Research article

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

  • Received: 18 March 2020 Accepted: 15 July 2020 Published: 07 August 2020
  • MSC : 58C40, 53C42, 35P15

  • In this paper, first, we find the integrability theorems for the invariant and slant distributions which appeared in the concept of semi-slant submanifolds. Utilizing these theorems, we prove that a semi-slant submanifold reduces to be a warped product semi-slant submanifold, provided some necessary and sufficient conditions concerning the shape operators. Also, it is shown that a few earlier results are exceptional cases of this paper results.

    Citation: Rifaqat Ali, Nadia Alluhaibi, Khaled Mohamed Khedher, Fatemah Mofarreh, Wan Ainun Mior Othman. Role of shape operator in warped product submanifolds of nearly cosymplectic manifolds[J]. AIMS Mathematics, 2020, 5(6): 6313-6324. doi: 10.3934/math.2020406

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  • In this paper, first, we find the integrability theorems for the invariant and slant distributions which appeared in the concept of semi-slant submanifolds. Utilizing these theorems, we prove that a semi-slant submanifold reduces to be a warped product semi-slant submanifold, provided some necessary and sufficient conditions concerning the shape operators. Also, it is shown that a few earlier results are exceptional cases of this paper results.


    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=Mϑ×fMT whith Mϑ and MT are proper slant and invariant submanifolds of the nearly cosymplectic manifold. Using the warped product semi-slant submanifold of type M=MT×fMϑ, 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=MT×fMϑ 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 ˜M.

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

    Theorem 1.1. An invariant distribution D<ζ> of a SSS M in a NCM ˜M is integrable if and only if the following equality holds

    2g(W0W1,W2)=csc2ϑ(g(h(W0,φW1),FW2)2g(h(W0,W1),FTW2)+g(h(W1,φW0),FW1)), (1.1)

    for any W0,W1Γ(Dζ) and W2Γ(Dϑ).

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

    Theorem 1.2. The slant distribution Dϑ of a SSS M in a NCM ˜M is integrable if and only if

    sin2ϑg(W2W3,W1)=12{g(h(W1,W2),FTW3)+g(h(W1,W3),FTW2)g(h(φW1,W2),FW3)g(h(φW1,W3),FW2)+η(W1)g(˜W3ζ,W2)}, (1.2)

    for any W1Γ(Dζ) and W2,W3Γ(Dϑ).

    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 ˜M having the integrable distributions D and Dϑ, is locally a WPSSS of the type M=MT×fMϑ if and only if

    (i)AFW2φW1=(W1λ)W2,and(ii)AFTW2W1=13cos2ϑ(W1λ)W2, (1.3)

    for all W1Γ(Dζ) and W2Γ(Dϑ). For all W3Γ(Dϑ), it is also satisfied that W3λ=0, for a positive function λ=lnf on MT.

    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 ϑ=π2. Therefore, we substitute ϑ=π2 in (1.3), then we get following result.

    Corollary 1.1. A CR-submanifold M of NCM ˜M having the integrable distributions D and D, is locally the CR-warped product of type M=MT×fM if and only if

    (i)AφW2W1=(φW1λ)W2, (1.4)

    for all W1Γ(Dζ) and W2Γ(D). For all W3Γ(D), it is also satisfied that Wλ=0, for the positive function λ=lnf on MT.

    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=MT×fMϑ in a NCM ˜M is a usual Riemannian product manifold of MT and Mϑ.

    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.

    The odd-dimensional C-manifold (˜M,g) associated to almost contact structure (φ,ζ,η) is referred to as the almost contact metric manifold fulfilling coming properties:

    φ2=I+ηζ,η(ζ)=1,φ(ζ)=0,ηφ=0, (2.1)
    g(φW1,φW2)=g(W1,W2)η(W1)η(W2),η(W1)=g(W1,ζ), (2.2)

    W1,W2Γ(T˜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

    (˜W1φ)W1=0, (2.3)

    It follows for a nearly cosymplectic manifold

    (˜W1φ)W2+(˜W2φ)W1=0, (2.4)

    for all vector fields W1,W2 are tangent to ˜M. The Gauss and Weingarten formulas which specifying the relation between Levi-Civitas connections on a submanifold M and ˜ on ambient manifold ˜M are given by (for more detail see [16])

    ˜W1W2=W1W2+h(W1,W2) (2.5)
    ˜W1ξ=AξW1+W1ξ, (2.6)

    for every W1,W2Γ(TM) and ξΓ(TM), in which h and Aξ have this next relation

    g(h(W1,W2),ξ)=g(AξW1,W2) (2.7)

    The next relation is that

    φW1=TW1+FW1, (2.8)

    in which FW1 and TW1are normal and tangential elements of φW1, respectively. If M is invariant and anti-invariant then FW1 as well as TW1 are zero, in the same order. Similarly, we have

    φξ=tξ+fξ (2.9)

    where tξ (resp. fξ) are tangential (resp. normal) components of φξ. The covariant derivative of the endomorphism φ is explained by

    (˜W1φ)W2=˜W1φW2φ˜W1W2,W1,W2Γ(T˜M). (2.10)

    In case the tangential and normal elements of (˜W1φ)W2 using PW1W2 and QW1W2, for a nearly cosymplectic manifold, it is satisfied that

    (i)PW1W2+PW2W1=0,(ii)QW1W2+QW2W1=0, (2.11)

    where W1,W2 are tangential to ˜M. For more details on properties of P and Q, see [22].

    There is a motivating class of submanifolds presented as slant submanifolds class. For any not zero vector W1 tangential to M about p, in which W1 is not proportional to ζp, 0ϑ(W1)π/2 is referred to the angle between φW1 and TpM which is named as Wirtinger angle. If ϑ(W1) is constant for any W1TpM<ζ> at point pM, therefore M is referred to as the slant submanifold [13] and ϑ 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

    T2=λ(I+ηζ), (2.12)

    for a constant λ[0,1] in which λ=cos2ϑ, where T is an endomorphism defined in (2.8). The following alliances are resulted from Eq (2.12).

    g(TW1,TW2)=cos2ϑ{g(W1,W2)η(W1)η(W1)} (2.13)
    g(FW1,FW2)=sin2ϑ{g(W1,W2)η(W1)η(W2)}, (2.14)

    W1,W2Γ(TM). The following result which was derived in [9] is the necessary and sufficient condition M to remain slant if

    (a)tFW1=sin2ϑ(W1+η(W1)ζ)and(b)fFW1=FTW1, (2.15)

    for any W1Γ(TM).

    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 ˜M, in case it is spanned by two perpendicular distributions D and Dϑ that is TM=DϑDζ. For more classification see in [9,22].

    Remark 3.1. A semi-slant submanifold is referred to as a mixed totally geodesic if h(W1,W2)=0, for any W1Γ(Dϑ) and W2Γ(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

    g([W0,W1],W2)=g(˜W0W1,W2)g(˜W1W0,W2).

    From (2.2) and ζ is orthogonal to Dϑ, we get

    g([W0,W1],W2)=g(˜W0W1,W2)g(φ˜W1W0,φW2).

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

    g([W0,W1],W2)=g(˜W0W1,W2)+g(˜W1W0,φTW2)+g((˜W1φ)W0,FW2)(˜W1φW0,FW2).

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

    g([W0,W1],W2)=g(˜W0W1,W2)+g(˜W1W0,T2W2)+g(˜W1W0,FTW2)g((˜W0φ)W1,FW2)g(h(W1,φW0),FW2).

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

    g([W0,W1],W2)=g(˜W0W1,W2)cos2ϑg(˜W1W0,W2)+g(h(W0,W1),FTW2)g(˜W0φW1,FW2)g(˜W0W1,φFW2)g(h(W1,φW0),FZ).

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

    g([W0,W1],W2)=g(˜W0W1,W2)cos2ϑg(˜W1W0,W2)+g(h(W0,W1),FTW2)g(h(W0,φW1),FW2)g(˜W0W1,tFW2)g(˜W0W1,fFW2)g(h(W1,φW0),FW2).

    Finally, from (2.15), we achieve that

    g([W0,W1],W2)=g(˜W0W1,W2)cos2ϑg(˜W1W0,W2)+2g(h(W0,W1),FTW2)+sin2ϑg(˜W0W1,W2)g(h(W0,φW1),FW2)g(h(W1,φW0),FW2).

    That is the result which we wanted.

    Corollary 3.1. The distribution Dζ is the totally geodesic foliation of a SSS M in NCM ˜M if and only if

    g(h(W0,W1),FTW2)=12{g(h(W0,φW1),FW2)+g(h(W1,φW0),FW2)},

    for all W0,W1Γ(Dζ) and W2Γ(Dϑ).

    Proof. From the total geodesic folition defintion, for every W0,W1Γ(Dζ), then W0W1Γ(Dζ). Putting this in the Eq (1.1), we get required result.

    Proof of Theorem 1.2

    By the property of Lie bracket, we have

    g([W2,W3],W0)=g(˜W2W3,W0)g(˜W3W2,W0).

    By (2.2), we get

    g([W2,W3],W0)=g(˜W2W3,W0)g(φ˜W3W2,φW0)+η(W0)g(˜W3ζ,W2).

    Using (2.11), we obtain

    g([W2,W3],W0)=g(˜W2W3,W0)+g((˜W3φ)W2,φW0)g(˜W3φW2,φW0)+η(W0)g(˜W3ζ,W2).

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

    g([W2,W3],W0)=g(˜W2W3,W0)g((˜W2φ)W3,φW0)g(˜W3TW2,φW0)g(˜W3FW2,φW0)+η(W0)g(˜W3ζ,W2).

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

    g([W2,W3],W0)=2g(˜W2W3,W0)g(˜W2φW3,φW0)+g(φˉW3TW2,W0)+g(h(φW0,W3),FW2)+η(W0)g(˜W3ζ,W2)

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

    g([W2,W3],W0)=2g(˜W2W3,W0)+g(h(φW0,W3),FW2)+g(φ˜W2TW3,W0)g(˜W2FW3,φW0)g((˜W3φ)TW2,W0)+g(˜W3φTW2,W0)+η(W0)g(˜W3ζ,W2).

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

    g([W2,W3],W0)=2g(˜W2W3,W0)+g(h(φW0,W3),FW2)+g(˜W2T2W3,W0)+g(˜W2FTW3,W0)+g(h(φW0,W2),FW3)cos2ϑg(˜W3W2,W0)g(AFTW2W0,W3)g(PW2TW3,W0)g(PW3TW2,W0)+η(W0)g(˜W3ζ,W2).

    Using (2.13) and (2.6), we derive

    g([W2,W3],W0)=2g(˜W2W3,W0)+g(h(φW0,W3),FW2)+g(h(φW0,W2),FW3)g(AFTW2W0,W3)g(AFTW3W0,W2)cos2ϑg(˜W3W2,W0)cos2ϑg(˜W2W3,W0))+η(W0)g(˜W3ζ,W2)g(PW2TW3,W0)g(PW3TW2,W0). (3.1)

    Now we compute last two terms of above equation by using property P (see [13]) as follows

    g(PW2TW3,W0)=g(TW3,PW2W0)=(φW3,PW2W0)=g(W3,φPW2W0)=g(W3,PW2φW0)=(PW2W3,φW0).

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

    sin2ϑg([W2,W3],W0)=2g(˜W2W3,W0)+g(h(φW0,W3),FW2)+g(h(φW0,W2),FW3)g(AFTW2W0,W3)g(AFTW3W0,W2)+η(W0)g(˜W3ζ,W2)2cos2ϑg(ˉW2W3,W0))g(PW2W3+PW3W2,φW0).

    Applying the structure equation (2.12), we obtain

    sinθg([W2,W3],W0)=2sin2ϑg(˜W2W3,W0)+g(h(φW0,W3),FW2)+g(h(φW0,W2),FW3)g(AFTW2W0,W3)g(AFTW3W0,W2)+η(W0)g(˜W3ζ,W2).

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

    An application of Theorem 1.2, we introduce

    Corollary 3.2. The slant distribution Dϑ of SSS M in a NCM ˜M, is the totally geodesic foliation in M if and only if

    g(h(φW0,W3),FZ2)+g(h(φW0,W2),FW3)=g(AFTW2W0,W3)+g(AFTW3W0,W2).

    for any W0Γ(Dζ) and W2,W3Γ(Dϑ).

    Proof. From the definition totally geodesic folition, for every W2,W3Γ(Dϑ), then W2W3Γ(Dϑ). Inserting this in the Eq (1.2), we get required result.

    Suppose M=M1×fM2 is the smooth Riemannian manifold associating to warped product metric g=g1+f2g2 in which f is a WF on M. This idea was given in [12] and derived the formula;

    W2W0=W0W2=(W0lnf)W2, (4.1)

    For all W0,W1Γ(TM1) and W2,W3Γ(TM2). The gradient of lnf is denoted by lnf and defined as:

    g(lnf,W0)=W0lnf.

    Remark 4.1. The warped product manifold M=M1×fM2 becomes trivial if f is a constant function.

    Remark 4.2. The base M1 is totally geodesic and fiber M2 is totally umbilical in WPM M=M1×fM2.

    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:

    (i) M=Mϑ×fMT

    (ii) M=MT×fMϑ,

    where Mϑ and MT 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=Mϑ×fMT does not exist in a NCM ˜M. Therefore, we shall consider non-trivial WPSSS in terms of M=MT×fMϑ in a NCM ˜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=MT×fMϑ in a NCM ˜M has the coming relations

    (i) ζlnf=0,

    (ii) g(h(W0,W1),FW2)=0,

    (iii) g(h(W2,φW0),FW2)=(W0λ)||W2||2,

    (iv) g(PW0W2,TW2)=2g(h(W0,W2),FTW2),

    for all W2Γ(TMϑ) and W0,W1Γ(TMT).

    There is a another interesting lemma.

    Lemma 4.2. [22] A WPSSS M=MT×fMϑ in a NCM ˜M satisfying the relation

    g(h(W0,W2),FTW2)=g(h(W0,TW2),FW2),=13(W0lnf)cos2ϑ||W2||2,

    for all W2Γ(TMϑ and W0Γ(TMT).

    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=MT×fMϑ is a WPSSS in a NCM ˜M with Mϑ and MT are proper slant and invariant submanifolds of ˜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 Dζ is

    2sin2ϑg(W0W1,W2)=g(AFW2φW0,W1)+g(AFW2φW1,W0)2g(AFTW2W0,W1).

    for W2Γ(Dϑ) and W0,W1Γ(Dζ). Thus relation (1.3)(i)-(ii) imply that

    2sin2ϑg(W0W1,W2)=(W0λ)g(W2,W1)+(W1λ)g(W0,W2)23cos2ϑ(W0λ)g(W1,W2).

    which implies that

    sin2ϑg(W0W1,W2)=0.

    As we have seen that Dϑ is proper slant then sinϑ0, which means that W0W1Γ(Dζ) for any W0,W1Γ(Dζ), it is concluded that the leaves of Dζ are totally geodesic in M into an immersion of ˜M. Also from Theorem 1.2, it can be seen that the slant distribution Dϑ is integrable if and only if the following equation holds

    sin2ϑg(W2W3,W0)=12{g(h(W0,W2),FTW3)+g(h(W0,W3),FTW2)g(h(φW0,W2),FW3)g(h(φW0,W3),FW2)+η(W0)g(˜W3ζ,W2)}.

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

    2sin2ϑg(W2W3,W0)=g(AFTW2W0,W3))g(AFW2φW0,W3)+g(AFTW3W0,W2)g(AFW3φW0,W2)+η(W0)g(˜W3ζ,W2).

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

    2sin2ϑg(W2W3,W0)=23cos2ϑ(W0λ)g(W2,W3)2(W0λ)g(W3,W2)+η(W0)g(˜W3ζ,W2).

    It implies that

    sin2ϑg(W2W3,W0)=(cos2ϑ33)(W0λ)g(W2,W3)+12η(W0)g(˜W3ζ,W2). (4.2)

    Moreover, we also assumed that Dϑ to be integrable, thus we consider a integral manifold Mϑ of Dϑ, that is, Mϑ is leaf of integrable distribution Dϑ and hϑ is the second fundamental form resulting from immersion Mϑ in M. Therefore, equation (4.2) becomes

    g(hϑ(W2W3,W0)=13{cot2ϑ3csc2ϑ}(W0λ)g(W2,W3)+12csc2ϑη(W0)g(˜W3ζ,W2). (4.3)

    By interchanging W2 and W3 in above equation, we obtain

    g(hϑ(W2W3,W0)=13{cot2ϑ3csc2ϑ}(W0λ)g(W2,W3)+12csc2ϑη(W0)g(˜W2ζ,W3). (4.4)

    Using (4.3) and (4.4), we attain

    2g(hϑ(W2W3,W0)=23{cot2ϑ3csc2ϑ}(W0λ)g(W2,W3)+12csc2ϑη(W0){g(˜W2ζ,W3)+g(˜W3ζ,W2)}.

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

    g(hθ(W2W3,W0)=13{cot2ϑ3csc2ϑ}g(W2,W3)g(λ,W0),

    which leads to

    hϑ(W2,W3)=13{cot2ϑ3csc2ϑ}g(W2,W3)λ.

    Hence, it is concluded that Mϑ is totally umbilical in M having the following mean curvature vector

    Hϑ=13(cot2ϑ3csc2ϑ)λ,

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

    g(ϑW2λ,W0)=g(λ,ϑW0W2). (4.5)

    Furthermore, λΓ(TMT) as MT is a totally geodesic in M by Remark 4.2, consequently

    ϑW0W2Γ(TMϑ),

    for any W0Γ(Dζ) and W2Γ(Dϑ). By equation (4.5), we get

    g(ϑW2λ,W0)=0.

    This shows that the mean curvature vector Hθ of Mϑ is parallel reciprocal to the normal connection ϑ of Mϑ in M. Therefore, the spherical condition is fulfilled, such that Mϑ 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=MT×fMϑ, in which MT and Mϑ are the integral manifold of Dζ and Dϑ, in the same order. It completes this proof.

    Proof of Theorem 1.4

    Thus from Lemma 4.2, for all W2Γ(TMϑ and W1Γ(TMT), we get

    g(h(W1,W2),FTW2)=g(h(W1,TW2),FW2)+13(W1lnf)cos2ϑ||W2||2.

    As we assumed that M is mixed totally geodesic submanifold, that is h(W1,W2)=h(W1,TW2)=0, for all W2Γ(Dϑ) and W0Γ(Dζ).

    This implies cos2ϑ(W1lnf)||W2||2=0. But M is proper slant submanifold, then cosϑ0, that is (W1lnf)||W2||2=0. Hence (W1lnf)=0, i.e, the warping function f is a constant on M and the proof is completed via Remark 4.1.

    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.

    There is no conflict of interest between the authors.



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