Role of shape operator in warped product submanifolds of nearly cosymplectic manifolds

1.   Introduction

The warped product is used to form a different semi-Riemannian manifolds. Such construction is of benefit in General Relativity, black holes study and cosmological models. The product manifold metric, at this case, turns to non-degenerate. The warped product manifolds were inaugurated by Bishop and O'Neill ^[12] to extend the Riemannian product manifolds. This idea concerning warped product submanifolds was given by Chen ^[15,16]. Extending such a concept, recently, Uddin et al. ^[22] studied the non existence case of warped product semi-slant submanifold in terms of $M = M_{\vartheta}\times_fM_T$ whith $M_{\vartheta}$ and $M_T$ are proper slant and invariant submanifolds of the nearly cosymplectic manifold. Using the warped product semi-slant submanifold of type $M = M_T\times_fM_{\vartheta}$, the geometric inequality is obtained in ^[22] which disclosed the connection between the second fundamental form, slant immersion and warping function. Similarly, in ^[2,4] it has been classified the warped product submanifolds at Sasakian (cosymplectic) manifolds with pointwise slant embedding. Characterization theorems for warped product submanifold at Kenmotsu manifold having the fiber is the slant submanifold, were proved in ^[3,5]. A number of authors extended the warped product submanifolds idea in almost contact manifolds [as in ^[1,6,7,8,16] and references therein]. Studying the previous articles, an important question is arisen. Why we choose nearly cosymplectic structure? Answer is that cosymplectic structure does not admits any type of warped product semi-slant submanifold ^[19], while a near cosymplectic manifold has a nontrivial warped product semi-slant submanifolds in the shape of $M = M_T\times_fM_{\vartheta}$ see ^[22,24]. Therefore such a class force to us to study in a nearly cosymplectic manifold. At the first part of this present paper, we derive integrability theorems under some restrictions.

Remark 1.1. We will use the following abberiation throughout the paper: "WPSSS" for Warped product sem-slant submanifold, "WF" for warping function, "RM" for Riemannian manifold, "SSS" for semi-slant submanifold and "NCM" for nearly cosymplectic manifold $\widetilde{M}$.

More precisely, we prove the following finding for invariant distribution.

Theorem 1.1. An invariant distribution $\mathcal{D}\oplus < \zeta >$ of a SSS $M$ in a NCM $\widetilde{M}$ is integrable if and only if the following equality holds

for any $W_0, W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2\in\Gamma(\mathcal{D}^\vartheta)$.

Similarly, for a slant distribution we get the next integrability result.

Theorem 1.2. The slant distribution $\mathcal{D}^\vartheta$ of a SSS $M$ in a NCM $\widetilde{M}$ is integrable if and only if

for any $W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2, W_3\in\Gamma(\mathcal{D}^\vartheta)$.

The results Theorem 1.1 and 1.2 will required for proof of main theorem of this paper. Next, we provide the characterizations of a class of SSS for being the class of WPSSS in a NCM by using the result of Hiepko ^[18]. Hence, we give the characterization theorem for a WPSSS of a NCM which is an important result of this study.

Theorem 1.3. A semi-slant submanifold $M$ of a NCM $\widetilde{M}$ having the integrable distributions $\mathcal{D}$ and $\mathcal{D}^\vartheta,$ is locally a WPSSS of the type $M = M_T\times_fM_\vartheta$ if and only if

for all $W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2\in\Gamma(\mathcal{D}^\vartheta)$. For all $W_3\in\Gamma(\mathcal{D}^\vartheta)$, it is also satisfied that $W_3\lambda = 0$, for a positive function $\lambda = \ln f$ on $M_T$.

As a direct result of Theorem 1.3 in a sense of Papaghiuc ^[21] which shown that the class of semi-slant submanifold is generality of class of CR-submanifold with slant angle $\vartheta = \frac{\pi}{2}$. Therefore, we substitute $\vartheta = \frac{\pi}{2}$ in (1.3), then we get following result.

Corollary 1.1. A CR-submanifold $M$ of NCM $\widetilde{M}$ having the integrable distributions $\mathcal{D}$ and $\mathcal{D}^\perp,$ is locally the CR-warped product of type $M = M_T\times_fM_\perp$ if and only if

for all $W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2\in\Gamma(\mathcal{D}^\perp)$. For all $W_3\in\Gamma(\mathcal{D}^\perp)$, it is also satisfied that $W\lambda = 0$, for the positive function $\lambda = \ln f$ on $M_T$.

Remark 1.2. Interestingly to notice that Corollary 1.1 coincides with Theorem 3.1 in ^[24] and hence Theorem 1.3 is generalized Theorem 3.1 in ^[24]. Also, we give immediately consequences of our results.

Now, we give another interesting theorem

Theorem 1.4. A mixed totally geodesic WPSSS $M = M_T\times_fM_\vartheta$ in a NCM $\widetilde M$ is a usual Riemannian product manifold of $M_T$ and $M_\vartheta$.

The paper is organized as follows: In section 2, we highlight some preliminaries and formulas which are useful for our literature. In section 3, we give the definition of semi-slant submanifolds and provide the proofs of integrability theorems. In section 4, we define warped product manifolds and give the proof of characterization theorem.

2.   Preliminaries

The odd-dimensional $C^{\infty}$-manifold $(\widetilde M, g)$ associated to almost contact structure $(\varphi, \zeta, \eta)$ is referred to as the almost contact metric manifold fulfilling coming properties:

$\forall$ $W_1, W_2\in\Gamma(T\widetilde M)$ (see, for instance^[10,11]). A cosymplectic manifold ^[17,22,23] regarding Riemannian connection is contained the almost contact metric manifold which satisfied the next equation

It follows for a nearly cosymplectic manifold

for all vector fields $W_1, W_2$ are tangent to $\widetilde M$. The Gauss and Weingarten formulas which specifying the relation between Levi-Civitas connections $\nabla$ on a submanifold $M$ and $\widetilde{\nabla}$ on ambient manifold $\widetilde M$ are given by (for more detail see ^[16])

for every $W_1, \; W_2\in\Gamma(TM)$ and $\xi\in \Gamma(T^\perp M)$, in which $h$ and $A_\xi$ have this next relation

The next relation is that

in which $FW_1$ and $TW_1$are normal and tangential elements of $\varphi W_1$, respectively. If $M$ is invariant and anti-invariant then $FW_1$ as well as $TW_1$ are zero, in the same order. Similarly, we have

where $t\xi$ (resp. $f\xi$) are tangential (resp. normal) components of $\varphi\xi$. The covariant derivative of the endomorphism $\varphi$ is explained by

In case the tangential and normal elements of $(\widetilde\nabla_{W_1}\varphi)W_2$ using $\mathcal{P}_{W_1}W_2$ and $\mathcal{Q}_{W_1}W_2$, for a nearly cosymplectic manifold, it is satisfied that

where $W_1, W_2$ are tangential to $\widetilde M$. For more details on properties of $\mathcal{P}$ and $\mathcal{Q}$, see ^[22].

There is a motivating class of submanifolds presented as slant submanifolds class. For any not zero vector $W_1$ tangential to $M$ about $p$, in which $W_1$ is not proportional to $\zeta_p$, $0\leq\vartheta(W_1)\leq\pi/2$ is referred to the angle between $\varphi W_1$ and $T_pM$ which is named as Wirtinger angle. If $\vartheta(W_1)$ is constant for any $W_1\in{T_pM- < \zeta > }$ at point $p\in M$, therefore $M$ is referred to as the slant submanifold ^[13] and $\vartheta$ is then slant angle of $M$. The following necessary and sufficient condition is an important for this paper which known as characterization slant submanifold and was proved in ^[13], a submanifold $M$ is slant if and only if the equality holds

for a constant $\lambda\in[0, 1]$ in which $\lambda = \cos^2\vartheta$, where $T$ is an endomorphism defined in (2.8). The following alliances are resulted from Eq (2.12).

$\forall$ $W_1, W_2\in\Gamma(TM)$. The following result which was derived in ^[9] is the necessary and sufficient condition $M$ to remain slant if

for any $W_1\in\Gamma(TM)$.

3.   Semi-slant submanifolds

A generality of CR-submanifold in a almost Hermitian manifold by utilizing slant distribution was described using N. Papaghiuc ^[21]. Such submanifolds are referred to as semi-slant submanifold. This is determined by Cabererizo ^[14] in an almost contact manifold, i.e.,

Definition 3.1. The Riemannian submanifold $M$ is called a semi-slant in $\widetilde M$, in case it is spanned by two perpendicular distributions $\mathcal{D}$ and $\mathcal{D}^{\vartheta}$ that is $TM = \mathcal{D}^\vartheta\oplus\mathcal{D}\oplus\langle\zeta\rangle$. For more classification see in ^[9,22].

Remark 3.1. A semi-slant submanifold is referred to as a mixed totally geodesic if $h(W_1, W_2) = 0$, for any $W_1\in\Gamma(\mathcal{D}^\vartheta)$ and $W_2\in\Gamma(\mathcal{D})$.

Now, we construct the integrability conditions of the distributions involving in the meaning of a semi-slant submanifold in a nearly cosymplectic manifold.

Proof of Theorem 1.1 and 1.2

Proof of Theorem 1.1 From the Lie bracket, we get

From (2.2) and $\zeta$ is orthogonal to $\mathcal{D}^\vartheta$, we get

Then by utilizing (2.8) and (2.11), we attain

Thus by (2.3), (2.8) as well as (2.5), we have

Utilizing (2.13), (2.5) as well as (2.11), resulted in

Using (2.9) and (2.5), then we derive

Finally, from (2.15), we achieve that

That is the result which we wanted.

Corollary 3.1. The distribution $\mathcal{D}\oplus\zeta$ is the totally geodesic foliation of a SSS $M$ in NCM $\widetilde{M}$ if and only if

for all $W_0, W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2\in\Gamma(\mathcal{D}^\vartheta)$.

Proof. From the total geodesic folition defintion, for every $W_0, W_1\in\Gamma(\mathcal{D}\oplus\zeta)$, then $\nabla_{W_0}W_1\in\Gamma(\mathcal{D}\oplus\zeta)$. Putting this in the Eq (1.1), we get required result.

Proof of Theorem 1.2

By the property of Lie bracket, we have

By (2.2), we get

Using (2.11), we obtain

Thus from (2.3) and (2.8), then above equation take the form

Utilizing (2.2), (2.11) and (2.6), we achieve that

Utilizing (2.8), (2.2) and (2.11), we get

From (2.13), (2.6) (2.11) and (2.8), we obtain

Using (2.13) and (2.6), we derive

Now we compute last two terms of above equation by using property $\mathcal{P}$ (see ^[13]) as follows

Thus by the hypothesis and property of Lie Bracket in Eq (3.1), we arrive at

Applying the structure equation (2.12), we obtain

Hence, our assertion has got proven. The proof is completed.

An application of Theorem 1.2, we introduce

Corollary 3.2. The slant distribution $\mathcal{D}^\vartheta$ of SSS $M$ in a NCM $\widetilde{M},$ is the totally geodesic foliation in $M$ if and only if

for any $W_0\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2, W_3\in\Gamma(\mathcal{D}^\vartheta)$.

Proof. From the definition totally geodesic folition, for every $W_2, W_3\in\Gamma(\mathcal{D}^\vartheta)$, then $\nabla_{W_2}W_3\in\Gamma(\mathcal{D}^\vartheta)$. Inserting this in the Eq (1.2), we get required result.

4.   Warped product submanifolds

Suppose $M = M_1\times_fM_2$ is the smooth Riemannian manifold associating to warped product metric $g = g_1+f^2g_2$ in which $f$ is a WF on $M$. This idea was given in ^[12] and derived the formula;

For all $W_0, W_1\in\Gamma(TM_1)$ and $W_2, W_3\in\Gamma(TM_2).$ The gradient of $\ln f$ is denoted by $\nabla\ln f$ and defined as:

Remark 4.1. The warped product manifold $M = M_1\times_fM_2$ becomes trivial if $f$ is a constant function.

Remark 4.2. The base $M_1$ is totally geodesic and fiber $M_2$ is totally umbilical in WPM $M = M_1\times_fM_2$.

The two kinds of the warped product submanifolds are described as the products between proper slant submanifolds and invariant submanifolds follows by Definition 3.1, and it can be expressed as:

$\rm (i)$ $M = M_{\vartheta}\times_fM_T$

$\rm (ii)$ $M = M_T\times_fM_{\vartheta}$,

where $M_{\vartheta}$ and $M_{T}$ are slant as well as invariant submanifolds, in the same order. For the first case, we recall the following finding which obtained by Uddin et al. ^[22] which stats that a WPSSS of type $M = M_{\vartheta}\times_fM_T$ does not exist in a NCM $\widetilde{M}$. Therefore, we shall consider non-trivial WPSSS in terms of $M = M_T\times_fM_{\vartheta}$ in a NCM $\widetilde{M}$. The following results were proved in ^[22] related to such type warped product semi-slant in nearly cosymplectic manifold.

Lemma 4.1. ^[22] A WPSSS $M = M_T\times_fM_{\vartheta}$ in a NCM $\widetilde{M}$ has the coming relations

(i) $\zeta\ln f = 0,$

(ii) $g(h(W_0, W_1), FW_2) = 0,$

(iii) $g(h(W_2, \varphi W_0), FW_2) = (W_0\lambda)||W_2||^2,$

(iv) $g(\mathcal{P}_{W_0}W_2, TW_2) = 2g(h(W_0, W_2), FTW_2),$

for all $W_2\in\Gamma(TM_{\vartheta})$ and $W_0, W_1\in\Gamma(TM_T)$.

There is a another interesting lemma.

Lemma 4.2. ^[22] A WPSSS $M = M_T\times_fM_\vartheta$ in a NCM $\widetilde M$ satisfying the relation

for all $W_2\in\Gamma(TM_{\vartheta}$ and $W_0\in\Gamma(TM_T)$.

Recalling the result of S. Hiepko^[18], it is suitable to prove such characterization theorems for a WPSSS. The main result will be proved now.

Proof of Theorem 1.3

Let $M = M_T\times_fM_\vartheta$ is a WPSSS in a NCM $\widetilde M$ with $M_{\vartheta}$ and $M_T$ are proper slant and invariant submanifolds of $\widetilde M$. Therefore, the first part direct follows Lemmas 4.1 (iii) and Lemma 4.2 together Eq (2.7).

Conversely let $M$ is a SSS with condition (1.3) is satisfied. As we assumed that invariant distribution is integrable in hypothesis of theorem. From Theorem 1.1, we have necessary and sufficient condition of the integrability of $\mathcal{D}\oplus\zeta$ is

for $W_2\in\Gamma(\mathcal{D}^\vartheta)$ and $W_0, W_1\in\Gamma(\mathcal{D}\oplus\zeta)$. Thus relation (1.3)(i)-(ii) imply that

which implies that

As we have seen that $\mathcal{D}^\vartheta$ is proper slant then $\sin\vartheta\not = 0$, which means that $\nabla_{W_0}W_1\in\Gamma(\mathcal{D}\oplus\zeta)$ for any $W_0, W_1 \in\Gamma(\mathcal{D}\oplus\zeta)$, it is concluded that the leaves of $\mathcal{D}\oplus\zeta$ are totally geodesic in $M$ into an immersion of $\widetilde M$. Also from Theorem 1.2, it can be seen that the slant distribution $\mathcal{D}^\vartheta$ is integrable if and only if the following equation holds

Then above equation can be written by using (2.7).

Thus from (1.3)(i)-(ii), we obtain

It implies that

Moreover, we also assumed that $\mathcal{D}^\vartheta$ to be integrable, thus we consider a integral manifold $M_{\vartheta}$ of $\mathcal{D}^\vartheta$, that is, $M_{\vartheta}$ is leaf of integrable distribution $\mathcal{D}^\vartheta$ and $h^\vartheta$ is the second fundamental form resulting from immersion $M_{\vartheta}$ in $M$. Therefore, equation (4.2) becomes

By interchanging $W_2$ and $W_3$ in above equation, we obtain

Using (4.3) and (4.4), we attain

Thus from the definition of gradient and killing vector field $\xi$ for a nearly cosymplectic manifold [see (2.1) in ^[17]], we get

which leads to

Hence, it is concluded that $M_{\vartheta}$ is totally umbilical in $M$ having the following mean curvature vector

where $\nabla\lambda$ is the gradient of $\lambda$. However, by direct computations as we known that $Z(\lambda) = 0$, we derive

Furthermore, $\nabla\lambda\in\Gamma(TM_T)$ as $M_T$ is a totally geodesic in $M$ by Remark 4.2, consequently

for any $W_0\in\Gamma(\mathcal{D}\oplus\zeta)$ and $W_2\in\Gamma(\mathcal{D}^\vartheta)$. By equation (4.5), we get

This shows that the mean curvature vector $H^\theta$ of $M_{\vartheta}$ is parallel reciprocal to the normal connection $\nabla^\vartheta$ of $M_{\vartheta}$ in $M$. Therefore, the spherical condition is fulfilled, such that $M_{\vartheta}$ is an extrinsic sphere in $M$. Using the result of Hiepko (cf. ^[18]), $M$ is the non-trivial warped product submanifold of the form $M = M_T\times_fM_{\vartheta}$, in which $M_T$ and $M_{\vartheta}$ are the integral manifold of $\mathcal{D}\oplus\zeta$ and $\mathcal{D}^\vartheta$, in the same order. It completes this proof.

Proof of Theorem 1.4

Thus from Lemma 4.2, for all $W_2\in\Gamma(TM_{\vartheta}$ and $W_1\in\Gamma(TM_T)$, we get

As we assumed that $M$ is mixed totally geodesic submanifold, that is $h(W_1, W_2) = h(W_1, TW_2) = 0$, for all $W_2\in\Gamma(\mathcal{D}^\vartheta)$ and $W_0\in\Gamma(\mathcal{D}\oplus\zeta)$.

This implies $\cos^2\vartheta(W_1\ln f)||W_2||^2 = 0$. But $M$ is proper slant submanifold, then $\cos\vartheta\not = 0$, that is $(W_1\ln f)||W_2||^2 = 0$. Hence $(W_1\ln f) = 0$, i.e, the warping function $f$ is a constant on $M$ and the proof is completed via Remark 4.1.

Acknowledgements

The authors thank the referees for their valuable and constructive comments for modifying the presentation of this work. They are grateful to Akram Ali for giving idea for the work and his help for finalizing the manuscript. The authors extend their appreciation to the deanship of scientific research at King Khalid University for funding this work through research groups program under grant number R.G.P.2/56/40.

This research also was funded by the Deanship of Scientific Research at Princess Nourah bint Abdulrahman University through the Fast-track Research Funding Program.

Conflict of interest

There is no conflict of interest between the authors.