# A Study of Doubly Warped Product Immersions in a Nearly Trans-Sasakian Manifold with Slant Factor

Authors: Ali H. Alkhaldi; Aliya Naaz Siddiqui; Kamran Ahmad; Akram Ali
Journal: Advances in Mathematical Physics
 (2021)

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## Abstract

In this article, we discuss the de Rham cohomology class for bislant submanifolds in nearly trans-Sasakian manifolds. Moreover, we give a classification of warped product bislant submanifolds in nearly trans-Sasakian manifolds with some nontrivial examples in the support. Next, it is of great interest to prove that there does not exist any doubly warped product bislant submanifolds other than warped product bislant submanifolds in nearly trans-Sasakian manifolds. Some immediate consequences are also obtained.


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## Body

## 1. Introduction and Motivations

The most inventive topic in the field of differential geometry currently is the theory of warped product manifolds. These manifolds are the most fruitful and natural generalization of Riemannian product manifolds. Due to the important roles of the warped product in mathematical physics and geometry, it has become the most active and interesting topic for researchers, and many nice results are available in the literature (see [1–3]).Chen [4, 5] initiates the concept of warped product submanifolds by proving the nonexistence result of warped product CR-submanifolds of type N⊥×fNT in Kähler manifolds, where N⊥ and NT are anti-invariant and invariant submanifolds, respectively. Moreover, he considers warped product CR-submanifolds of type NT×fN⊥ and gives an inequality involving a warping function f and the squared norm of the second fundamental form h2.On the other hand, the concept of ordinary warped products can be extended to doubly warped products. By using this generalization, Sahin [6] shows that there exist no doubly warped product CR-submanifolds in Kähler manifolds other than warped product CR-submanifolds. He also investigates the existence of doubly twisted product CR-submanifolds in the same ambient. Many geometers have obtained several results on warped products and doubly warped products [7–12].The concept of bislant submanifolds is defined by Cabrerizo et al. [13] as the natural generalization of contact CR-, slant, and semislant submanifolds. Such submanifolds generalize invariant, anti-invariant, and pseudoslant submanifolds as well. Recently, the warped product bislant submanifolds in nearly trans-Sasakian manifolds is studied by Siddiqui et al. in [1]. They obtain several inequalities for the squared norm of the second fundamental form in terms of a warping function f.In this paper, firstly, we discuss the de Rham cohomology class for closed bislant submanifolds in a nearly trans-Sasakian manifold. Secondly, in view of embedding theorem of Nash [14], we study an isometric immersion of a warped product bislant submanifold into an arbitrary nearly trans-Sasakian manifold. Then, we investigate the existence of doubly warped products in the same ambient.


## 2. Nearly Trans-Sasakian Manifolds and their Submanifolds

Definition 1 (see [15]).
A2m+1-dimensional differentiable manifold N¯ is said to have an almost contact structure ϕ,ξ,η,g if there exists on N¯, where
(i)
a tensor fieldϕ of type 1,1(ii)
a vector fieldξ(iii)
a 1-formη(iv)
a Riemannian metricg
such that(1)ϕ2=−I+η⊗ξ,ϕξ=0,ηξ=1,η∘ϕ=0,ηX=gX,ξ,gϕX,ϕY=gX,Y−ηXηY,gϕX,Y+gX,ϕY=0,
for anyX,Y∈TN¯.The covariant derivative of the tensor fieldϕ is given by
(2)∇¯XϕY=∇¯XϕY−ϕ∇¯XY,for anyX,Y∈TN¯.In 2000, Gherghe introduced a notion of nearly trans-Sasakian structure of typeα,β, which generalizes the trans-Sasakian structure. A nearly trans-Sasakian structure of type α,β is called nearly α-Sasakian (resp. nearly β-Kenmotsu) if β=0 (resp. α=0).Definition 2 (see [16]).
An almost contact metric structureϕ,ξ,η,g on N¯ is called a nearly trans-Sasakian structure if
(3)∇¯XϕY+∇¯YϕX=α2gX,Yξ−ηYX−ηXY−βηYϕX+ηXϕY,
for anyX,Y∈TN¯.Remark 3.
(i)
A nearly trans-Sasakian structure of typeα,β is(a)
nearly Sasakian if β=0,α=1 [17](b)
nearly Kenmotsu if α=0,β=1 [18](c)
nearly cosymplectic if α=β=0 [19](ii)
Remark that every Kenmotsu manifold is a nearly Kenmotsu manifold but the converse is not true. Also, a nearly Kenmotsu manifold is not a Sasakian manifold. On another hand, every nearly Sasakian manifold of dimension greater than five is a Sasakian manifold.We put dimN=n and dim N¯=2m+1. The Riemannian metric for N and N¯ is denoted by the same symbol g. Let TN and T⊥N denote the Lie algebra of the vector field and set of all normal vector fields on N, respectively. The operator of covariant differentiation with respect to the Levi-Civita connection in N and N¯ is denoted by ∇ and ∇¯, respectively. The Gauss and Weingarten formulae are respectively given as [15]
(4)∇¯XY=∇XY+hX,Y,(5)∇¯XV=−AVX+∇X⊥Y,for anyX,Y∈TN and V∈T⊥N. Here, h is the second fundamental form, A is the shape operator, and ∇⊥ is the operator of covariant differentiation with respect to the linear connection induced in the normal bundle T⊥N.The second fundamental form and the shape operator are related as [15]
(6)ghX,Y,V=gAVX,Y,for anyX,Y∈TN and V∈T⊥N. Here, g denote the induced metric on N as well as the Riemannian metric on N¯.Letx∈N and E1,⋯,En be a local orthonormal frame of TxN and En+1,⋯,E2m+1 be a local orthonormal frame of Tx⊥N. The mean curvature vector H of a submanifold N at x is given by [15]
(7)H=1n∑i=1nhEi,Ei.A submanifoldN of N¯ is said to be [15]
(i)
totally umbilical if hX,Y=gX,YH, for any X,Y∈TN(ii)
totally geodesic if hX,Y=0, for any X,Y∈TN(iii)
minimal if H=0, that is, trace h≡0For anyX∈TN, we put [15]
(8)ϕX=PX+FX,wherePX=tangentϕX and FX=normalϕX. Then P is an endomorphism of TN, and F is the normal bundle valued 1-form on TN. In the same way, for any V∈T⊥N, we put [15]
(9)ϕV=BV+CV,whereBV=tangentϕV and CV=normalϕV. It is easy to see that P and C are skew-symmetric and
(10)gFX,V=−gX,BV,for anyX∈TN and V∈T⊥N.Definition 4.
A submanifoldN of an almost contact metric manifold N¯ is said to be invariant if F≡0, that is, ϕX∈TN, and anti-invariant if P≡0, that is, ϕX∈T⊥N, for any X∈TN.In contact geometry, Lotta introduced slant immersions as follows [20].Definition 5.
LetN be a submanifold of an almost contact metric manifold N¯. For each nonzero vector X∈TxN−ξx and x∈N, the angle θp∈0,π/2 between ϕX and PX is called slant angle of N. If slant angle is constant for each X∈TxN−ξx, then the submanifold is called the slant submanifold.For slant submanifolds, the following facts are known:(11)P2X=cos2θ−X+ηXξ,gPX,PY=cos2θgX,Y−ηYηX,(12)gFX,FY=sin2θgX,Y−ηYηX,for anyX,Y∈TN. Here, θ is slant angle of N.Remark 6.
If we assume(i)
θ=0, then N is an invariant submanifold(ii)
θ=π/2, then N is an anti-invariant submanifold(iii)
θp∈0,π/2, then N is a proper slant submanifoldThere are some other important classes of submanifolds which are determined by the behavior of tangent bundle of the submanifold under the action of an almost contact metric structureϕ of N¯ [1]:
(i)
A submanifoldN of N¯ is called a contact CR-submanifold of N¯ if there exists a differentiable distribution D on N whose orthogonal complementary distribution D⊥ is anti-invariant(ii)
A submanifoldN of N¯ is called a semislant submanifold of N¯ if there exists a pair of orthogonal distributions D and Dθ such that D is invariant and Dθ is proper slant(iii)
A submanifoldN of N¯ is called pseudoslant submanifold of N¯ if there exists a pair of orthogonal distributions D⊥ and Dθ such that D⊥ is anti-invariant and Dθ is proper slantDefinition 7 (see [13]).
A submanifoldN of an almost contact metric manifold M¯ is said to be a bislant submanifold if there exists a pair of orthogonal distributions Dθ1 and Dθ2 of N such that
(13)TN=Dθ1⊕Dθ2⊕ξ.(i)
PDθ1⊥Dθ2 and PDθ2⊥Dθ1(ii)
Each distributionDθi is slant with slant angle θi for i=1,2Remark 8.
If we assume(i)
θ1=0 and θ2=π/2, then N is a CR-submanifold(ii)
θ1=0 and θ2≠0,π/2, then N is a semislant submanifold(iii)
θ1=π/2 and θ2≠0,π/2, then N is a pseudoslant submanifold(iv)
θ1,θ2∈0,π/2, then N is a proper bislant submanifold
For a bislant submanifoldN of an almost contact metric manifold, the normal bundle of N is decomposed as
(14)T⊥N=FDθ1⊕FDθ2⊕μ,
whereμ is a ϕ-invariant normal subbundle of N.


## 3. Cohomology Class for Bislant Submanifolds of Nearly Trans-Sasakian Manifolds

Chen [21] introduces a canonical de Rham cohomology class for a closed CR-submanifold in a Kähler manifold. So, in this section, we discuss the de Rham cohomology class for a closed bislant submanifold of a nearly trans-Sasakian manifold N¯,ϕ,ξ,η,g with minimal horizontal distribution Dθ1⊕ξ. We put dimN=m and dimDθ1⊕ξ=2a+1. Let us assume the following:
(i)
E1,⋯,Ea,Ea+1=secθ1PE1,⋯,E2a=secθ1PEa,E2a+1=ξ,E2a+2=E1∗,⋯,E2a+b+1=Eb∗,E2a+b+2=Eb+1∗=secθ2PE1∗,⋯,Em=E2a+2b+1=E2b∗=secθ2PEb∗ is a local orthonormal frame of N(ii)
E1,⋯,Ea,Ea+1=secθ1PE1,⋯,E2a=secθ1PEa,E2a+1=ξ is a local orthonormal frame of Dθ1⊕ξ(iii)
E2a+2=E1∗,⋯,E2a+b+1=Eb∗,E2a+b+2=Eb+1∗=secθ2PE1∗,⋯,Em=E2a+2b+1=E2b∗=secθ2PEb∗ is a local orthonormal frame of Dθ2We chooseς1,⋯,ς2a+1,ς2a+2,⋯,ςm as the dual frame of 1-forms to the above local orthonormal frame. Then, we define a 2a+1-form ϖ on N by ϖ=ς1∧ς2∧⋯∧ς2a+1. It is globally defined on N. In the same way, we again define a m−2a−1-form Ω on N by Ω=ς2a+2∧ς2a+3∧⋯∧ςm, which is globally defined on N.We prepare some preliminary lemmas.Lemma 9.
LetN be a submanifold of an arbitrary nearly trans-Sasakian manifold N¯, then
(15)∇XPY−AFYX−P∇YX−2BhX,Y+∇YPX−AFXY−P∇XY=α2gX,Yξ−ηYX−ηXY−βηYPX+ηXPY,(16)hX,PY+∇X⊥FY−F∇XY−2ChX,Y+hY,PX+∇Y⊥FX−F∇YX=−βηYFX+ηXFY,
for anyX,Y∈TN.Proof.
For any vector fieldsX,Y∈TN, making use of the structure equation and (2), we obtain
(17)∇¯XϕY−ϕ∇¯XY+∇¯YϕX−ϕ∇¯YX=α2gX,Yξ−ηYX−ηXY−βηYϕX+ηXϕY,
which gives(18)∇XPY+hPY,X−AFYX+∇X⊥FY−P∇XY−2BhX,Y−F∇XY−2ChX,Y+∇XPY+hPX,Y−AFXY+∇Y⊥FX−P∇YX−F∇YX=α2gX,Yξ−ηYX−ηXY−βηYPX+ηXPY+ηYFX+ηXFY.
Comparing the tangential and normal components of the above equation, we get the desired relations (15) and (16).
The next lemma gives the integrability condition of slant distributionDθ2.☐Lemma 10.
LetN be a bislant submanifold of an arbitrary nearly trans-Sasakian manifold N¯,ϕ,ξ,η,g. Then, slant distribution Dθ2 is integrable if and only if
(19)−2F∇YX+hX,PY+hY,PX−2ChX,Y+∇X⊥FY+∇Y⊥FX∈FDθ2,
for anyX,Y∈Dθ2.Proof.
Making use of Lemma9, we obtain
(20)gFX,Y,FZ=−2gF∇YX,FZ+ghX,PY,FZ+ghY,PX,FZ−g2ChX,Y,FZ+g∇X⊥FY,FZ+g∇Y⊥FX,FZ,
for anyZ∈Dθ1⊕ξ. Thus, the assertion follows from the fact that FDθ1 and FDθ2 are mutually perpendicular. In this way, we proved the integrability condition of slant distribution Dθ2.☐We prove the following.Theorem 11.
For any closed bislant submanifoldN of an arbitrary nearly trans-Sasakian manifold N¯,ϕ,ξ,η,g with minimal Dθ1⊕ξ and
(21)−2F∇YX+hX,PY+hY,PX−2ChX,Y+∇X⊥FY+∇Y⊥FX∈FDθ2,
for anyX,Y∈Dθ2, the 2a+1-form ϖ is closed and defines a canonical de Rham cohomology class ϖ∈H2a+1N,ℝ, where dimDθ1⊕ξ=2a+1.
Moreover, the cohomology groupH2a+1N,ℝ is nontrivial if Dθ2 is minimal and Dθ1⊕ξ is integrable.Proof.
From the definition ofϖ, we have dϖ=∑i=12a+1−1i−1ς1∧⋯∧dςi∧⋯∧ς2a+1, which implies that dϖ=0 if and only if
(22)dϖX2,Y2,X1,⋯,X2a=0,(23)dϖX2,X1,⋯,X2a+1=0,
for anyX2,Y2∈Dθ2 and X1,⋯,X2a+1∈Dθ1⊕ξ. Thus, by simple computation, we find that (22) is satisfied if and only if Dθ2 is integrable. On the other hand, (23) is satisfied if and only if Dθ1⊕ξ is minimal. However, the integrability condition of Dθ2 holds due to Lemma 10, and by the hypothesis of the theorem, we have Dθ1⊕ξ is minimal. Hence, the form ϖ is closed. It defines a canonical de Rham cohomology class ϖ∈H2a+1N,ℝ.
Next, we prove that the cohomology classϖ is nontrivial. Since Dθ2 is minimal and Dθ1⊕ξ is integrable, then in this case, we need to show that ϖ is harmonic. By definition of Ω and the similar argument for ϖ, we see that dΩ=0, that is, Ω is closed, if Dθ1⊕ξ is integrable and Dθ2 is minimal. This further proves that δϖ=0, that is, ϖ is coclosed. From dϖ=0,δϖ=0, and N is a closed submanifold, we deduce that ϖ is harmonic 2a+1-form. Hence, the cohomology group H2a+1N,ℝ is nontrivial if Dθ2 is minimal and Dθ1⊕ξ is integrable.☐


## 4. Warped Product Bislant Submanifolds

Definition 12 (see [22]).
LetN1,g1 and N2,g2 be two Riemannian manifolds and f>0 be a differentiable function on N1. Consider two projections on N1×N2, ρ:N1×N2⟶N1 and δ:N1×N2⟶N2. The projection maps given by ρp,q=p and δp,q=q for p,q∈N1×N2. Then, the warped product N=N1×fN2 is the product manifold N1×N2 equipped with the Riemannian structure such that
(24)gX,Y=g1ρ∗X,ρ∗Y+f∘ρ2g2δ∗X,δ∗Y,
for anyX,Y∈TN, where ∗ is the symbol for the tangent maps. The function f is called the warping function of N.Example 13.
A surface of revolution is a warped product manifold.Example 14.
The standard space-time models of the universe are warped products as the simplest models of neighbourhoods of stars and black holes.Remark 15.
In particular, a warped product manifold is said to be trivial if its warping function is constant. In such a case, we call the warped product manifold a Riemannian product manifold. IfN=N1×fN2 is a warped product manifold, then N1 is totally geodesic and N2 is totally umbilical submanifold of N [22].
LetN=N1×fN2 be a warped product manifold with a warping function f. Then,
(25)∇XZ=∇ZX=XlnfZ,
for anyX∈TN1 and Z∈TN2, where ∇lnf is the gradient of lnf and ∇ and ∇N2 denote the Levi-Civita connections on N and N2, respectively.The definition of warped product bislant submanifolds in a nearly trans-Sasakian manifold is as follows.Definition 16.
A warped productN1×fN2 of two slant submanifolds N1 and N2 of a nearly trans-Sasakian manifold N¯ is called a warped product bislant submanifold.Remark 17.
A warped product bislant submanifoldN1×fN2 is called proper if N1 and N2 are proper slant in N¯. Otherwise, the warped product bislant submanifold N1×fN2 is called nonproper.For a warped product bislant submanifold in a nearly trans-Sasakian manifold such thatξ∈TN1, we have the following result.Theorem 18.
LetN=N1×fN2 be a warped product bislant submanifold with bislant angles θ1,θ2 in a nearly trans-Sasakian manifold N¯ such that ξ∈TN1. If, for any X1∈TN1 and X2,Y2∈TN2,
(26)ghX1,X2,FY2=ghX1,Y2,FX2,
holds, then one of the following cases must occur:(i)
N is a warped product pseudoslant submanifold such that N2 is a totally real submanifold N⊥ of N¯(ii)
IfN¯ is nearly Sasakian manifold, that is, β=0, then N is a Riemannian product(iii)
Ifβ≠0, then βηX1=−X1lnfProof.
For any vector fieldsX1∈TN1 and X2,Y2∈TN2, we have
(27)ghX1,X2,FY2=g∇¯X1X2,ϕY2−g∇X1X2,PY2=g∇¯X1ϕX2,Y2−g∇¯X1ϕX2,Y2−X1lnfgX2,PY2.
On the other hand, we have(28)ghX1,X2,FY2=g∇¯X2X1,ϕY2−g∇X2X1,PY2=g∇¯X2ϕX1,Y2−g∇¯X2ϕX1,Y2−X1lnfgX2,PY2.
By adding (27) and (28), we get
(29)2ghX1,X2,FY2=ghX1,Y2,FX2+ghX2,Y2,FX1−PlnfgX2,Y2−X1lnfgX2,PY2−αηX1gX2,Y2+βηX1gPX2,Y2.
InterchangingX2 by Y2 in (29), we find
(30)2ghX1,Y2,FX2=ghX1,X2,FY2+ghX2,Y2,FX1−PlnfgX2,Y2−X1lnfgY2,PX2−αηX1gX2,Y2+βηX1gPY2,X2.
By subtracting (30) from (29) and by applying our assumption, we obtain
(31)gPX2,Y2X1lnf+βηX1=0.
ForY2=PY2, we get
(32)cos2θ2gY2,X2X1lnf+βηX1=0.
From the last expression, any one of the following holds: ifβ=0, then f is constant, or if β≠0, then βηX1=−X1lnf or θ2=π/2. Thus, our assertions follow.
Now, we have the following theorem for a warped product bislant submanifold in a nearly trans-Sasakian manifold such thatξ∈TN2.☐Theorem 19.
LetN=N1×fN2 be a warped product bislant submanifold with bislant angles θ1,θ2 in a nearly trans-Sasakian manifold N¯ such that ξ∈TN2. If, for any X1∈TN1 and X2,Y2∈TN2,
(33)ghX1,X2,FY2=ghX1,Y2,FX2,
holds, then one of the following cases must occur:(i)
N is a warped product pseudoslant submanifold such that N2 is a totally real submanifold N⊥ of N¯(ii)
N is a Riemannian productProof.
For any vector fieldsX1∈TN1 and X2,Y2∈TN2, we have
(34)ghX1,X2,FY2=g∇¯X1X2,ϕY2−g∇X1X2,PY2=g∇¯X1ϕX2,Y2−g∇¯X1ϕX2,Y2.
On the other hand, we have(35)ghX1,X2,FY2=g∇¯X2X1,ϕY2−g∇X2X1,PY2=g∇¯X2ϕX1,Y2−g∇¯X2ϕX1,Y2.
By adding (34) and (35), we get
(36)2ghX1,X2,FY2=ghX1,Y2,FX2+ghX2,Y2,FX1−PlnfgX2,Y2−X1lnfgX2,PY2.
InterchangingX2 by Y2 in (36), we find
(37)2ghX1,Y2,FX2=ghX1,X2,FY2+ghX2,Y2,FX1−PlnfgX2,Y2−X1lnfgY2,PX2.
By subtracting (37) from (36) and by applying our assumption, we obtain
(38)X1lnfgPX2,Y2=0.
ForY2=PY2, we get
(39)cos2θ2X1lnfgY2,X2−ηX2ηY2=0.
Therefore, eitherf is constant or cosθ2=0 holds. Consequently, either N is a Riemannian product or θ2=π/2. In the latter case, N is a warped product pseudoslant submanifold.☐We give some nontrivial examples of warped product bislant submanifold of the formN=Nθ×fN⊥ whose bislant angles θ1≠0,π/2 and θ2=π/2. Such warped product bislant submanifolds are called pseudoslant submanifolds.Example 20.
Letℂ4 be the complex Euclidean space with its usual Kähler structure and the real global coordinates x1,y1,x2,y2,x3,y3,x4,y4 and N¯=ℝ×fℂ4 be a warped product manifold between the product real line of ℝ and the complex space ℂ4. Let <,> be the Euclidean metric tensor of ℝ9. An almost contact structure ϕ of N¯ is defined by
(40)ϕ∂∂xi=∂∂yi,ϕ∂∂yj=−∂∂xj,ϕ∂∂t=0,1≥i,j≥4
such that(41)ξ=et∂∂t,η=etdt,g=et<,>.
On the other hand, we define a submanifoldN by immersion g as follows:
(42)gu,v,w,s,t=u,v,0,0,vcosr,vsinr,scosw,ssinw,t.
Therefore, it is easy to choose tangent bundle ofN which is spanned by the following:
(43)X1=∂∂x1,X2=cosr∂∂y1+sinr∂∂y2,X3=cosw∂∂y3+sinw∂∂y4,X5=∂∂z.
Thus,Dθ1=SpanX1,X2 is a slant distribution with slant angle π/4. Also, it is easy to verify that Dθ2=SpanX3,X4 is a totally real distribution. Hence, the submanifold N defined by f is a bislant submanifold, which is tangent to the structure vector ξ and whose bislant angles satisfy θ1≠0,π/2 and θ2=π/2. It is easy to check that the distributions Dθ1 and Dθ2 are integrable. Then, it can be verified that N=Nθ×fN⊥ is a warped product bislant submanifold of N¯ with warping function f=et, t∈ℝ.Example 21.
We consider any submanifoldN in a nearly trans-Sasakian manifold ℝ7(44)fu,v,w,q=ucosv,wcosv,usinv,wsinv,w−u,w+u,q.
The tangent bundle ofN is spanned by
(45)E1=cosv∂∂x1+sinv∂∂x2−∂∂x3+∂∂y3,E2=−usinv∂∂x1+ucosv∂∂x2−wsinv∂∂y1+wcosv∂∂y2,E3=∂∂x3+cosv∂∂y1+sinv∂∂y2+∂∂y3,E4=∂∂q.
Furthermore, we have(46)ϕE1=cosv∂∂y1+sinv∂∂y2−∂∂y3−∂∂x3,ϕE2=−usinv∂∂y1+ucosv∂∂y2+wsinv∂∂x1−wcosv∂∂x2,ϕE3=∂∂y3−cosv∂∂x1−sinv∂∂x2−∂∂x3,ϕE4=0.
It is easy to check thatϕE2 is orthogonal to TN. Then, the proper slant and anti-invariant distributions of N are respectively defined by Dθ=SpanE1,E3 with slant angle θ=arccos1/3 and D⊥=SpanE2. Also, E4=ξ is tangent to Dθ. Hence, f defines a proper 4-dimensional pseudoslant submanifold (bislant submanifold with bislant angles arccos1/3,π/2) N in ℝ7. It is easy to check that the distributions Dθ⊕ξ and D⊥ are integrable.Now, we assume thatNθ and N⊥ are the integral manifolds of Dθ and D⊥, respectively. Then, it follows from Definition 12 and (44) that the induced metric tensor g of N is given by
(47)g=cos2v+sin2v+2du2+u2sin2v+u2cos2v+w2sin2v+w2cos2vdv2+cos2v+sin2v+2dw2+dq2=3du2+dw2+dq2+u2+w2dv2=g1+g2,whereg1=3du2+dw2+dq2 and g2=u2+w2dv2 are respectively the metric tensors of Nθ and N⊥. As a consequence, N=Nθ×fN⊥ is a warped product pseudoslant submanifold of ℝ7 with a warping function, that is, f=u2+w2 such that ξ is tangent to Nθ.


## 5. Doubly Warped Product Bislant Submanifolds

In general, doubly warped products can be considered as a generalization of warped products.Definition 22 (see [23, 24]).
LetN1,g1 and N2,g2 be Riemannian manifolds. A doubly warped product N,g is a product manifold which is of the form N=f2N1×f1N with the metric g=f12g1⊕f22g2, where f1:N1×N2⟶0,∞ and f2:N1×N2⟶0,∞ are smooth maps. More precisely, if ρ:N1×N2⟶N1 and δ:N1×N2⟶N2 are natural projections, the metric g is defined by
(48)gX,Y=f2∘δ2g1ρ∗X,ρ∗Y+f1∘ρ2g2δ∗X,δ∗Y,
for anyX,Y∈TN, where ∗ is the symbol for the tangent maps. The functions f1 and f2 are called the warping functions of N.Remark 23.
If we assume(i)
eitherf1≡1 or f2≡1, but not both, then we obtain a warped product(ii)
bothf1≡1 and f2≡1, then we have a product manifold(iii)
neitherf1 nor f2 is constant, then we have a nontrivial doubly warped productFor doubly warped product manifoldN=f2N1×f1N with warping functions f and g, we have the following:
(49)∇YX=∇XY=Ylnf1X+Xlnf2Y,for anyX∈TN1 and Y∈TN2.Now, we define the notion of doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds as follows.Definition 24.
The doubly warped product of two slant submanifolds,f2N1×f1N2, is called the doubly warped product bislant submanifold of slant submanifolds N1 and N2 with slant angles θ1 and θ2, respectively, of a nearly trans-Sasakian manifold with warping functions f1 and f2 if only depend on the points of N1 and N2, respectively.First we have the following theorem for doubly warped product submanifoldsN=f2N1×f1N2 in nearly trans-Sasakian manifolds such that ξ∈TN1.Theorem 25.
LetN=f2N1×f1N2 be a doubly warped product submanifold in a nearly trans-Sasakian manifold N¯, where N1 and N2 are Riemannian submanifolds of N¯ and ξ∈TN1. Then, N is a warped product bislant submanifold of type N1×f1N2 if and only if
(50)ghX,Y,FX=ghX,X,FY,
for anyX∈TN1 and Y∈TN2.Proof.
From Lemma9, we get
(51)∇XPY−AFYX−P∇YX−2BhX,Y+∇YPX−AFXY−P∇XY=−αηXY−βηXPY,
for anyX∈TN1 and Y∈TN2. Applying (49), we derive
(52)PYlnf2X−Ylnf2PX−Xlnf1PY+PXlnf1Y−AFYX−2BhX,Y−AFXY=−αηXY−βηXPY.
Taking the inner product withX∈TN1, we obtain
(53)PYlnf2X2−ghX,X,FY−2gBhX,Y,X−ghY,X,FX=0.
Using relation (10) in the above equation, we get
(54)PYlnf2X2=ghX,X,FY−ghY,X,FX=0.
Thus, from (54), we conclude that PYlnf2=0 if and only if
(55)ghX,Y,FX=ghX,X,FY,
for anyX∈TN1 and Y∈TN2. PYlnf2=0 shows that f2 is constant, that is, f2 depends only on the points of N1. Thus, it follows that N is a warped product bislant submanifold of type N1×f1N2. This proves the theorem completely.☐Secondly, we prove the following theorem for doubly warped product bislant submanifoldsN=f2N1×f1N2 in nearly trans-Sasakian manifolds such that ξ∈TN2.Theorem 26.
LetN=f2N1×f1N2 be a doubly warped product bislant submanifold in a nearly trans-Sasakian manifold N¯, where N1 and N2 are proper slant submanifolds with respect to θ1 and θ2, respectively, and ξ∈TN2. Then, N is a warped product bislant submanifold of type N1×f1N2 if and only if
(56)ghX,Y,FY=ghY,Y,FX,
for anyX∈TN2 and Y∈TN1.Proof.
For any vector fieldsX∈TN2 and Y∈TN1, we have
(57)ghPX,Y,FY=g∇¯YPX,ϕY=−gϕ∇¯YPX,Y=g∇¯YϕPX,Y−g∇¯YϕPX,Y=−g∇¯YϕY,PX−g∇¯YP2X,Y−g∇¯YFPX,Y=cos2θ1g∇YX,Y+ghY,Y,FPX=cos2θ1Xlnf2Y2+ghY,Y,FPX.
ReplacingX by PX in the last relation, we obtain
(58)PXlnf2Y2=ghY,Y,FX−ghX,Y,FY.
Thus, from (54), we conclude that PXlnf2=0 if and only if
(59)ghY,Y,FX=ghX,Y,FY,
for anyX∈TN2 and Y∈TN1.
PXlnf2=0 implies that f2 is constant, that is, f2 depends only on the points of N1. Hence, N is a warped product bislant submanifold of type N1×f1N2. This proves the theorem completely.☐


## 6. Conclusion

From Theorems25 and 26, we conclude that there exist no doubly warped product bislant submanifolds in nearly trans-Sasakian manifolds, other than warped product bislant submanifolds, under some additional conditions.


## 7. Some Applications of Theorem25 for Different Kinds of Ambient Manifolds

LetN=f2N1×f1N2 be a doubly warped product submanifold, where N1 and N2 are Riemannian submanifolds of N¯ and ξ∈TN1. The following corollaries are the immediate consequences of Theorem 25.Corollary 27.
There does not exist any doubly warped product submanifoldN=f2N1×f1N2 in a nearly Sasakian manifold N¯, other than the warped product bislant submanifold, if and only if (50) holds.Corollary 28.
There does not exist a doubly warped product submanifoldN=f2N1×f1N2 in a nearly Kenmotsu manifold N¯, other than the warped product bislant submanifold, if and only if (50) holds.Corollary 29.
There does not exist a doubly warped product submanifoldN=f2N1×f1N2 in a nearly cosymplectic manifold N¯, other than the warped product bislant submanifold, if and only if (50) holds.


## 8. Some Applications of Theorem26 for Different Kinds of Ambient Manifolds

LetN=f2N1×f1N2 be a doubly warped product bislant submanifold, where N1 and N2 are proper slant submanifolds with respect to θ1 and θ2, respectively, and ξ∈TN2. The following corollaries are the immediate consequences of Theorem 26.Corollary 30.
There is no doubly warped product bislant submanifoldN=f2N1×f1N2 in a nearly Sasakian manifold N¯, other than the warped product bislant submanifold, if and only if (56) holds.Corollary 31.
There is no doubly warped product bislant submanifoldN=f2N1×f1N2 in a nearly Kenmotsu manifold N¯, other than the warped product bislant submanifold, if and only if (56) holds.Corollary 32.
There is no doubly warped product bislant submanifoldN=f2N1×f1N2 in a nearly cosymplectic manifold N¯, other than the warped product bislant submanifold, if and only if (56) holds.



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*Source: 5065333-2021-10-04.xml*