# Inequalities for the Class of Warped Product Submanifold of Para-Cosymplectic Manifolds

Authors: Fatemah Mofarreh; Sachin Kumar Srivastava; Mayrika Dhiman; Wan Ainun Mior Othman; Akram Ali
Journal: Advances in Mathematical Physics
 (2022)

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## Abstract

The aim of this paper is to study the warped product pointwise semislant submanifolds in the para-cosymplectic manifold with the semi-Riemannian metric. For which, firstly we provide the more generalized definition of pointwise slant submanifolds and related characterization results followed by the definition of pointwise slant distributions and pointwise semislant submanifolds. We also derive some results for different foliations on distribution, and lastly, we defined pointwise semislant warped product submanifold, given existence and nonexistence results, basic lemmas, theorems, and optimal inequalities for the ambient manifold.


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## Body

## 1. Introduction

To generalize the Riemannian product manifolds, Bishop and O’Neil [1] introduced the concept of warped product for the manifolds with negative curvature and showed the surface of revolution as the simplest example of warped product manifold. The authors of [2–5] studied the warped product submanifolds for different manifolds. Warped product plays the beneficial role in encoding the universe, and the inequalities related to the second fundamental form with the warping function cover the wide as well as important section of it. These were firstly formulated by Chen in [6, 7]. Warped product for lightlike manifolds for the first time was studied in [8] and for semi-Riemannian manifold under the name PR-warped product on para-Kähler manifold in [9], where he derived the aforesaid inequalities for the case of semi-Riemannian metric. From there, the study on warped product escalates among geometers with also in view that the same has so many applications in the physics mainly in general relativity and black hole theory [10].Beside this, the name slant submanifolds were introduced as the generalized version of holomorphic and totally real cases of submanifolds by Chen in [11]. Further, the theory extended to various manifolds with Riemannian as well as semi-Riemannian metric by many geometers. Later, in 2017, the authors in [12] defined the slant submanifolds irrespective of the writinger angle for the semi-Riemannian manifold and formulated three cases which are separately explained and achieved some effective results with bunch of examples. They defined it in terms of quotient gtX,tX/gJX,JX which is constant for the case of slant submanifolds for every vector field X (spacelike or timelike) on the submanifold M of manifold M¯,J,g. As slant and semislant submanifolds generalized to pointwise slant submanifolds (former called quasi slant) by Etayo in [13], Chen and Garay studied the same for the almost Hermitian case [14]. Sahin [15] defined pointwise semislant notion of submanifolds with an example. Recently, there are many interesting papers related with submanifold theory, singularity theory, classical differential geometry, etc. The readers can find more details about those techniques and theories in a series of papers [16–29]. Moreover, interdisciplinary research is one of the hottest trends in science; in the future work, we intend to apply and combine the techniques and results presented in [16–25] alongside with the methods in this paper to obtain more new results.The paper is structured as follows: Section2 contains the preliminary knowledge about ambient manifold, submanifold, and warped product with some important lemmas. Section 3 defines the pointwise slant submanifold, characterization lemma, and an example. Section 4 and Section 5 deal with the study of the pointwise slant distributions and pointwise semislant submanifolds, respectively. Section 6 includes the definition of warped product, some nonexistence results, lemmas, and theorems provided with an example. Finally, inequalities for the same submanifold are given in Section 7.


## 2. Preliminaries

Definition 1.
A2m+1-dimensional smooth manifold M^ admits φ,ξ,η,g structure with φ as a 1,1-tensor field, ξ as a characteristic vector field, η as a globally differential 1-form, and g as a semi-Riemannian metric named as an almost paracontact semi-Riemannian manifold M^,φ,ξ,η,g which satisfies
(1)φ2=I−η⊗ξ,ηξ=1,(2)g·,·=−gφ·,φ·+η·η·,where I represents an identity transformation of tangent space of M^ and ⊗ represents a tensor product. A structure compatible semi-Riemannian metric “g” relates to η as [30].
(3)g·,ξ=η·.Equations (1) and (2) easily ensure the following:
(4)rankφ=2m,φξ=0,η∘ϕ=0,(5)gφ·,·+g·,φ·=0.LetΦ be the fundamental 2-form on M^; then,
(6)Φ·,·=dη·,·=g·,φ·.Basis. An almost paracontact semi-Riemannian manifold always exists with aφ−basisEi,Eiå,ξ, a certain type of local pseudoorthonormal basis which includes Ei, ξ as space-like, and Eiå=φEi as timelike vector fields.Definition 2 (see [31]).
An almost paracontact semi-Riemannian manifoldM^ is termed as para-cosymplectic if the forms η and Φ are parallel with respect to the Levi-Civita connection ∇^ by
(7)∇^η=0,∇^Φ=0.Lemma 3.
LetM^ be a para-cosymplectic manifold with structure vector field ξ∈ΓTM^; then,
(8)∇^Xξ=0,
∀X∈ΓTM^.Proof.
Directly follow with the help of Equations (4) and (7) and covariant differentiation.


### 2.1. Submanifold

LetM be an isometrically immersed submanifold of a para-cosymplectic manifold M^ with an induced nondegenerate metric g (denoted metric by same symbol as on M^), denoting ∇ as Levi-Civita connection and h as the second fundamental form on M. Thus, the Gauss-Weingarten formulas are
(9)∇^XY=∇XY+hX,Y,(10)∇^Xζ=−AζX+∇X⊥ζ,for X,Y∈ΓTM (tangent bundle), and ζ∈ΓTM⊥ (normal bundle); ∇⊥ denotes normal connection, and Aζ denotes shape operator associated with the normal section on M. The metric relation of Aζ and h is given as
(11)gAζX,Y=ghX,Y,ζ.EveryX∈ΓTM is split as
(12)φX=tX+nX.Similarly, everyζ∈ΓTM⊥ is split as
(13)φζ=t⊥ζ+n⊥ζ,where tX and t⊥ζ (nX and n⊥ζ) are the tangential parts (normal parts) of φX and φζ, respectively. Based on Equation (12), the submanifold M classifies as anti-invariant if t=0 or invariant if n=0 on M. After using Equation (12) in Equation (5), we get
(14)gX,tY=−gtX,Y.Now, from Lemma3 and Equation (11), we have our next result.Lemma 4.
IfM is a submanifold immersed in a para-cosymplectic manifold M^ with structure vector field ξ∈ΓTM, then
(15)∇Xξ=∇ξX=∇ξξ=0,hX,ξ=0,Aζξ=0,AζX⊥ξ,for every X∈ΓTM and ζ∈ΓTM⊥.Next, let us take two semi-Riemannian manifoldsMB,g1 and MF,g2 and a positive smooth function f on MB. Taking MB×MF as the product manifold along with canonical projections,
(16)σ1:MB×MF⟶MB,σ2:MB×MF⟶MF,such that σ1pB,pF=pB and σ2pB,pF=pF for any point p=pB,pF∈MB×MF. Then, the product manifold MW=MB×fMF is called warped product if metric g called the warped metric on MW can be formulated as
(17)gX,Y=g1σ1∗X,σ1∗Y+f∘σ12g2σ2∗X,σ2∗Y.For every X,Y∈ΓTMW, “∗” represents the derivation map, and we call f as a warping function. Abstractly, the metric can be written as
(18)g=g1+f2g2,where the warped product MW=MB×fMF is split into a product of the base space MB and the fiber space MF, except that the fiber MF is warped [1, 32].Proposition 5 (see [32]).
The warped product submanifoldMW=MB×fMF satisfies
(i)
∇XY=ΓTMB(ii)
∇XU=∇UX=Xf/fU(iii)
∇UV=∇V′U=gU,V/f∇ffor X,Y∈ΓTMB and U,V∈ΓTMF, where ∇ is the Levi-Civita connection on MW, ∇′ is the connection on MF, and ∇f is the gradient of f defined as g∇f,X=Xf.Further, lete1,⋯,eK,eK+1,⋯,e2m+1 be a local orthonormal basis on TM^ among which e1,⋯,eK+1 are tangent to M and eK+2,⋯,e2m+1 are normal to M. If we set
(19)hxyk=ghex,ey,ek,x,y∈e1,⋯,eK+1,k∈eK+2,⋯,e2m+1,then we get
(20)hex,ey=∑k=K+12m+1εkhxyker,εk=gek,ek,where hxyk are the coefficients of h. Accordingly, squared norm of the second fundamental form h is defined as
(21)h2=∑x,y=1K+1εxεyghex,ey,hex,ey.


## 2.1. Submanifold

LetM be an isometrically immersed submanifold of a para-cosymplectic manifold M^ with an induced nondegenerate metric g (denoted metric by same symbol as on M^), denoting ∇ as Levi-Civita connection and h as the second fundamental form on M. Thus, the Gauss-Weingarten formulas are
(9)∇^XY=∇XY+hX,Y,(10)∇^Xζ=−AζX+∇X⊥ζ,for X,Y∈ΓTM (tangent bundle), and ζ∈ΓTM⊥ (normal bundle); ∇⊥ denotes normal connection, and Aζ denotes shape operator associated with the normal section on M. The metric relation of Aζ and h is given as
(11)gAζX,Y=ghX,Y,ζ.EveryX∈ΓTM is split as
(12)φX=tX+nX.Similarly, everyζ∈ΓTM⊥ is split as
(13)φζ=t⊥ζ+n⊥ζ,where tX and t⊥ζ (nX and n⊥ζ) are the tangential parts (normal parts) of φX and φζ, respectively. Based on Equation (12), the submanifold M classifies as anti-invariant if t=0 or invariant if n=0 on M. After using Equation (12) in Equation (5), we get
(14)gX,tY=−gtX,Y.Now, from Lemma3 and Equation (11), we have our next result.Lemma 4.
IfM is a submanifold immersed in a para-cosymplectic manifold M^ with structure vector field ξ∈ΓTM, then
(15)∇Xξ=∇ξX=∇ξξ=0,hX,ξ=0,Aζξ=0,AζX⊥ξ,for every X∈ΓTM and ζ∈ΓTM⊥.Next, let us take two semi-Riemannian manifoldsMB,g1 and MF,g2 and a positive smooth function f on MB. Taking MB×MF as the product manifold along with canonical projections,
(16)σ1:MB×MF⟶MB,σ2:MB×MF⟶MF,such that σ1pB,pF=pB and σ2pB,pF=pF for any point p=pB,pF∈MB×MF. Then, the product manifold MW=MB×fMF is called warped product if metric g called the warped metric on MW can be formulated as
(17)gX,Y=g1σ1∗X,σ1∗Y+f∘σ12g2σ2∗X,σ2∗Y.For every X,Y∈ΓTMW, “∗” represents the derivation map, and we call f as a warping function. Abstractly, the metric can be written as
(18)g=g1+f2g2,where the warped product MW=MB×fMF is split into a product of the base space MB and the fiber space MF, except that the fiber MF is warped [1, 32].Proposition 5 (see [32]).
The warped product submanifoldMW=MB×fMF satisfies
(i)
∇XY=ΓTMB(ii)
∇XU=∇UX=Xf/fU(iii)
∇UV=∇V′U=gU,V/f∇ffor X,Y∈ΓTMB and U,V∈ΓTMF, where ∇ is the Levi-Civita connection on MW, ∇′ is the connection on MF, and ∇f is the gradient of f defined as g∇f,X=Xf.Further, lete1,⋯,eK,eK+1,⋯,e2m+1 be a local orthonormal basis on TM^ among which e1,⋯,eK+1 are tangent to M and eK+2,⋯,e2m+1 are normal to M. If we set
(19)hxyk=ghex,ey,ek,x,y∈e1,⋯,eK+1,k∈eK+2,⋯,e2m+1,then we get
(20)hex,ey=∑k=K+12m+1εkhxyker,εk=gek,ek,where hxyk are the coefficients of h. Accordingly, squared norm of the second fundamental form h is defined as
(21)h2=∑x,y=1K+1εxεyghex,ey,hex,ey.


## 3. Pointwise Slant Submanifolds

The semi-Riemannian manifold has difficulty of defining the Writinger angle as the vector fields may be timelike. Thus, the next definition is in the view of [12], generalizing the slant submanifold in our ambient semi-Riemannian manifold.Definition 6.
An isometrically immersed submanifoldM of an almost paracontact manifold M^ is termed as pointwise slant if at every point p∈M, the quotient gtX,tX/gφX,φX=λθ for θ≥0 is independent of the choice of any nonzero spacelike or timelike vector X∈Mp, where Mp=X∈TpM:gX,ξ=0. For slant angle θ, we say λθ a slant coefficient.Remark 7.
The value ofλθ can be
(i)
λθ=cosh2θ∈1,∞ for tX/φX>1; tX is timelike or spacelike of each spacelike or timelike vector field X adding θ>0(ii)
λθ=cos2θ∈0,1 for tX/φX<1; tX is timelike or spacelike of each spacelike or timelike vector field X adding 0≤θ≤2π(iii)
λθ=−sinh2θ∈−∞,0 for tX is timelike or spacelike for any timelike or spacelike vector field X adding θ>0Remark 8.
The special cases are as follows:(i)
The constant value ofλθ throughout M implies M is slant submanifold [11, 12](ii)
The pointp∈M is called a complex point if t≡φ, which means that the slant coefficient λθ is equal to 1. The submanifold with every point as complex point is complex or holomorphic submanifold(iii)
The pointp∈M is called a totally real point if t≡0, which means that the slant coefficient λθ is equal to 0. The submanifold with every point as totally real point is totally real submanifoldFurthermore, let us take the union of allMp’s and denoting the same by
(22)T∗M=⋃p∈MX∈MpgX,ξ=0.Lemma 9.
The submanifoldM isometrically immersed in para-cosymplectic manifold M^ is a pointwise slant submanifold if and only if on every point p∈M; there exists λ∈−∞,∞ for some θ≥0 such that t2X=λθX for each spacelike (or timelike) vector field X∈Mp.Proof.
For each pointp of a pointwise slant submanifold M, the definition (22) follow as
(23)gtX,tX=λθgφX,φX,for X∈TpM. With the use of Equations (5) and (14) and the condition that X∈TpM in Equation (23), we get the desired result.Proceeding further with some results which are not hard to prove, any pointwise slant submanifoldM satisfies
(24)gtX,tY=λθgφX,φY=−λθgX,Y,gnX,nY=1−λθgφX,φY=−1−λθgX,Y,for X,Y∈T∗M.Proposition 10.
The submanifoldM of a para-cosymplectic manifold M^ is pointwise slant submanifold if and only if
(i)
t⊥nX=1−λθX and ntX=−n⊥nX for any spacelike (or timelike) vector field X∈ΓTM(ii)
n⊥2ζ=λθζ for nonlightlike normal vector field ζ, where λθ is the slant coefficient of MProof.
AssumeM as a pointwise slant submanifold.
(i)
Then for everyX∈T∗M, φ2X=X. On other way,
(25)φ2X=t2X+ntX+t⊥nX+n⊥nX.Equating tangential and normal parts and using Lemma9, we can attain the result(ii)
Sinceζ∈ΓTM⊥, thus there exists X∈ΓT∗M as M is pointwise slant submanifold such that nX=ζ. Now, n′2ζ=n⊥n⊥nX=−n⊥ntX=nt2X=λθζ. The converse can be easily derived using same equationsTheorem 11 (see [33]).
A totally geodesic and connected pointwise slant submanifoldM of a para-cosymplectic manifold M^ is a slant submanifold.


## 4. Pointwise Slant Distributions

Analogous to [34], we generalize slant distributions by defining pointwise slant distributions in M^. Furthermore, we study some basic characterizations for the distributions on our ambient manifold.Definition 12.
A pointwise slant distributionD on M is a differentiable distribution for which the quotient gtDX,tDX/gφX,φX=λDθ is independent of the choice of any spacelike or timelike vector field X∈Dp. Here,
(i)
Dp is the distribution at point p∈M(ii)
tDX is the projection of φX on the distribution D(iii)
λDθ is the slant coefficient corresponding to the distribution D on M for θ≥0, and the value of λDθ may be cosh2θ, cos2θ, or sinh2θRemark 13.
(1)
A pointwise slant distributionD is invariant if tDX≡φX with λDθ=1 or anti-invariant for tDX≡0 with λDθ=0. Other than these two cases, we call the distribution to be proper pointwise slant distribution [12](2)
The distributionD on M is as follows [9, 31]:
(i)
totally geodesic: ifhX,Y=0(ii)
involutive: ifX,Y∈Dfor every X,Y∈D.Corollary 14.
The distributionD on the submanifold M is pointwise slant distribution if and only if there exists λDθ for θ≥0 such that tD2X = λDθX for any nonlightlike vector field X∈Dp⊂TpM.Proof.
The result follows similar to Lemma9.


## 5. Pointwise Semislant Submanifold

Definition 15.
A submanifoldM of a para-cosymplectic M^ is named as pointwise semislant submanifold if the set of complementary orthogonal distributions DT,Dλ exists on M and fulfills the listed conditions:
(i)
TM=DT⊕Dλ(ii)
DT is φ-invariant distribution, i.e., DT⊆DT(iii)
Dλ is a pointwise slant distribution having λθ as a slant coefficient for θ≥0Remark 16.
Further, submanifoldM is
(i)
proper pointwise semislant when DT≠0,Dλ≠0 with nonconstant λθ(ii)
proper slant submanifold when DT=0 and Dλ≠0 with λθ globally constant for θ [35](iii)
proper semi-invariant when DT≠0 and Dλ≠0 such that tX≡0 for any X∈ΓDλ [12](iv)
invariant submanifold when Dλ=0 [35](v)
anti-invariant submanifold when DT=0 and tX≡0 for every X∈ΓDλ[35]Remark 17.
The decomposition of the tangent space can be expressed in two ways:(i)
Ifξ∈ΓTM, the TM=ξ⊕DT′⊕Dλ(ii)
Ifξ∈ΓTM⊥, the TM=DT′⊕Dλ. Here, DT′=X∈DT:gX,ξ=0⊆DT. Thus, we have either DT=DT′ or DT=ξ⊕DT′ [4]DenotePT and Pλ as the projections, respectively, on the distributions DT and Dλ. Then, any X∈ΓTM is split as
(26)X=PTX+PλX.Operatingφ, using Equation (12) and the case distribution DT which is φ-invariant on the previous equation, we concluded that
(27)tX=tPTX+tPλX∈ΓTM,nX=nPλX∈ΓTM⊥.AsDλ is pointwise slant distribution, by the consequences of Corollary 14, we obtain that
(28)t2X=λθX,for X∈ΓDλ with λθ as the slant coefficient. Clearly, for any point p∈M, if ξ∈TpM, then
(29)φX=tPT′X+tPλX+nPλX,where PT′ is the projection on the distribution DT′. But this does not affect our result as ξ disappears when φ operates on Z.However, the normal bundle denoted asTM⊥ may be written as
(30)TM⊥=nDλ⊕ν,where ν represents the subspace of normal bundle that is invariant under φ.Lemma 18 (see [31]).
The shape operatorA of a proper pointwise semislant submanifold M of para-cosymplectic manifold M^ ensures the listed conditions:
(31)gφAφUS,X=g∇SU,X,(32)AφUV=AφVU,(33)AζX=φAφζX,(34)gAζX,U=−gAφζφX,U,for S∈ΓTM, X∈ΓTDT, U,V∈ΓTDλ, and ζ∈ΓTM⊥.Both whenξ is normal or tangent to M, the integrability and geodesic conditions brought out to be same after calculations for both the distributions, thus denoting them as common DT.Lemma 19.
IfM is a proper pointwise semislant submanifold of para-cosymplectic manifold M^, for ξ∈ΓTM or ξ∈ΓTM⊥, the invariant distribution DT on M is
(i)
integrable if and only ifhtX,Y=hX,tY(ii)
totally geodesic if and only ifAntUY,=AnUtYfor X,Y∈ΓDT and U∈ΓDλ.Proof.
Equation (2) expands as
(35)gX,Y,U=−gφ∇^XY−∇^YX,φU,for every nonzero vector fields X,Y∈ΓDT and U∈ΓDλ. Using Equation (12) for the φU in Equation (35) and followed by using Equations (5), (7), and (9) and Lemma 9, we arrive at
(36)1−λθgX,Y,U=ghX,tY−hY,tX,nU.
Result (i) is clear using remark (28) as λθ is nonconstant in Equation (36). Again, from Gauss formula and Equation (2),
(37)g∇XY,U=g∇^XY,U=−gφ∇^XY,φU.
Employing Equations (7), (9), (11), (12), and (28) and Remark 16 in Equation (37), result (ii) follows.Lemma 20.
IfM is a proper pointwise semislant submanifold of para-cosymplectic manifold M^, for ξ∈ΓTM or ξ∈ΓTM⊥, the pointwise slant distribution Dλ on M is
(i)
involutive if and only if(38)gAnVU−AnUV,tX=gAntUV−AntVU,X(ii)
totally geodesic if and only if(39)gAnVtX,U=gAntVX,Ufor X∈ΓDT and U,V∈ΓDλ.Proof.
Equation (2) implies
(40)gU,V,X=−gφU,V,φX+ηU,VηX,
for every nonzero vector fieldsX∈ΓDT and U,V∈ΓDλ. Solving separately the term gφU,V,φX using Equations (5), (7), (10), (12), and (28), we receive
(41)gφU,V,φX=λθgU,V,X+gλ′θVθU−λ′θUθV,X−gAnVU−AnUV,φX+gAntVU−AntUV,X,where λ′θ is the first derivative of λθ. Surely, U,V are orthogonal to X after using this fact in Equation (41), and substituting in Equation (40), we get
(42)1−λθgU,V,X=g−AntVU+AntUV,X+gAnVU−AnUV,tX+ηU,VηX.
Forξ∈ΓTM, one can replace X by ξ in Equation (42), and consequently, we get
(43)−λθgU,V,ξ=ghU,ξ,ntV−ghV,ξ,ntU.
Using Lemma4 in Equation (43) and for reason that λθ a nonconstant, we get
(44)ηU,V=0.
Therefore, in Equation (42) using Equation (44) along with the facts that M is proper, we arrived at the desired result (i).
Further, using Gauss formula and employing Equations (2), (7), (9), (11), (12), and (28) give
(45)g∇UV,X=λθg∇XY,U+gλ′θUθV,X+gAntVX,U−gAnVtX,Uη∇^UVηX.
Sinceξ∈ΓTM, we can replace X by ξ in Equation (45), and consequently, we get
(46)1−λθη∇VU=−gAntUξ,V+η∇VU,−λθη∇VU=−gAntUξ,V.
Using Lemma4 in above expression, we get
(47)η∇VU=η∇^VU=0.
Hence, Equation (45) implies that
(48)1−λθ∇UV,X=gAntVX,U−gAnVtX,U.
Thus, from (48) and M as proper, X,Y,U as nonnull vector fields, the proof of the (ii) directly follows.


## 6. Pointwise Semislant Warped Product Submanifold

Definition 21.
A pointwise semislant warped product submanifoldM of a para-cosymplectic manifold M^ is a warped product of an invariant submanifold MT and a proper pointwise slant submanifold Mλ either in the form MT×fMλ or Mλ×fMT, where f is a positive smooth function taken on first submanifold in the product and slant coefficient of Mλ is λθ. A trivial product is the case of such submanifold for which warping function f is constant.Proposition 22 (see [33]).
A nontrivial pointwise semislant warped product submanifoldM of the form Mλ×fMT with ξ∈ΓTM⊥ does not exist on a para-cosymplectic manifold M^.Proposition 23 (see [33]).
A nontrivial pointwise semislant warped product submanifoldM of the form Mλ×fMT with ξ∈ΓTM does not exist on a para-cosymplectic manifold M^.Proposition 24.
A nontrivial pointwise semislant warped product submanifoldM of the form MT×fMλ with ξ∈ΓTMλ does not exist on a para-cosymplectic manifold M^.Proof.
Directly follow from Lemma4 and Proposition 5.Lemma 25.
For a nontrivial pointwise semislant warped product submanifoldM=MT×fMλ of a para-cosymplectic manifold M^,
(49)ghX,U,ntU−ghX,tU,nU=0,(50)ghX,tU,nU=λθXlnfgU,U,
∀X∈ΓDT and U∈ΓDλ.Proof.
(1)
SinceghX,U,ntU=g∇^XU,ntU for X∈ΓTMT and U∈ΓMλ, on right side, using Equations (5), (7), and (12), Proposition 5, and Lemma 9, we have
(51)ghX,U,ntU=−XlnfgtU,tU−g∇^XnU,tU−λθXlnfgU,U.Equations (10), (11), and (14) and Lemma 9 further help to achieve (49)(2)
AsghX,tU,nU=g∇^XtU,nU, next substituting nU=φU−tU applying Equations (5) and (7), Proposition 5, and Lemma 9 and the facts that MT is invariant, we get (50)Proposition 26.
IfM=MT×fMλ is a nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold M^, then
(52)1−λθg∇^XU,Y=ghX,tY,nU−ghX,Y,ntU,(53)1−λθg∇^UV,X=ghU,tX,nV−ghU,X,ntV,
∀X,Y∈ΓDT and U,V∈ΓDλ.Proof.
Asg∇^XU,Y=−gφ∇^XU,φY, η∇^XU=−g∇^Xξ,U=0. Using Equations (7), (9), and (12) and Lemma 9 in above expression gives (52). In similar way, we can prove (53).Lemma 27.
A nontrivial proper pointwise semislant submanifoldM=MT×fMλ of a para-cosymplectic manifold M^ satisfies
(54)ghX,Y,nU=0,(55)ghX,V,nU=−φXlnfgV,U−XlnfgtV,U,(56)ghX,tV,nU=−φXlnfgtV,U−λθXlnfgV,U,(57)ghX,V,ntU=−φXlnfgV,tU+λθXlnfgV,U,
∀X,Y∈ΓDT and U,V∈ΓDλ.Proof.
Result (54) is not hard to prove using Equations (7), (12), and (14) and Proposition 5. Substituting tU=V in Equation (49) gives ghX,V,nU=ghX,U,nV; one can replace nV=φV−tV, and using Equations (5) and (7) and Proposition 5 gives (55). Putting V=tV and U=tU, respectively, in Equation (55) gives results (56) and (57).Lemma 28.
IfM=MT×fMλ is a nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold M^, then
(i)
forξ∈ΓTMT,
(58)ghtX,V,nU=−X−ηXξlnfgV,U−φXlnfgtV,U(ii)
forξ∈ΓTM⊥,
(59)ghtX,V,nU=−XlnfgV,U−φXlnfgtV,U
∀X∈ΓDT and U,V∈ΓDλ.Proof.
ReplacingX=φX in Equation (55) and having the fact that submanifold MT is invariant, both results directly follow.Proposition 29.
LetM=MT×fMλ be nontrivial pointwise semislant warped product submanifold of a para-cosymplectic manifold M^, then
(60)ghX,V,ntU−ghX,tU,nV=λ′θXθgU,V,
∀X∈ΓDT and U,V∈ΓDλ, and λ′θ is the first derivative of slant coefficient.Proof.
Using metric and para-cosymplectic condition(61)g∇^XU,V=−g∇^XφU,φV,this expression under the effect of Equations (5), (9), (12), and (14) turns as
(62)g∇^XU,V=−g∇^XtU,tV−ghX,tU,nV+g∇^Xt⊥nU+n⊥nU,V.
Further, using Propositions10 and (5) ended with the desired result.Proposition 30.
A nontrivial pointwise semislant warped product submanifoldM=MT×fMλ of a para-cosymplectic manifold M^ satisfies the following:
(i)
Forξ∈ΓTM,
(63)ghtX,V,nU+ghX,V,ntU=−1−λθX−ηXξlnfgV,U(ii)
Forξ∈ΓTM⊥,
(64)ghtX,V,nU+ghX,V,ntU=−1−λθXlnfgV,U∀X∈ΓTMT and U,V∈ΓTMλ.Proof.
Lemma28 and Equation (57) of Proposition 26 directly give the results.Definition 31.
The submanifoldM=MT×fMλ is named as mixed totally geodesic if for every X∈ΓDT and U∈ΓDλ,
(65)hX,U=0.Theorem 32.
IfM=MT×fMλ is a mixed totally geodesic pointwise semislant warped product submanifold of a para-cosymplectic manifold M^, following cases arise:
(i)
Ifξ∈ΓTM, then M is either a trivial product or a warped product of a holomorphic (complex) submanifold and a totally real submanifold(ii)
Ifξ∈ΓTM⊥, then M is either a trivial product or a warped product of two complex submanifoldsProof.
Using definition (59), M satisfies hX,U=0 as well as hφX,U=0 (as MT is φ-invariant) for X∈ΓTMT and U∈ΓTMλ. Using this condition in proposition (58) when ξ∈ΓTM, we get
(66)1−λθX−ηXξlnfgV,U=0.
Indicate eitherlnf=0 implies the trivial case or 1−λθX=ηXξ, after taking inner product with ξ, and in the view of Remark 8, the condition for the totally real holds for the submanifold Mλ. Following similar way for the second case, we ended up with λθ=1 which is the condition for complex submanifold.Theorem 33.
A mixed totally geodesic pointwise semislant warped product submanifoldM=MT×fMλ of a para-cosymplectic manifold M^ satisfies
(67)εntUlnfX2=g∇^XtX,tU,
∀ spacelike (or timelike) vector fields X∈ΓTMT and U∈ΓTMλ.Proof.
Asg∇^XtX,tU=g∇^XtX,tU+gt∇^XX,tU, under the effects of Equation (14), Proposition 5, and Lemma 9, it turns
(68)g∇^XtX,tU=g∇^XtX,tU+λθUlnfgX,X.
Other way,g∇^XtX,tU+g∇^XnX,tU=g∇^XφX,tU; with this expression under the use of Equations (5), (7), and (12), Proposition 5, and Lemma 9, g∇^XnX,tU=0 and M as mixed totally geodesic, we have
(69)g∇^XtX,tU=λθUlnfgX,X+εntUlnfX2.
This expression with the use of Equation (68) and for the reason vector field X can be spacelike or timelike yields the result.Example 34.
Consider a7-dimensional smooth manifold M^=ℝ6×ℝ+⊂ℝ7 having standard Cartesian coordinates as y1,y2,y3,y4,y5,y6,t and defining a structure φ,ξ,η,g as
(70)φe1=e4,φe2=e5,φe3=e6,φe4=e1,φe5=e2,φe6=e3,φe7=0,ξ=e7,η=dtandge1,e1=ge2,e2=ge3,e3=ge7,e7=1,ge4,e4=ge5,e5=ge6,e6=−1,for e1,⋯,e7 as a local orthonormal frame on ΓTM^. Obviously, M^ over φ,ξ,η,g fufills the condition of para-cosymplectic manifold. Let M be a submanifold of M^ with ξ tangent to it and defined for x,y∈ℝ, u,v≥0 and some k(constant) as
(71)Mx,y,u,v,z=x+ycoshu,x+ysinhv,u,x+ysinhu,x+ycoshv,v,kx−ky+z.The vector fields that generates the tangent bundleTM are
(72)X=coshue1+sinhve2+sinhue4+coshve5+ke7,Y=coshue1+sinhve2+sinhue4+coshve5−ke7,U=x+ysinhue1+e3+x+ycoshue4,V=x+ycoshve2+x+ysinhve5+e6,Z=e7.After calculations, it is found that the invariant distributionsDT⊕⊲ξ⊳ is the span of subspace X,Y,Z and pointwise slant distribution Dλ is the span of subspace U,V with t2=1/1−x2−y22Id such that x2+y2≠1; then, the slant coefficient is
(i)
λθ=cosh2θ for x2+y2<1(ii)
λθ=cos2θ for x2+y2>1As the distributionsDT⊕⊲ξ⊳ and Dλ are integrable, let MT and Mλ be their respective integral manifolds such that M=MT×fMλ turns a nontrivial 5-dimensional pointwise semislant warped product submanifold of M^ with induced metric g(semi-Riemannian) as
(73)g=k2dX2+k2dY2+dZ2+1−x2−y2dU2−dV2,with warping function f=1−x2−y2.LetM be an another submanifold of M^ with ξ normal to it and defined for x,y,u,v∈ℝ as
(74)Mx,y,u,v=xu,x,u+yv,yu,u+v,xv,0.Then, the vector fields that generatesTM are
(75)X=ue1+e2+ve6,Y=ve3+ue4+e5,U=xe1+e3+ye4+e5,V=ye3+xe6.The invariant distributionDT is the span of subspace X,Y, and pointwise slant distribution Dλ is the span of subspace U,V with t2=x2/x2−y22I such that x2≠y2; then, the slant coefficient is
(i)
λθ=cosh2θ for x2>y2(ii)
λθ=cos2θ for y2>x2As the distributionsDT and Dλ are integrable, let MT and Mλ be their respective integral manifolds such that M=MT×fMλ turns a nontrivial 4-dimensional pointwise semislant warped product submanifold of M^ with induced metric g(semi-Riemannian) as
(76)g=1+u2−v2dX2−1+u2−v2dY2+x2−y2dU2−dV2,with warping function f=x2−y2.


## 7. Inequalities

This section includes the geometric sharp inequalities for the aforesaid submanifoldM=MT×fMλ for the case ξ tangent and normal to M.Lemma 35 (see [31]).
LetM=MT×fMλ be a pointwise semislant warped product submanifold of a para-cosymplectic manifold M^. Then, M ensures
(77)ghX,U,ζ=−ghX,φU,φζ,(78)ghX,U,φζ=−g∇X⊥φU,ζ,(79)ghX,U,φζ=−g∇Z⊥φU,ζ,
∀X∈ΓDT, U∈ΓDλ, and ζ∈Γν.Theorem 36.
LetM=MT×fMλ be a pointwise semislant warped product submanifold of a para-cosymplectic manifold M^ with ξ∈ΓTM. If MT is an invariant submanifold of 2n1+1-dimension and Mλ is a proper pointwise slant submanifold 2n2-dimension satisfying ∇⊥φDλ⊆φDλ, the succeeding inequalities holds for h(80)h2≥n21+λ2θ∇lnf2+hνDT2forS1≥S2,h2≤n21+λ2θ∇lnf2+hνDT2forS1≤S2,where λθ is the slant coefficient corresponding to Mλ, ∇lnf is the gradient of lnf, hνDT2=ghνDT,DT,hνDT,DT with its ν component and invariant distribution DT, S1=hrts2+hr′t′s2, and S2=hr′ts2+hrt′s2.Proof.
Forξ∈ΓTM, choose the local orthonormal frame on the following:
(a)
MT by ei,ei′=φei for i=1,⋯,n1 and ei=ξ for i=2n1+1 such that gei,ei=εi=1 implies gei′,ei′=εi′=−1 and gξ,ξ=ε0=1(b)
Mλ by e¯r,e¯r′=1/−λθte¯r for r=1,⋯,n2 and such that ge¯r,e¯r=ε¯r=1 implies ge¯r′,e¯r′=ε¯r′=−1(c)
nMλ by e¯s=1/−1−λθne¯r for r=1,⋯,n2, having ge¯s,e¯s=ε¯s=−1 and on ν by ζl,ζl′=φζ such that gζl,ζl=εl=1 implies gζl′,ζl′=εl′=−1
Computeh2 which is given as
(81)h2=hDT,DT2+2hDT,Dλ2+hDλ,Dλ2.
The first termhDT,DT2 can be expanded as
(82)hDT,DT2=ghDT,DT,hDT,DT=∑i,j=0n1εiεjghei,ej,hei,ej+εi′εjghei′,ej,heii,ej+εiεj′ghei,ej′,hei,ej′+εi′εj′ghei′,ej′,hei′,ej′+∑i=0n1ε0εighe0,ei,he0,ei+εiε0ghei,e0,hei,eo.
AsDT is totally geodesic and Equation (54) of Lemma 27 directs that hDT,DT∈ν using which, we can write
(83)hei,ej=hijlζl+hijl′ζl′,hei′,ej=hi′jlζl+hi′jl′ζl′,hei,ej′=hij′lζl+hij′l′ζl′,hei′,ej′=hi′j′lζl+hi′j′l′ζl′,he0,ei=h0ilζl+h0il′ζl′,hei,e0=hi0lζl+hi0l′ζl′,he0,e0=h00lζl+h00l′ζl′.
Simplifying these expressions in Equation (82) and using Equation (19) and Lemma 4 and in view of orthonormal frame, we get
(84)hDT,DT2=∑i,j=1n1∑l=1n3hijl2−hijl′2−hij′l2−hi′jl′2−hij′l2−hij′l′2+hi′j′l2−hi′j′l′2.
The integrable condition of theDT and Equation (77) of the Lemma 35 implies that
(85)hi′jl2=hij′l2,hi′jl′2=hij′l′2,hi′j′l2=hi′j′l2,hi′j′l′2=hijl′2,(86)hi′jl2=hi′jl′2,hijl′2=hi′jl2.
After substitution of Equation (84) in (86), we get
(87)hDT,DT2=4∑i,j=1n1∑l=1n2hijl2−hijl′2=hνDT2.
For the second part, we have(88)hDT,Dλ2=ghDT,Dλ,hDT,Dλ=∑i=1n1∑r=1n2εiε¯rghei,e¯r,hei,e¯r+εi′ε¯rghei′,e¯r,hei′,e¯r+εiε¯r′ghei,e¯r′,hei,e¯r′+εi′ε¯r′ghei′,e¯r′,hei′,e¯r′+ε0ε¯rghe0,e¯r,hei,e¯r+ε0ε¯r′ghe0,e¯r′,he0,e¯r′,where
(89)hei,e¯r=hirse¯s+hirlζl+hirl′ζl′,hei′,e¯r=hi′rse¯s+hi′rlζl+hi′rl′ζl′,(90)hei,e¯r′=hir′se¯s+hir′lζl+hir′l′ζl′,hei′,e¯r′=hi′r′se¯s+hi′r′lζl+hi′r′l′ζl′,(91)he0,e¯r=h0rse¯s+h0rlζl+h0rl′ζl′,he0,e¯r′=h0r′se¯s+h0r′lζl+h0r′l′ζl′.
After simplifying Equation (88) using expressions in Equation (91), we get
(92)hDT,Dλ2=∑i=1n1∑r,s=1n2hirs2+hi′r′s2−hir′s2−hir′s2+hirl2+hi′r′l2−hi′rl2−hir′l2−hirl′2+hi′r′l′2−hi′rl′2−hir′l′2.
Using Equations (55), (56), (58), (77), and (78), we have
(93)hirs=−ei′lnfge¯r,e¯s,hi′rs=−eilnfge¯r,e¯s,hir′s=−λθeilnfge¯r,e¯s,hi′r′l′=−λθei′lnfge¯r,e¯s,hirl2=hir′l′2,hi′rl2=hi′r′l′2andhirl′2=hir′l2.
Substituting above values in Equation (92), we have
(94)hDT,Dλ2=n21+λ2θ∑i=1n1eilnf2−ei′lnf2+2hi′rl2−hi′rl′2−hirl2−hirl′2.
Since∑i=1n1eilnf2−ei′lnf2=g∇lnf,∇lnf=∇lnf2 and using the condition that ∇⊥φDλ⊆φDλ in formula (78), we concluded that hDT,Dλ⊆φDλ, above equation leads to
(95)hDT,Dλ2=n21+λ2θ∇lnf2.
Lastly,(96)hDλ,Dλ2=ghDλ,Dλ,hDλ,Dλ=∑r,t=1n2ε¯rε¯tghe¯r,e¯t,he¯r,e¯t+ε¯r′ε¯tghe¯r′,e¯t,he¯r′,e¯t+ε¯rε¯t′ghe¯r,e¯t′,he¯r,e¯t′+ε¯r′ε¯t′ghe¯r′,e¯t′,he¯r′,e¯t′,where the included expressions are as below:
(97)he¯r,e¯t=hrtse¯s+hrtlζl+hrtl′ζl′,he¯r′,e¯t=hr′tse¯s+hr′tlζl+hr′tl′ζl′he¯r,e¯t′=hrt′se¯s+hrt′lζl+hrt′l′ζl′,he¯r′,e¯t′=hr′t′se¯s+hr′t′lζl+hr′t′l′ζl′.
Employing these expressions in Equation (96) in view of the chosen frame and simplifying, we get
(98)hDλ,Dλ2=∑r,s,t=1n2hrts2+hrtl2−hrtl′2−hr′ts2+hr′tl2−hr′tl′2−hrt′s2+hrt′l2−hrt′l′2+hr′t′s2+hr′t′l2−hr′t′l′2.
Using the condition that∇⊥φDλ⊆φDλ in formula (79), we concluded that hDλ,Dλ⊆φDλ, which implies the Equation (98) with
(99)hDλ,Dλ2=∑r,s,t=1n2hrts2+hr′t′s2−hr′ts2+hrt′s2.
Result directly follows by lettingS1=hrts2+hr′t′s2 and S2=hr′ts2+hrt′s2.Remark 37.
Equality holds ifS1=S2.Theorem 38.
LetM=MT×fMλ be a pointwise semislant warped product submanifold of a para-cosymplectic manifold M^ with ξ normal to M such that ξ∈Γν. If MT is an invariant submanifold of 2n1-dimension and Mλ is a proper pointwise slant submanifold of 2n2-dimension satisfying ∇⊥φDλ⊆φDλ, the succeeding inequalities holds for h(100)h2≥n21+λ2θ∇lnf2+hνDT2forS1≥S2,h2≤n21+λ2θ∇lnf2+hνDT2forS1≤S2,where λθ is the slant coefficient corresponding to Mλ, ∇lnf is the gradient of lnf, hνDT2=ghνDT,DT,hνDT,DT with its ν component and invariant distribution DT, S1=hrts2+hr′t′s2, and S2=hr′ts2+hrt′s2.Proof.
Forξ∈ΓTM⊥, choose the local orthonormal frame on the following:
(a)
MT by ei,ei′=φei for i=1,⋯,n1 such that gei,ei=εi=1 implies gei′,ei′=εi′=−1(b)
Mλ by e¯r,e¯r′=1/−λθte¯r for r=1,⋯,n2 and such that ge¯r,e¯r=ε¯r=1 implies ge¯r′,e¯r′=ε¯r′=−1(c)
nMλ by e¯s=1/−1−λθne¯r for r=1,⋯,n2, having ge¯s,e¯s=ε¯s=−1 and on ν by ζl,ζl′=φζ for l=1,⋯,n3 and ζl=ξ for l=2n3+1 such that gζl,ζl=εl=1 implies gζl′,ζl′=εl′=−1 and gξ,ξ=ε0=1
Further, result can be acquired carrying the same steps as above proof and using Equations (33) and (34) of Lemma 18.



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