}_{V_1}{E}_1,\phi G_1)=-g({\tilde{\nabla }}_{V_1}\phi E_1,G_1)+g(\left({\tilde{\nabla }}_{V_1}\phi \right)E_1,G_1).\end{array}$$

From (3), (5) (i), and (21), we get

$$\begin{array}{c}\hfill g(\mathfrak{B}(V_1,E_1),\phi G_1)=-\phi E_1(lnf)g(V_1,G_1)=0,\end{array}$$

which is (i). For (ii), we have

$$\begin{array}{cc}\hfill g(\mathfrak{B}(E_1,G_1),F{V}_1)& =g({\tilde{\nabla }}_{G_1}{E}_1,\phi V_1)-g({\tilde{\nabla }}_{G_1}{E}_1,T{V}_1)\hfill \\ \hfill & =-g({\tilde{\nabla }}_{G_1}\phi E_1,V_1)+g(\left({\tilde{\nabla }}_{G_1}\phi \right)E_1,V_1)+G_1(lnf)g(T{V}_1,E_1)\hfill \\ \hfill & =g(\phi E_1,{\tilde{\nabla }}_{G_1}{V}_1).\hfill \end{array}$$

Using (5) (i) and (21), we get the second relation. Hence, the proof is complete. □

Lemma 4. 

Let $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ be a contact skew CR-warped product submanifold of order 1 in a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$. Then, we have

(i)

$g(\mathfrak{B}(E_1,E_2),F{V}_1)=0$,

(ii)

$g(\mathfrak{B}(G_1,G_2),F{V}_1)=g(\mathfrak{B}(V_1,G_1),\phi G_2)$,

• for any $E_1,E_2\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle ),V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1,G_2\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$.

Proof. 

For any $E_1,E_2\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle ),V_1\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$, we have

$$\begin{array}{cc}\hfill g(\mathfrak{B}(E_1,E_2),F{V}_1)& =g({\tilde{\nabla }}_{E_1}{E}_2,\phi V_1)+g(E_2,{\tilde{\nabla }}_{E_1}T{V}_1)\hfill \\ \hfill & =-g({\tilde{\nabla }}_{E_1}\phi E_2,V_1)+g(\left({\tilde{\nabla }}_{E_1}\phi \right)E_2,V_1)\hfill \\ \hfill & =g(\phi E_2,{\tilde{\nabla }}_{E_1}{V}_1).\hfill \end{array}$$

The first result is obtained by applying (21) with the orthogonality of the vector fields. Similarly, statement (ii) can also be established. □

First, we give the following frame field to establish our second main result.

Let $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ be a $m={\mathrm{Q}}_1+{\mathrm{Q}}_2+{\mathrm{Q}}_3$-dimensional contact skew CR-warped product submanifold of a $2n+1$-dimensional locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$ such that $\xi$ is tangent to ${\mathfrak{D}}^{\perp }$ and $dim\mathfrak{D}={\mathrm{Q}}_1=q_1=2s$, $dim{\mathfrak{D}}^{\perp }={\mathrm{Q}}_2=q_2=a_1+1$ and $dim{\mathfrak{D}}^{\gamma }=q_3={\mathrm{Q}}_2=2\beta$. Then, we consider that $\{{\alpha }_1,{\alpha }_2,\cdots {\alpha }_{a_1},{\alpha }_{a_1+1}=\xi ,\}$, $\{{\alpha }_{a_1+2}={\alpha }_1^{*},\cdots {\alpha }_{a_1+\beta +1}={\alpha }_{\beta }^{*},{\alpha }_{a_1+\beta +2}={\alpha }_{\beta +1}^{*}=sec\gamma T{\alpha }_1^{*},\cdots {\alpha }_{a_1+1+2\beta }={\alpha }_{2\beta }^{*}=sec\gamma T{\alpha }_{\beta }^{*}\}$ and $\{{\alpha }_{a_1+2\beta +2}={\tilde{\alpha }}_1,\cdots {\alpha }_{2\beta +a_1+s+1}={\tilde{\alpha }}_s,{\alpha }_{2\beta +a_1+s+2}={\tilde{\alpha }}_{s+1}=\phi {\tilde{\alpha }}_1,\cdots {\alpha }_m={\tilde{\alpha }}_{2s}=\phi {\tilde{\alpha }}_s\}$ are orthonormal frame fields of ${\mathfrak{D}}^{\perp }$, ${\mathfrak{D}}^{\gamma }$ and $\mathfrak{D}$ respectively. Thus, the frame fields of normal subbundles $\phi {\mathfrak{D}}^{\perp }$, $F{\mathfrak{D}}^{\gamma }$ and $\nu$, respectively are $\{{\alpha }_{m+1}={\overline{\alpha }}_1=\phi {\alpha }_1,\cdots {\alpha }_{m+a_1}={\overline{\alpha }}_{a_1}=\phi {\alpha }_{a_1}\}$, $\{{\alpha }_{m+a_1+1}={\overline{\alpha }}_{a_1+1}={\widehat{\alpha }}_1=csc\gamma F{\alpha }_1^{*},\cdots {\alpha }_{m+a_1+\beta }={\overline{\alpha }}_{a_1+\beta }={\widehat{\alpha }}_{\beta }=csc\gamma F{\alpha }_{\beta }^{*},{\alpha }_{m+a_1+\beta +1}={\overline{\alpha }}_{a_1+\beta +1}={\widehat{\alpha }}_{\beta +1}=csc\gamma sec\gamma FT{\alpha }_1^{*},\cdots {\alpha }_{m+a_1+2\beta }={\overline{\alpha }}_{a_1+2\beta }={\widehat{\alpha }}_{2\beta }=csc\gamma sec\gamma FT{\alpha }_{\beta }^{*}\}$ and $\{{\alpha }_{m+a_1+2\beta +1}={\overline{\alpha }}_{a_1+2\beta +1},\cdots {\alpha }_{2n+1}={\overline{\alpha }}_{2n+1-m}\}$.

Theorem 2. 

Let ξ be tangent to ${\mathrm{Q}}_1={\mathrm{Q}}_T\times {\mathrm{Q}}_{\perp }$ and a contact skew CR-warped product submanifold $\mathrm{Q}={\mathrm{Q}}_1{\times }_f{\mathrm{Q}}_{\gamma }$ be ${\mathfrak{D}}^{\perp }-{\mathfrak{D}}^{\gamma }$ mixed totally geodesic in a locally conformal almost cosymplectic manifold ${\tilde{\mathrm{Q}}}^{2n+1}$. Then

(i)

If $\xi \in {\mathrm{Q}}_{\perp }$ and ${\mathfrak{D}}^{\gamma }$ is the proper slant, the following inequality holds for the length ${\parallel \mathfrak{B}\parallel }^2$ of $\mathrm{Q}$

$$\begin{array}{c}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge 2\beta \left\{{cos}^2\gamma \left(\left|\right|{\overrightarrow{\nabla }}^{\perp }(lnf){\left|\right|}^2-{\lambda }^2\right)+2\left({csc}^2\gamma +{cot}^2\gamma \right)\left|\right|{\nabla }^T(lnf){\left|\right|}^2\right\}\end{array}$$

(32)

where $\mathfrak{B}$ is the second fundamental form of $\mathrm{Q}$.

(ii)

Similarly, if $\xi \in {\mathrm{Q}}_T$ and ${\mathfrak{D}}^{\gamma }$ is the proper slant, the following inequality holds for the length ${\parallel \mathfrak{B}\parallel }^2$ of $\mathrm{Q}$

$$\begin{array}{c}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge 2\beta \left\{{cos}^2\gamma \left|\right|{\overrightarrow{\nabla }}^{\perp }(lnf){\left|\right|}^2+2\left({csc}^2\gamma +{cot}^2\gamma \right)\left(\left|\right|{\nabla }^T(lnf){\left|\right|}^2-{\lambda }^2\right)\right\}\end{array}$$

(33)

where $dim{\mathfrak{D}}^{\gamma }=2\beta$ and ${\overrightarrow{\nabla }}^{\perp }(lnf)$ and ${\nabla }^T(lnf)$ are the gradient components along ${\mathrm{Q}}_{\perp }$ and ${\mathrm{Q}}_T$, respectively.

(iii)

If the equality holds in (32) and (33), then ${\mathrm{Q}}_1$ is totally geodesic in $\tilde{\mathrm{Q}}$ and ${\mathrm{Q}}_{\gamma }$ is a totally umbilical submanifold of $\tilde{\mathrm{Q}}$.

Proof. 

As a consequence of the definition of the second fundamental form gives

$$\begin{array}{cc}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2=& \sum _{c,d=1}^{m}g(\mathfrak{B}({\alpha }_c,{\alpha }_d),\mathfrak{B}({\alpha }_c,{\alpha }_d))=\sum _{r=m+1}^{2n+1}\sum _{c,d=1}^{m}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\alpha }_r)}^2.\hfill \end{array}$$

Decompose the above relation for the normal submandles of $\mathrm{Q}$, we rewrite

$$\begin{array}{cc}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2=& \sum _{r=m+1}^{m+a_1}\sum _{c,d=1}^{m}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\alpha }_r)}^2+\sum _{r=m+a_1+1}^{m+a_1+2\beta }\sum _{c,d=1}^{m}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\alpha }_r)}^2\hfill \\ \hfill & +\sum _{r=m+a_1+2\beta +1}^{2n+1}\sum _{c,d=1}^{m}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\alpha }_r)}^2.\hfill \end{array}$$

(34)

Since we could not find any relation for the $\nu$-components, then leaving the $\nu$-components terms. Then, we have

$$\begin{array}{cc}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge & \sum _{r=1}^{a_1}\sum _{c,d=1}^{a_1+1}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),\phi {\alpha }_r)}^2+\sum _{r=1}^{a_1}\sum _{c,d=1}^{2\beta }g{(\mathfrak{B}({\alpha }_c^{*},{\alpha }_d^{*}),\phi {\alpha }_r)}^2\hfill \\ \hfill & +\sum _{r=1}^{a_1}\sum _{c,d=1}^{2s}g{(\mathfrak{B}({\tilde{\alpha }}_c,{\tilde{\alpha }}_d),\phi {\alpha }_r)}^2+2\sum _{r=1}^{a_1}\sum _{i=1}^{a_1+1}\sum _{d=1}^{2\beta }g{(\mathfrak{B}({\alpha }_c,{\alpha }_d^{*}),\phi {\alpha }_r)}^2\hfill \\ \hfill & +2\sum _{r=1}^{a_1}\sum _{c=1}^{a_1+1}\sum _{d=1}^{2s}g{(\mathfrak{B}({\alpha }_c,{\tilde{\alpha }}_d),\phi {\alpha }_r)}^2+2\sum _{r=1}^{a_1}\sum _{c=1}^{2\beta }\sum _{d=1}^{2s}g{(\mathfrak{B}({\alpha }_c^{*},{\tilde{\alpha }}_d),\phi {\alpha }_r)}^2\hfill \\ \hfill & +\sum _{r=1}^{2\beta }\sum _{c,d=1}^{a_1+1}g{(\mathfrak{B}({\alpha }_c,{\alpha }_d),{\widehat{\alpha }}_r)}^2+\sum _{r=1}^{2\beta }\sum _{c,d=1}^{2\beta }g{(\mathfrak{B}({\alpha }_c^{*},{\alpha }_d^{*}),{\widehat{\alpha }}_r)}^2\hfill \\ \hfill & +\sum _{r=1}^{2\beta }\sum _{c,d=1}^{2s}g{(\mathfrak{B}({\tilde{\alpha }}_c,{\tilde{\alpha }}_d),{\widehat{\alpha }}_r)}^2+2\sum _{r=1}^{2\beta }\sum _{c=1}^{a_1+1}\sum _{d=1}^{2\beta }g{(\mathfrak{B}({\alpha }_c,{\alpha }_d^{*}),{\widehat{\alpha }}_r)}^2\hfill \\ \hfill & +2\sum _{r=1}^{2\beta }\sum _{c=1}^{a_1+1}\sum _{d=1}^{2s}g{(\mathfrak{B}({\alpha }_c,{\tilde{\alpha }}_d),{\widehat{\alpha }}_r)}^2+2\sum _{r=1}^{2\beta }\sum _{c=1}^{2\beta }\sum _{d=1}^{2s}g{(\mathfrak{B}({\alpha }_c^{*},{\tilde{\alpha }}_d),{\widehat{\alpha }}_r)}^2.\hfill \end{array}$$

(35)

We could not find any relation for the first, third, fifth, and eighth terms on the right-hand side of (35), therefore we shall leave these positive terms. Also, fourth, sixth, seventh, ninth, tenth, and eleventh terms vanish identically by using the ${\mathfrak{D}}^{\perp }-{\mathfrak{D}}^{\gamma }$ mixed totally geodesic condition, Lemma 3 and Lemma 4. Then, the second and twelfth terms can be expanded as

$$\begin{array}{cc}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2& \ge \sum _{r=1}^{a_1}\sum _{c,d=1}^{\beta }g{(\mathfrak{B}({\alpha }_c^{*},{\alpha }_d^{*}),\phi {\alpha }_r)}^2+{sec}^2\gamma \sum _{r=1}^{a_1}\sum _{c,d=1}^{\beta }g{(\mathfrak{B}({\alpha }_c^{*},T{\alpha }_d^{*}),\phi {\alpha }_r)}^2\hfill \\ \hfill & +{sec}^2\gamma \sum _{r=1}^{a_1}\sum _{c,d=1}^{\beta }g{(\mathfrak{B}(T{\alpha }_c^{*},{\alpha }_d^{*}),\phi {\alpha }_r)}^2+{sec}^4\gamma \sum _{r=1}^{a_1}\sum _{c,d=1}^{\beta }g{(\mathfrak{B}(T{\alpha }_c^{*},T{\alpha }_d^{*}),\phi {\alpha }_r)}^2\hfill \\ \hfill & +2{csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}({\alpha }_c^{*},{\tilde{\alpha }}_d),F{\alpha }_r^{*})}^2+2{sec}^2\gamma {csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}(T{\alpha }_c^{*},{\tilde{\alpha }}_d),F{\alpha }_r^{*})}^2\hfill \\ \hfill & +2{csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}({\alpha }_c^{*},\phi {\tilde{\alpha }}_d),F{\alpha }_r^{*})}^2+2{sec}^2\gamma {csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}(T{\alpha }_c^{*},\phi {\tilde{\alpha }}_d),F{\alpha }_r^{*})}^2\hfill \\ \hfill & +2{sec}^2\gamma {csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}({\alpha }_c^{*},{\tilde{\alpha }}_d),FT{\alpha }_r^{*})}^2+2{sec}^4{csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}(T{\alpha }_c^{*},{\tilde{\alpha }}_d),FT{\alpha }_r^{*})}^2\hfill \\ \hfill & +2{sec}^2\gamma {csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}({\alpha }_c^{*},\phi {\tilde{\alpha }}_d),FT{\alpha }_r^{*})}^2\hfill \\ \hfill & +2{sec}^4\gamma {csc}^2\gamma \sum _{r=1}^{\beta }\sum _{c=1}^{\beta }\sum _{d=1}^{s}g{(\mathfrak{B}(T{\alpha }_c^{*},\phi {\tilde{\alpha }}_d),FT{\alpha }_r^{*})}^2.\hfill \end{array}$$

(36)

Thus, (22) and (27), giving the following:

$$\begin{array}{cc}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge & 2\beta {cos}^2\gamma \sum _{r=1}^{a_1}{\left(\lambda \eta \left({\alpha }_r\right)-{\alpha }_r(lnf)\right)}^2+4\beta \left({csc}^2\gamma +{cot}^2\gamma \right)\sum _{d=1}^s{\left(\phi {\tilde{\alpha }}_d(lnf)\right)}^2\hfill \\ \hfill & +4\beta \left({csc}^2\gamma +{cot}^2\gamma \right)\sum _{d=1}^s{\left({\tilde{\alpha }}_d(lnf)\right)}^2\hfill \\ \hfill =& 2\beta {cos}^2\gamma \sum _{r=1}^{a_1+1}\left({\lambda }^2\eta {\left({\alpha }_r\right)}^2+{\left({\alpha }_r(lnf)\right)}^2-2\lambda \eta \left({\alpha }_r\right)\left({\alpha }_r(lnf)\right)\right)\hfill \\ \hfill & -2\beta {cos}^2\gamma {\left(\eta \left({\alpha }_{a_1+1}\right)-{\alpha }_{a_1+1}(lnf)\right)}^2\hfill \\ \hfill & +4\beta \left({csc}^2\gamma +{cot}^2\gamma \right)\sum _{d=1}^{2s}{\left({\tilde{\alpha }}_d(lnf)\right)}^2.\hfill \end{array}$$

Thus, the inequality (i) follows from the above relation. Inequality (ii) can be derived; we just set the frame field of $\mathfrak{D}$ such that $\xi$ is in $\mathfrak{D}$. If the equality holds in (32) and (33), then from leaving third term in (34), we derive

$$\begin{array}{c}\hfill \mathfrak{B}(T\mathrm{Q},T\mathrm{Q})\perp \nu ,\mathrm{i}.\mathrm{e}.,\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\lambda })\perp \nu .\end{array}$$

(37)

From left first, third and fifth terms in (35), we find

$$\begin{array}{c}\hfill \mathfrak{B}(\mathfrak{D},\mathfrak{D})\perp \phi {\mathfrak{D}}^{\perp },\mathfrak{B}({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp })\perp \phi {\mathfrak{D}}^{\perp },\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\perp })\perp \phi {\mathfrak{D}}^{\perp }.\end{array}$$

(38)

Also, from the vanishing sixth term in (35), we get

$$\begin{array}{c}\hfill \mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\gamma })\perp \phi {\mathfrak{D}}^{\perp }.\end{array}$$

(39)

From (37), (38) and (39), we conclude that

$$\begin{array}{c}\hfill \mathfrak{B}(\mathfrak{D},\mathfrak{D}),\mathfrak{B}({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp }),\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\perp }),\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\gamma })\mathrm{lie}\mathrm{in}F{\mathfrak{D}}^{\gamma }.\end{array}$$

(40)

But from leaving the eighth term and vanishing the seventh, ninth and eleventh terms in (35), we find

$$\begin{array}{c}\hfill \mathfrak{B}(\mathfrak{D},\mathfrak{D})\perp F{\mathfrak{D}}^{\gamma },\mathfrak{B}({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp })\perp F{\mathfrak{D}}^{\gamma },\mathfrak{B}({\mathfrak{D}}^{\gamma },{\mathfrak{D}}^{\gamma })\perp F{\mathfrak{D}}^{\gamma },\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\perp })\perp F{\mathfrak{D}}^{\gamma }.\end{array}$$

(41)

Then, from (40) and (41), we deduce that

$$\begin{array}{c}\hfill \mathfrak{B}(\mathfrak{D},\mathfrak{D})=0,\mathfrak{B}({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\perp })=0,\mathfrak{B}(\mathfrak{D},{\mathfrak{D}}^{\perp })=0.\end{array}$$

(42)

Also, from the hypothesis of the theorem $\mathrm{Q}$ is ${\mathfrak{D}}^{\perp }-{\mathfrak{D}}^{\gamma }$ mixed totally geodesic. Then we have

$$\begin{array}{c}\hfill \mathfrak{B}({\mathfrak{D}}^{\perp },{\mathfrak{D}}^{\gamma })=0.\end{array}$$

(43)

From the fact that ${\mathrm{Q}}_1$ is totally geodesic in $\mathrm{Q}$ [5,6] with (42) and (43), ${\mathrm{Q}}_1$ is totally geodesic in $\tilde{\mathrm{Q}}$. Since ${\mathrm{Q}}_{\gamma }$ is totally umbilical in $\mathrm{Q}$ [5,6], using this fact with the third relation of (41) and the last relation of (40) and (43), we find that ${\mathrm{Q}}_{\gamma }$ is totally umbilical in $\tilde{\mathrm{Q}}$. Hence, the theorem is proved completely. □

6. Conclusions

This work develops a systematic study of contact skew CR-warped product submanifolds in locally conformal almost cosymplectic manifolds. A complete characterization is obtained for when a contact skew CR-submanifold admits a warped product structure under the integrability of the slant distribution. Moreover, sharp geometric inequalities involving the squared norm of the second fundamental form, the warping function, and the slant angle are established. The equality cases lead to strong rigidity phenomena, showing that the base manifold is totally geodesic while the fiber becomes totally umbilical in the ambient space. Several known results for CR-, semi-slant, and pseudo-slant warped product submanifolds are recovered as special cases, illustrating the unifying nature of the present framework and its relevance to contact metric geometry.

Applications of Theorem 1 and Theorem 2

Some special cases of Theorem 1 and Theorem 2, we have the following consequences as applications.

In Theorem 1, if $dim\mathfrak{D}=0$ and $\lambda =1$, then a contact skew CR-warped product submanifold reduces to a warped product pseudo-slant slant submanifold. The statement of Theorem 1 will be as follows:

• Let $\mathrm{Q}$ be a proper pseudo-slant submanifold of a locally conformal almost cosymplectic manifold $\mathrm{Q}$. Then $\mathrm{Q}$ is locally a warped product submanifold if and only if

  $$\begin{array}{c}\hfill A_{\phi G_1}T{V}_1-A_{FT{V}_1}{G}_1={cos}^2\gamma \left(\eta \left(G_1\right)-G_1\left(\mu \right)\right)V_1,\end{array}$$

  such that the slant distribution ${\mathfrak{D}}^{\gamma }$ is integrable and μ is a smooth function on $\mathrm{Q}$ with $V_2\left(\mu \right)=0$, for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $G_1\in \Gamma ({\mathfrak{D}}^{\perp }\oplus \langle \xi \rangle )$. This is Theorem 3.1 of [20].
• Similarly, if we consider $dim{\mathfrak{D}}^{\perp }=0$ and $\lambda =1$ in Theorem 1, then $\mathrm{Q}$ becomes a semi-slant warped product submanifold. In this case, Theorem 1 can be stated as follows: Let $\mathrm{Q}$ be a proper semi-slant submanifold of a locally conformal almost cosymplectic manifold $\tilde{\mathrm{Q}}$ such that the slant distribution ${\mathfrak{D}}^{\gamma }$ is integrable. Then, $\mathrm{Q}$ is locally a warped product submanifold if and only if

  $$\begin{array}{c}\hfill A_{F{V}_1}\phi E_1-A_{FT{V}_1}{E}_1={sin}^2\gamma \left(E_1\left(\mu \right)-\eta \left(E_1\right)\right)V_1,\end{array}$$

  for some smooth functions μ on $\mathrm{Q}$ such that $V_2\left(\mu \right)=0$, for any $V_1,V_2\in \Gamma \left({\mathfrak{D}}^{\gamma }\right)$ and $E_1\in \Gamma (\mathfrak{D}\oplus \langle \xi \rangle )$. This is a statement of Theorem 2 of [22].
• If $dim\mathfrak{D}=0$ and $\lambda =1$ substituting in Theorem 2, then the squared norm of the second fundamental form of $\mathrm{Q}={\mathrm{Q}}_{\perp }{\times }_f{\mathrm{Q}}_{\gamma }$ satisfies

  $$\begin{array}{c}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge 2\beta {cos}^2\gamma \left(\left|\right|{\overrightarrow{\nabla }}^{\perp }(lnf){\left|\right|}^2-1\right),\end{array}$$

  where $dim{\mathfrak{D}}^{\gamma }=2\beta$ and ${\overrightarrow{\nabla }}^{\perp }(lnf)$ is the gradient along ${\mathrm{Q}}_{\perp }$. Therefore, Theorem 4.1 of [20] is a special case of Theorem 2. Also, generalized to [21].
• Similarly, if we consider $dim{\mathfrak{D}}^{\perp }=0$ and $\lambda =1$ in Theorem 2, then the squared norm of the second fundamental form of $\mathrm{Q}={\mathrm{Q}}_T{\times }_f{\mathrm{Q}}_{\gamma }$ satisfies

  $$\begin{array}{c}\hfill {\left|\right|\mathfrak{B}\left|\right|}^2\ge 4\beta \left({csc}^2\gamma +{cot}^2\gamma \right)\left(\left|\right|{\nabla }^T(lnf){\left|\right|}^2-1\right),\end{array}$$

  where $dim{\mathfrak{D}}^{\gamma }=2\beta$ and ${\nabla }^T(lnf)$ is the gradient of $lnf$ along ${\mathrm{Q}}_T$. Thus, Theorem 2 generalized to Theorem 3 of [22].

Remark 2. 

If we consider $\lambda =0$ and $\lambda =1$ throughout the paper, then our results generalize the work in [14] and [4], respectively.