Ricci curvature of contact CR-warped product submanifolds in generalized Sasakian space forms admitting nearly Sasakian structure

1.   Introduction

Due to applications of warped product manifolds in Physics and theory of relativity ^[10], the study of warped product manifolds has been a fascinating topic of research. The warped products provide many basic solutions to the Einstein field equations ^[10]. One of the most important example of warped product manifolds is the modeling of space time near black holes in the universe. The Robertson-Walker model is a warped product which represents a cosmological model to the model of universe as a space time ^[27].

Bishop and Neill ^[11] explored the geometry of Riemannian manifolds of negative curvature and introduced the notion of warped product for these manifolds (see the definition in section 2). The warped product manifolds are the natural generalization of Riemannian product manifolds. Some natural properties of warped product were investigated in ^[11].

In 1981, Chen first used the idea of warped products for CR-submanifolds of Kaehler manifolds (^[13,15]). Basically, Chen proved the existence of CR-warped product submanifolds of the type $N_T\times_f N_\perp$ in the setting of Kaehler manifold, where $N_T$ and $N_\perp$ are the holomorphic and totally real submanifolds. Further, Hasegawa and Mihai ^[19] extended the study of Chen for the contact CR-warped product submanifolds of the Sasakian manifolds. Mihai ^[22] obtained an estimate for the squared norm of the second fundamental form in terms of the warping function for the contact CR-warped product submanifolds in the frame of Sasakian space form. Since then, many authors have studied warped product submanifolds in the different settings of Riemannian manifolds and numerous existence results have been explored (see the survey article ^[17]).

In 1999, Chen ^[14] discovered a relationship between Ricci curvature and squared mean curvature vector for an arbitrary Riemannian manifold. On the line of Chen a series of articles have been appeared to formulate the relationship between Ricci curvature and squared mean curvature in the setting of some important structures on Riemannian manifolds (see ^{[6,12,21,22,23,29]}). Recently Ali et al. ^[2] established a relationship between Ricci curvature and squared mean curvature for warped product submanifolds of a sphere and provide many physical applications.

In this paper our aim is to obtain a relationship between Ricci curvature and squared mean curvature for contact CR-warped product submanifolds in the setting of generalized Sasakian space form admitting a nearly Sasakian structure. Further, we provide some applications in terms of Hamiltonians and Euler-Lagrange equation. In the last we also worked out some applications of Obata's differential equation.

2.   Some basic results

A $(2n +1)-$dimensional $C^\infty-$manifold $\bar M$ is said to have an almost contact structure if there exist on $\bar M$ a tensor field $\phi$ of the type $(1, 1),$ a vector field $\xi$ and a 1-form $\eta$ satisfying

There always exists a Riemannian metric $g$ on an almost contact metric manifold $\bar M$ satisfying the following conditions

for all $X, Y\in T\bar M.$

An almost contact metric manifold is said to be nearly Sasakian manifold, if

for all $X, Y\in T\bar M.$

Moreover, the structure vector field $\xi$ is Killing vector field on a Riemannian manifold if it satisfies the following equation

In ^[1] Alegre et al. introduced the notion of generalized Sasakian space form as that an almost contact metric manifold $(\bar M, \phi, \xi, \eta, g)$ whose curvature tensor $\bar R$ satisfies

for all vector fields $X, Y, Z, W$ and certain differentiable functions $f_1, f_2, f_3$ on $\bar M.$ A generalized Sasakian space form with functions $f_1, f_2, f_3$ is denoted by $\bar M(f_1, f_2, f_3)$. If $f_1 = \frac{c+3}{4}$, $f_2 = f_3 = \frac{c-1}{4},$ then $\bar M(f_1, f_2, f_3)$ is a Sasakian space form $\bar M(c)$ ^[1]. If $f_1 = \frac{c-3}{4},$ $f_2 = f_3 = \frac{c+1}{4}$, then $\bar M(f_1, f_2, f_3)$ is a Kenmotsu space form $\bar M(c)$ ^[1] and if $f_1 = f_2 = f_3 = \frac{c}{4},$ then $\bar M(f_1, f_2, f_3)$ is a cosymplectic space form $\bar M(c)$ ^[1].

Now, we discuss some examples of space forms. Let $M(c)$ be a complex space form with the complex structure $(J, g_M)$, consider the product manifold $N^{2n+1} = M(c)\times R$ and define the following tensors on $N^{2n+1}$

where $\pi: M(c)\times R \to M(c)$ is the projection map and $t$ is the standard coordinate function on the real axis. Then $(N^{2n+1}, \phi, \xi, \eta, g_N)$ is a cosymplectic space form with constant $\phi-$sectional curvature equal to $c$ (^[1,7]).

The Hyperbolic space $H(-1)$ of constant sectional curvature -1 is an example of Kenmotsu space form.

Nearly Sasakian structure was introduced on the 5-dimensional sphere $S^5(2)$ of constant sectional curvature 2 as totally umbilical hypersurface of nearly Kaehler 6-sphere $S^6$, which is not a Sasakian structure ^[8].

One of the present author Ibrahim Al-Dayel ^[18] proved that there exists two other structures which are Sasakian as well as nearly cosymplectic structure. More precisely,

Theorem 2.1. ^[18] There are two structures $(\phi_i, \xi, \eta, g), i = 1, 2$ related to the nearly Sasakian structure $(\phi, \xi, \eta, g)$ on the 5-sphere $S^5(2)$ such that $S^5(2)(\phi_1, \xi, \eta, g)$ is homothetic to Sasakian manifold and $S^5(2)(\phi_2, \xi, \eta, g)$ is a nearly cosymplectic manifold.

Another example of generalized Sasakian space form admitting the nearly cosymplectic structure is totally geodesic hypersurface $S^5(\psi_1, \xi, \eta, g)$ of the nearly Kaehler 6-sphere $(S^6, J, g)$ ^[9]. Further extending this study one of the present author Ibrahim Al-Dayel in ^[18] obtained two more structures $\psi_2$ and $\psi_3$ such that $S^5(\psi_2, \xi, \eta, g)$ and $S^5(\psi_3, \xi, \eta, g)$ are Sasakian and nearly cosymplectic manifolds respectively.

Let $(M^n, g)$ be an $n-$dimensional Riemannian manifold isometrically immersed in a $m-$dimensional Riemannian manifold $\bar M.$ Then the Gauss and Weingarten formulas are $\bar \nabla_XY = \nabla_XY + h(X, Y)$ and $\bar\nabla_XN = -A_N X+\nabla^\perp_XN$ respectively, for all $X, Y\in TM$ and $N \in T^\perp M.$ Where $\nabla$ is the induced Levi-Civita connection on $M,$ $N$ is a vector field normal to $M$, $h$ is the second fundamental form of $M$, $\nabla^\perp$ is the normal connection in the normal bundle $T^\perp M$ and $A_N$ is the shape operator of the second fundamental form. The second fundamental form $h$ and the shape operator are associated by the following formula

The equation of Gauss is given by

for all $X, Y, Z, W\in TM.$ Where, $\bar R$ and $R$ are the curvature tensors of $\bar M$ and $M$ respectively.

For any $X\in TM$ and $N\in T^\perp M,$ $\phi X$ and $\phi N$ can be decomposed as follows

and

where $PX$ (resp. $tN$) is the tangential and $FX$ (resp. $fN$) is the normal component of $\phi X$ (resp. $\phi N$).

For any orthonormal basis $\{e_1, e_2, \dots, e_n\}$ of the tangent space $T_xM$, the mean curvature vector $H(x)$ and its squared norm are defined as follows

where $n$ is the dimension of $M$. If $h = 0$ then the submanifold is said to be totally geodesic and minimal if $H = 0.$ If $h(X, Y) = g(X, Y) H$ for all $X, Y\in TM,$ then $M$ is called totally umbilical.

The scalar curvature of $m-$ dimensional Riemannian manifold $\bar M$ is denoted by $\bar\pi (\bar M)$ and is defined as

where $\bar\kappa_{pq} = \bar\kappa(e_p \wedge e_q)$. Throughout this study, we shall use the equivalent version of the above equation, which is given by

In a similar way, the scalar curvature $\bar \pi(L_x)$ of a $L-$plane is expressed as

Let $\{e_1, \dots, e_n\}$ be an orthonormal basis of the tangent space $T\!_xM$ and if $e_r$ belongs to the orthonormal basis $\{e_{n+1}, \dots e_m\}$ of the normal space $T^\perp M$, then we have

and

Let $\kappa_{pq}$ and $\bar\kappa_{pq}$ be the sectional curvatures of the plane sections generated by $e_p$ and $e_q$ at the point $x \in M^n$ and in the Riemannian space form $\bar M^m(c)$, respectively. Thus by Gauss equation, we have

The global tensor field for orthonormal frame of vector field $\{e_1, \dots, e_n\}$ on $M^n$ is defined as

for all $X, Y\in T_x M^n.$ The above tensor is named Ricci tensor. When we fix a specific vector $e_u$ from $\{e_1, \dots, e_n\}$ on $M^n$, which is designated by $\chi$. Therefore the Ricci curvature is defined by

Let $(N_1, g_1)$ and $(N_2, g_2)$ be two Riemannian manifolds with Riemannian metrics $g_1$ and $g_2$ respectively and let $\psi$ be a positive differentiable function on $N_1$. If $\pi: N_1\times N_2\to N_1$ and $\eta: N_1\times N_2\to N_2$ are the projection maps given by $\pi(p, q) = p$ and $\eta(p, q) = q$ for every $(p, q)\in N_1\times N_2,$ then the warped product manifold is the product manifold $N_1\times N_2$ equipped with the Riemannian structure such that

for all $X, Y\in TM.$ The function $\psi$ is called the warping function of the warped product manifold ^[11]. If the warping function is constant, then the warped product is trivial i.e., simply Riemannian product. Then from Lemma 7.3 of ^[11], we have

where $\nabla$ is the Levi-Civita connection on $M$. For a warped product $M = N_1\times_\psi N_2$ it is easy to observe that

for $X\in TM_1$ and $Z\in TM_2.$

We denote $\nabla \psi$ the gradient of $\psi$ and it is defined as

for all $X\in TM.$

Let $\{e_1, e_2, \dots, e_n\}$ be an orthogonal basis of the tangent space $TM$ of a $n-$dimensional Riemannian manifold $M.$ Then (2.9) provides the following,

The Laplacian of $\psi$ is defined by

The Hessian tensor for a differentiable function $\psi$ is a symmetric covariant tensor of rank 2 and is defined as

where $H^{\psi}$ is the Hessian of $\psi.$

For the warped product submanifolds, we observed the well known result, which is described as follows ^[16]

For a compact orientable Riemannian manifold $M$ with or without boundary and as a consequences of the integration theory of manifolds, we have

where $\psi$ is a function on $M$ and $dV$ is the volume element of $M$.

3.   Contact CR-warped product submanifolds

Suppose $M$ be a $n-$dimensional submanifold isometrically immersed in an almost contact metric manifold $\bar M(g, \phi, \xi, \eta)$ such that the structure vector field $\xi$ is tangent to $M$. The submanifold $M$ is called contact CR-submanifold if it admits an invariant distribution $D$ whose orthogonal complementary distribution $D^\perp$ is anti-invariant such that $TM = D\oplus D^\perp\oplus \langle\xi\rangle,$ where $\phi D\subseteq D$, $\phi D^\perp \subseteq T^\perp M$ and $\langle\xi\rangle$ is the 1-dimensional distribution spanned by $\xi$ ^[24]. If $\mu$ is the invariant subspace of the normal bundle $T^\perp M$, then in the case of contact CR- submanifold, the normal bundle $T^\perp M$ can be decomposed as $T^\perp M = \mu\oplus \phi D^\perp$. A contact CR-submanifold is called contact CR-product submanifold if the distributions $D$ and $D^\perp$ are parallel on $M.$ Moreover, a contact CR-submanifold is said to be mixed totally geodesic if $h(D, D^\perp) = 0.$ As a generalization of the product manifold submanifolds one can consider warped product submanifolds. In ^[19] Hasegawa and Mihai studied contact CR-warped product submanifolds in Sasakian manifolds basically, they proved the existence of the contact CR-warped product submanifolds of the type $N_T\times_f N_\perp$ such that the structure vector field $\xi$ is tangent to $N_T$. After that Mihai ^[22] and Munteanu ^[26] investigated contact CR-warped product in Sasakian space forms and obtained an inequality for squared norm of second fundamental form and warping function. Moreover, Atceken ^[5] explored the existence of contact CR-warped product submanifolds in the expressions of some inequalities this study was extended to the setting of trans-Sasakian generalized Sasakian spaceforms by Sular and Ozgur ^[28]. More recently, Ishan and Khan ^[20] generalized contact CR-warped product submanifolds in the setting of generalized Sasakian manifolds admitting nearly Sasakian structure. Throughout, this study we consider $n-$dimensional contact CR-warped product submanifold $M^n = N_T^{n_1}\times_\psi N_\perp^{n_2}$, such that the structure vector field $\xi$ is tangential to $N_T$, where $n_1$ and $n_1$ are the dimensions of the invariant and anti-invariant submanifold respectively.

Now, we start with some initial results.

Lemma 3.1. Let $M = N_T^{n_1}\times_\psi N_\perp^{n_2}$ be a contact CR-warped product submanifold isometrically immersed in a nearly Sasakian manifold $\bar M$. Then

(i) $g(h(X, Y), \phi Z) = 0,$

(ii) $g(h(\phi X, \phi X), N) = - g(h(X, X), N),$

for any $X, Y\in TN_T,$ $Z\in TN_\perp$ and $N\in\mu.$

Proof. By using Gauss and Weingarten formulae in Eq (2.1), we have

taking inner product with $Y$ and using (2.3), we get the result that needed.

To show $(ii)$, for any $X\in TN_T$ we have

using Gauss formula and (2.1), we get

taking inner product with $\phi N$, above equation yields

replacing $X$ by $\phi X$ and using the fact that $\xi$ is Killing vector field ^[9], the last equation gives

From (3.1) and (3.2), we get the required result.

By the Lemma 3.1 it is evident that the isometric immersion $N_T^{n_1}\times_\psi N_\perp^{n_2}$ into a nearly Sasakian manifold is $D-$ minimal. The $D-$ minimal property provides us a useful relationship between the contact CR-warped product submanifold $N_T\times_\psi N_\perp$ and the equation of Gauss.

Definition 3.1 The warped product $N_1\times_\psi N_2$ isometrically immersed in a Riemannian manifold $\bar M$ is called $N_i$ totally geodesic if the partial second fundamental form $h_i$ vanishes identically. It is called $N_i$-minimal if the partial mean curvature vector $H^i$ becomes zero for $i = 1, 2$.

Let $\{e_1, \dots, e_\beta, e_{\beta+1} = \phi e_1\dots, \dots e_{n_1-1} = \phi e_\beta, e_{n_1} = \xi, e_{n_1+1}, \dots, e_n\}$ be a local orthonormal frame of vector fields on the contact CR-warped product submanifold $M^n = N_T^{n_1}\times_\psi N_\perp^{n_2}$ such that $\{\xi, e_1, \dots, e_{n_1}\}$ are tangent to $N_T$ and $\{e_{n_1+1}, \dots e_n\}$ are tangent to $N_\perp.$ Moreover, $\{ e_1^* = \phi e_{n_1+1}, \dots, e_n^* = \phi e_n, e_{n+1}^*, \dots, e_m^*\}$ is a local orthonormal frame of the normal space $T^\perp M.$

From Lemma 3.1, one can observe

Therefore, it knows that the trace of $h$ due to $N_T$ becomes zero. Hence in view of the Definition 3.1, we have the following important result.

Theorem 3.2. Let $M^n = N_T^{n_1} \times_\psi N_\perp^{n_2}$ be a contact CR-warped product submanifold isometrically immersed in a nearly Sasakian manifold. Then $M^n$ is $D-$ minimal.

So, it is easy to conclude the following

where $\|H\|^2$ is the squared mean curvature.

4.   Ricci curvature for contact CR-warped product submanifold

The present section deals to formulate the Ricci curvature in the expressions of mean curvature vector and warping function $\psi.$

Theorem 4.1. Let $M = N_T ^{n_1}\times_\psi N_\perp ^{n_2}$ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form $\bar M(f_1, f_2, f_3)$ admitting nearly Sasakian structure. If for each orthogonal unit vector field $\chi\in T_xM$ orthogonal to $\xi$, either tangent to $N_T$ or $N_\perp$. Then we have

(1) The Ricci curvature satisfy the following inequality

(i) If $\chi$ is tangent to $N_T^{n_1}$, then

(ii) $\chi$ is tangent to $N_\perp^{n_2}$, then

(2) If $H(x) = 0$ for each point $x\in M^n.$ Then there is a unit vector field $X$ which satisfies the equality case of (1) if and only if $M^n$ is mixed totally geodesic and $\chi$ lies in the relative null space $N_x$ at $x$.

(3) For the equality case we have

(a) The equality case of (4.1) holds identically for all unit vector fields tangent to $N_T$ at each $x\in M^n$ if and only if $M^n$ is mixed totally geodesic and $D-$totally geodesic contact CR-warped product submanifold in $\bar M^m(f_1, f_2, f_3)$.

(b) The equality case of (4.2) holds identically for all unit vector fields tangent to $N_\perp$ at each $x\in M^n$ if and only if $M$ is mixed totally geodesic and either $M^n$ is $D^\perp$- totally geodesic contact CR-warped product or $M^n$ is a $D^\perp$ totally umbilical in $\bar M^m(f_1, f_2, f_3)$ with dim $D^\perp = 2$.

(c) The equality case of (1) holds identically for all unit tangent vectors to $M^n$ at each $x\in M^n$ if and only if either $M^n$ is totally geodesic submanifold or $M^n$ is a mixed totally geodesic totally umbilical and $D-$ totally geodesic submanifold with dim $N_\perp = 2,$

where $n_1$ and $n_2$ are the dimensions of $N_T$ and $N_\perp$ respectively.

Proof. Suppose that $M = N_T ^{n_1}\times_\psi N_\perp ^{n_2}$ be a contact CR-warped product submanifold of a generalized Sasakian space form. From Gauss equation, we have

Let $\{e_1, \dots, e_{n_1}, e_{n_1+1}, \dots, e_n\}$ be a set of orthonormal vector fields on $M^n$ such that the frame $\{e_1, \dots, e_{n_1}\}$ is tangent to $N_T$ and $\{e_{n_1+1}, \dots, e_n\}$ is tangent to $N_\perp$. So, the unit tangent vector $\chi = e_A \in \{e_1, \dots, e_n\}$ can be expanded (4.3) as follows

The above expression can be expanded as

In view of the Lemma 3.1, the preceding expression takes the form

By Eq (2.7), we have

Substituting the values of Eq (4.5) in (4.4), we discover

Since, $M^n = N_T^{n_1} \times_\psi N_\perp^{n_2}$, then from (2.5), the scalar curvature of $M^n$ can be defined as follows

Using (2.5) and (2.10), we derive

Utilizing (4.8) together with (2.2) in (4.6), we have

Considering $\chi = e_a$, we got two options: $\chi$ may be tangent to the submanifold $N_T^{n_1}$ or to the fibre $N^{n_2}_\perp$.

Option 1: If $e_a$ is tangent to $N^{n_1}_T$, then fix a unit tangent vector from $\{e_1, \dots, e_{n_1}\}$ suppose $\chi = e_a = e_1$, then from (4.9) and (2.8), we find

From (2.2), (2.5) and (2.6), we have

Using in (4.10), we have

Further, the seventh and eighth terms on right hand side of (4.14) can be written as

Likewise, we get

Utilizing above two values in (4.14), we get

Since $M^n = N^{n_1}_T \times_\psi N^{n_2}_\perp$ is $N^{n_1}_T$-minimal then we can observe the following

and

Simultaneously, we can conclude

Using (4.16) and (4.17) in (4.15), after the assessment of (4.18), we finally get

Further, using the fact that $\sum_{r = n+1}^m (h^r_{n_1+1 n_1+1}+\dots+h^r_{n n}) = n^2\|H\|^2,$ we get

From the above inequality, we can conclude the inequality (4.1).

Option 2: If $e_a$ is tangent to $N_\perp^{n_2}$, then we select the unit vector from $\{e_{n_1+1}, \dots, e_n\}$, suppose that the unit vector is $e_n$ i.e., $\chi = e_n.$ Then from (2.2), (2.5) and (2.6), we have

Now, in a similar way as in option $1$ using (4.19), we have

Using similar steps of option $i$, the above inequality takes the form

By the Lemma 3.1, one can observe that

Utilizing this in (4.21), we get

The last term of the above inequality can be written as

Moreover, the fifth term on right hand side of (4.23) can be expanded as

Using last two values in (4.23), we have

or equivalently

On applying similar techniques as in the proof of option $1$, we arrive at

which gives the inequality (4.2).

Next, we explore the equality cases of the inequality (4.1). First, we redefine the notion of the relative null space ${\cal N}_x$ of the submanifold $M^n$ in the generalized Sasakian space form $\bar M^{m} (f_1, f_2, f_3)$ at any point $x\in M^n,$ the relative null space was defined by B. Y. Chen ^[14], as follows

For $A\in \{1, \dots, n\}$ a unit vector field $e_A$ tangent to $M^n$ at $x$ the equality sign of $(4.1)$ holds identically iff

such that $r\in \{{n+1}, \dots m\}$ the condition $(i)$ indicates that $M^n$ is mixed totally geodesic contact CR-warped product submanifold. Combining statements $(ii)$ and $(iii)$ with the fact that $M^n$ is contact CR-warped product submanifold, we get that the unit vector field $\chi = e_A$ belongs to the relative null space ${\cal N}_x$. The converse is straightforward and statement (2) is proved.

For a contact CR-warped product submanifold, the equality satisfies in (4.1)if for all unit tangent vector belong to $N_T$ at $x$ iff

where $p\in\{1, \dots, n_1\}$ and $r \in\{n+1, \dots, m\}.$ Since $M^n$ is contact CR-warped product submanifold, the third condition says that $h^r_{pp} = 0, \; \; p \in\{1, \dots, n_1\}$. Using this in the condition $(ii)$, we shall say that $M^n$ is $D-$totally geodesic contact CR-warped product submanifold in $\bar M^{m} (f_1, f_2, f_3)$ and mixed totally geodesicness inheres from the condition $(i)$. Which demonstrates $(a)$ in $(3).$

For a contact CR-warped product submanifold, the equality sign of (4.1) holds identically for all unit tangent vector fields tangent to $N_\perp$ at $x$ if and only if

such that $K\in\{ n_1+1, \dots, n\}$ and $r\in \{n+1, \dots, m\}.$ From the condition $(iii)$ two cases emerge, that is

If the first case of (4.25) is satisfied, then by virtue of condition $(ii)$, it is easy to conclude that $M^n$ is a $D^\perp-$ totally geodesic contact CR-warped product submanifold in $\bar M^m(c)$. This is the first case of part $(b)$ of statement $(3)$.

On the other hand, let $M^n$ is not $D^\perp-$totally geodesic contact CR-warped product submanifold and dim $N^\perp = 2.$ Then condition $(ii)$ of (4.25) implies that $M^n$ is $D^\perp-$ totally umbilical contact CR-warped product submanifold in $\bar M^{m} (f_1, f_2, f_3)$, it is case second of the present part. Thus part $(b)$ of $(3)$ is verified.

To prove $(c)$ using parts $(a)$ and $(b)$ of $(3)$, we combine (4.24) and (4.25). For the first case of this part, assume that $dim N_\perp \neq 2.$ Since from parts $(a)$ and $(b)$ of $(3)$ we conclude that $M^n$ is $D-$totally geodesic and $D^\perp-$ totally geodesic submanifold in $\bar M^{m} (f_1, f_2, f_3).$ Therefore, $M^n$ is a totally geodesic submanifold in $\bar M^m(c).$

Another case, suppose that first case does not satisfies. Then parts $(a)$ and $(b)$ provide that $M^n$ is mixed totally geodesic and $D-$ totally geodesic submanifold of $\bar M^{m} (f_1, f_2, f_3)$ with $dim N_\perp = 2.$ From the condition $(b)$ it is clear that $M^n$ is $D^\perp-$totally umbilical contact CR-warped product submanifold and from $(a)$ it is $D-$totally geodesic, which is part $(c)$. This proves the theorem.

In view of (2.10), we have another version of the theorem 4.1 as follows.

Theorem 4.2. Let $M^n = N_T ^{n_1}\times_\psi N_\perp ^{n_2}$ be a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form $\bar M^{m} (f_1, f_2, f_3)$ admitting nearly Sasakian structure. Then for each orthogonal unit vector field $\chi\in T_xM$ orthogonal to $\xi$, either tangent to $N_T$ or $N_\perp$ the Ricci curvature satisfies the following inequalities:

(i) If $\chi$ is tangent to $N_T$, then

(ii) If $\chi$ is tangent to $N_\perp$, then

The equality cases are similar as in the theorem 4.1.

Remark 4.3. In particular, it is straightforward to see that the example given in Proposition 3.3 of ^[26] satisfies the inequalities of Theorem 4.2.

5.   Application of Obata's differential equation

This section is based on the study of Obata ^[26]. Basically, Obata characterized a Riemannian manifolds by a specific ordinary differential equation and derived that an $n-$dimensional complete and connected Riemannian manifold $(M^n, g)$ to be isometric to the n-dimensional sphere if and only if there exists a non-constant smooth function $\tau$ on $M^n$ that is the solution of the differential equation $H^\tau = -c\tau g,$ where $H^\tau$ is the Hessian of $\tau$. Moreover, for the warped product submanifolds the Obata's differential equation is used in (^[3,25]). Recently, Alodan et al. ^[4] applied Obata's work in the study of hypersurface of Sasakian manifold. Inspired by these studies, we obtain the following characterization.

Theorem 5.1. Let $M^n = N^{n_1}_T\times_\psi N^{n_2}_\perp$ be a compact orientable contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form $M^m(f_1, f_2, f_3)$ admitting nearly Sasakian structure with positive Ricci curvature and satisfying one of the following relation:

(i) $\chi\in TN_T$ orthogonal to $\xi$ and

(ii) $\chi\in TN_\perp$ and

where $\lambda_1 > 0$ is an eigenvalue of the warping function $\tau = ln\psi$. Then the base manifold $N^{n_1}_T$ is isometric to the sphere $S^{n_1}(\frac{\lambda_1}{n_1})$ with constant sectional curvature $\frac{\lambda_1}{n_1}$.

Proof. Let $\chi\in TN_T$. Consider that $\tau = ln\psi$ and define the following relation as

But we know that $\|I\|^2 = trace (II^*) = p,$ where $p$ is a real number and

Then Eq (5.3) transform to

Assuming $\lambda_1$ is an eigenvalue of the eigen function $\tau$ then $\Delta\tau = \lambda_1\tau.$ Thus we get

On the other hand, we obtain $\Delta\tau^2 = 2\tau\Delta\tau + \|\nabla\tau\|^2$ or $\lambda_1\tau^2 = 2\lambda_1\tau^2 + \|\nabla\tau\|^2$ which implies that $\tau^2 = - \frac{1}{\lambda_1} \|\nabla\tau\|^2,$ using this in Eq (5.4), we have

In particular $t = - \frac{\lambda_1}{n_1}$ on (5.5) and integrating with respect to $dV$

Integrating the inequality (4.26) and using the fact $\int_{M^n} \Delta\phi dV = 0,$ we have

From (5.6) and (5.7) we derive

According to assumption $Ric(\chi) \geq 0,$ the above inequality gives

From (5.1), we get

But we know that

Combining last two statements, we get

Since the warping function $\tau = ln\psi$ is not constant function on $M^n$ so equation (5.8) is Obata's differential equation ^[26] with constant $c = \frac{\lambda_1}{n_1} > 0.$ As $\lambda_1 > 0$ and therefore the base submanifold $N^{n_1}_T$ is isometric to the sphere $S^{n_1}(\frac{\lambda_1}{n_1})$ with constant sectional curvature $\frac{\lambda_1}{n_1}$. This proves the theorem.

Acknowledgments

The authors are highly thankful to anonymous referees for theirs valuable suggestions and comments which have improved the contents of the paper.

Conflict of interest

The authors declare that they have no Competing interests.