# Warped Product Pointwise Semi Slant Submanifolds of Sasakian Space Forms and their Applications

Authors: Nadia Alluhaibi; Meraj Ali Khan
Journal: Advances in Mathematical Physics
 (2020)

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## Abstract

In this study, we attain some existence characterizations for warped product pointwise semi slant submanifolds in the setting of Sasakian space forms. Moreover, we investigate the estimation for the squared norm of the second fundamental form and further discuss the case of equality. By the application of attained estimation, we obtain some classifications of these warped product submanifolds in terms of Ricci soliton and Ricci curvature. Further, the formula for Dirichlet energy of involved warping function is derived. A nontrivial example of such warped product submanifolds is also constructed. Throughout the paper, we will use the following acronyms: “WP” for warped product, “WF” for warping function, “AC” for almost contact, and “WP-PSS” for the warped product pointwise semi slant.


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## Body

## 1. Introduction

In the field of geometry, an increased interest is observed for the exploration of the warped product manifolds due to its applicability in the theory of relativity and physics [1]. Earlier researchers have shown that warped products facilitate the computation of the Einstein field equations [1]. In addition, the concept of warped product manifolds is used in the notion of modelling of space-time near black holes [2], such as Schwarzschild space-time P×rS2, where r>0, the base is a half plane P=R×R+, and the unit sphere S2 is the fibre. At some instances, Schwarzschild space-time also acts similar to the black hole. The Robertson-Walker model is an example of a warped product which is considered as a cosmological model adopted for the space-time modelling of the universe [3].Identifying Dirichlet energy of smooth functions is considered as an integral part in the field of physics and engineering. Moreover, Dirichlet energy is found to be an analog of kinetic energy. Letτ be any real valued smooth function on a compact manifold M, then the Dirichlet energy of τ is defined by
(1)Eτ=12∫M∇τ2dV,where ∇τ is the gradient of τ and dV is the volume element. It is clear that Eτ≥0. It is well recognized that a product manifold cannot be used for the representation of nonzero (constant) curvature manifolds. Therefore, the concept of the warped product manifold emerged due to the nonexistence of negative curvature for the Riemannian product of manifolds. Therefore, Bishop and O’Neill [4] initiated the concept of warped product manifolds for devising the class of manifolds for the negative or nonpositive curvature. A warped product manifold (see definition in Section 2) is a generality of a product manifold. As the warping function of warped product manifolds is a positive valued smooth function, the focus of the present study is on examining the Dirichlet energy of such functions.The study of Bishop and O’Neill [4] has enlightened a few intrinsic properties of the warped product manifolds. Initial extrinsic studies of the warped product manifold in the almost complex setting were performed by Chen [5, 6] while obtaining some results of existence for CR-submanifold as CR-warped product submanifold in Kaehler manifolds. Furthermore, contact CR-warped product submanifolds were investigated in the study of Hasegawa and Mihai [7] in the almost contact settings. Warped product manifolds are also investigated in the contact setting by many other geometers which have attained various existence results [8–11].Warped product pointwise semi slant submanifold is another generalized class of warped product semi slant submanifolds and contact CR-warped product submanifolds. In [12], Park studied such warped product submanifolds. After that, Mihai and Uddin extended this study in Sasakian manifolds and acquired some optimal inequalities related to warping function and second fundamental form. Warped product pointwise semi slant submanifolds for almost contact and almost complex manifolds were explored in [13–15].On the other hand, the gradient Ricci soliton is extensively investigated in physics, theory of relativity, and differential geometry. The classification results related to Ricci soliton and gradient Ricci solitons with the warped product structure have been established in [16–19]. Moreover, the Ricci curvature has a significant nature in Riemannian geometry; for example, Ricci flat is used to solve the Einstein field equation on a Riemannian manifold in which the cosmological constant vanishes. More clearly, in the general relativity theory, the Ricci tensor is correlated with Einstein’s field equation to study the material contents of the universe. So, in comparison with Riemannian curvature, the Ricci curvature is more significant in the theory of relativity and physics.In this study, warped product pointwise semi slant submanifolds are studied in the setting of Sasakian space forms and some associated inequalities are attained. The study also provides estimates for the squared norm of the second fundamental form concerning the slant function and warping function and then addresses the case of equality. Some applications of the case of equality and the derived inequality are explored. More precisely, Dirichlet energy of the warping functions is computed via the obtained inequality. Finally, a classification of the WP-PSS submanifolds admitting the gradient Ricci soliton, in respect of Ricci curvature and second fundamental form, is obtained, and some existence results are also established.The study consists of five sections in which the second section is dedicated to pave the way for the other sections by providing the fundamental definition, formulas, and preliminary outcomes needed for the study. Section3 investigates the WP-PSS submanifold existence in Sasakian space forms and provides the proof of the main findings of the study. In addition, it gives a nontrivial example for the WP-PSS submanifolds in the setting of a Sasakian manifold. The formulas for Dirichlet energy of warping function is attained in Section 4 by using the derived inequality. The paper is concluded in Section 5, followed by its bibliography.


## 2. Preliminaries

AC∞−manifold M¯ of dimension 2m+1 is said to admit an AC structure if on M¯ ∃ a tensor field Ψ of type 1,1, a vector field η, and a 1-form ζ fulfilling [20]
(2)Ψ2=−I+ζ⊗η,Ψη=0,ζ∘Ψ=0,ζη=1.The manifoldM¯ with the structure Ψ,η,ζ is called AC metric manifold. On an AC metric manifold M¯, ∃ a Riemannian metric g satisfying
(3)ζY=gY,η,gΨY,ΨV=gY,V−ζYζV,for all Y,V∈TM¯, where TM¯ is the tangent bundle of M¯.An AC metric structureΨ,η,ζ,g is referred to as Sasakian manifold if it fulfills the following relation [20]:
(4)∇¯YΨV=gY,Vη−ζVY,for any Y,V∈TM¯, where ∇¯ denotes the Riemannian connection of the metric g. Then, we have
(5)∇¯Yη=−ΨY.ASasakian space form M¯c [20] is a Sasakian manifold with constant Ψ-holomorphic sectional curvature c. Moreover, the following formula gives the curvature tensor R¯ of Sasakian space form, M¯c:
(6)R¯Y1,Y2V=c−34gY2,VY1−gY1,VY2+c−14gY1,ΨVΨY2−gY2,ΨVΨY1+2gY1,ΨY2ΨV+ζY1ζVY2−ζY2ζVY1+gY1,VζY2η−gY2,VζY1η,∀ vector fields Y1,Y2,V on M¯.LetM be a submanifold of an AC-metric manifold M¯ with induced metric g, and the vector field η is tangential to M. The Riemannian connection ∇¯ of M¯ induces canonically the connections ∇⊥ and ∇ on the normal bundle T⊥M and the tangent bundle TM of M, resp. Then, the Weingarten and Gauss formulas are as follows:
(7)∇¯Y1N=−ANY1+∇Y1⊥N,(8)∇¯Y1Y2=∇Y1Y2+σY1,Y2,∀ Y1,Y2∈TM and N∈T⊥M, where AN and σ are the shape operator and the second fundamental form, resp. For the immersion of M into M¯, AN and σ verify the relation
(9)gσY1,Y2,N=gANY1,Y2.IfM is a submanifold of a Riemmanian manifold M¯, then the equation of Codazzi is as follows:
(10)R¯Y1,Y2V⊥=∇Y1⊥σY2,V−∇Y2⊥σY1,V+σ∇Y2Y1,V−σ∇Y1Y2,V+σY1,∇Y2V−σY2,∇Y1V,where R¯Y1,Y2V⊥ denotes the normal component of the curvature tensor R¯Y1,Y2V.IfPY and FY denote the tangential and normal component of ΨY, respectively, then
(11)ΨY=PY+FY,∀ Y∈TM. Similarly, for any normal vector field N∈T⊥M,
(12)ΨN=tN+fN.Thus,F (resp., t) is a normal (resp., tangential) valued 1-form on TM (resp. T⊥M) and P (resp., f) is 1-1 tensor field on TM (resp., T⊥M). The covariant derivatives of Ψ, F, and P are given by
(13)∇¯Y1ΨY2=∇Y1ΨY2−Ψ∇Y1Y2,(14)∇¯Y1FY2=∇Y1⊥FY2−F∇Y1Y2,(15)∇¯Y1PY2=∇Y1PY2−P∇Y1Y2.From equations (4), (8), (7), (11), and (12), we have
(16)∇¯Y1PY2=AFY2Y1+tσY1,Y2−gY1,PY2η−ζY2PY1,(17)∇¯Y1FY2=fσY1,Y2−σY1,PY2−ζY2FY1.Lete1,e2,⋯,em be a local orthonormal basis of vector fields on m-dimensional submanifold M. Hence, the mean curvature vector H of M is obtained from the formula
(18)H=1m∑s=1mσes,es,and the squared norm of the second fundamental form σ is given by
(19)σ2=∑r,s=1mgσer,es,σer,es.A submanifoldM of M¯ is called totally umbilical if σY1,Y2=gY1,Y2H, and M is defined as totally geodesic if σY1,Y2=0, ∀Y1,Y2∈TM.Definition 1 (see [21]).
A submanifoldM of an AC metric manifold M¯ is slant if the angle between Y and ΨY is constant for any y∈M and Y∈TyM−η. Then the constant angle θ s.t. 0≤θ≤π/2 is called slant angle of M in M¯.Definition 2 (see [21]).
Ifθ=0, the submanifold M is invariant submanifold and is anti-invariant submanifold if θ=π/2. If θ is neither 0 nor π/2, M is proper slant submanifold.Etayo [22] introduced the concept of pointwise slant submanifolds as a natural generality of the slant submanifolds of almost Hermitian manifolds. Later, Chen and Garay [23] investigated pointwise slant submanifolds in the setting of almost Hermitian manifolds and attained some primary results. A step forward, Park [12] extended the notion of pointwise slant submanifolds in the setting of AC-metric manifolds. Recently, Uddin and Al-Khalidi [24] modified the definition of pointwise slant submanifolds for AC-metric manifolds. More precisely, a submanifold M of an AC-metric manifold M¯ is referred to as poinwise slant submanifold if ∀Y∈TyM such that η∈TM the angle θY between ΨY and TyM−0 is independent of the choice of nonzero vector field Y∈TpM−0. Hence, θ is dealt with as the function on M, which is known as a slant function of the pointwise slant function. Now, we attain the following characterizing theorem.Theorem 3 (see [24]).
Suppose thatM is a submanifold of an AC-metric manifold M¯ such that η∈TM. Then, M is pointwise slant if and only if
(20)P2=cos2θ−I+ζ⊗η,where θ:TM⟶R.Consequently, we have(21)gPY1,PY2=cos2θgY1,Y2−ζY1ζY2,(22)gFY1,FY2=sin2θgY1,Y2−ζY1ζY2,(23)∀Y1,Y2∈TM.Recently, Mihai and Uddin [15] defined and studied pointwise semi slant submanifold of Sasakian manifolds.Definition 4.
A submanifoldM of an AC-metric manifold M¯ is referred to as a pointwise semi slant submanifold if ∃ two orthogonal complementary distributions D and Dθ on M such that
(i)
TM=D⊕Dθ⊕η
(ii)
D is invariant
(iii)
Dθ is pointwise slant with a slant function θThe WP manifolds are the generalized form of the product manifolds and are defined as follows.LetC1,gC1 and C2,gC2 be two Riemannian manifolds with the Riemannian metrics gC1 and gC2, resp., and Φ be a positive differentiable map on C1. Let πC1:C1×C2⟶C1, πC2:C1×C2⟶C2 be the projection maps given by πC1c1,c2=c1 and πC2c1,c2=c2, ∀ c1,c2∈C1×C2. The WP M=C1×ΦC2 [4] is the manifold C1×C2 endowed with the Riemannian structure s.t.
(24)gY1,Y2=gC1πC1∗Y1,πC1∗Y2+Φ∘πC12gC2πC2∗Y1,πC2∗Y2,for all Y1,Y2∈TM, where ∗ denotes the tangential map (the differential on the tangent bundle). Here, Φ is known as the WF of the WP manifold. M is a trivial WP if the WF is constant.LetY1 be a vector field on C1 and Y2 be a vector field on C2, then we induce from Lemma 7.3 of [3]:
(25)∇Y1Y2=∇Y2Y1=Y1ΦΦY2,where ∇ is the Levi-Civita connection on M. For M=C1×ΦC2, it can be seen that
(26)∇Y1Y2=∇Y2Y1=Y1lnΦY2,for Y1∈TC1 and Y2∈TC2.The gradient of Φ is denoted by ∇Φ and is defined as
(27)g∇Φ,Y=YΦ,∀Y∈TM.LetM be a Riemannian manifold M of dimension m with e1,⋯,em as an orthogonal basis of TM and g a Riemannian metric of M. Then, as a result of (27), we get
(28)∇Φ2=∑s=1mesΦ2.The Laplacian of a smooth functionΦ on a Riemannian manifold is defined by the following equation:
(29)ΔΦ=∑s=1m∇esesΦ−esesΦ=−∑i=1mg∇eigradΦ,ei.For a smooth functionΦ on M, the Hessian of Φ is defined as
(30)HΦU,V=UVΦ−∇UVΦ=g∇UgradΦ,V,for any U,V∈TM.The Hessian tensor for a differential functionΦ is a symmetric covariant tensor of rank 2 and is defined as
(31)ΔΦ=−traceHΦ.Throughout the study, we used the following relation between the Hessian tensor and the Laplacian of a smooth functionΦ on a Riemannian manifold:
(32)HessΦ=−ΔΦ.Hopf’s Lemma [25]. If M is an m-dimensional connected compact Riemannian manifold and Φ is a differentiable function on M s.t. ΔΦ≤0 everywhere on M (or ΔΦ≥0 everywhere on M), then Φ is a constant function.For a compact orientable Riemannian manifoldM with or without boundary and as a consequence of the integration theory on manifolds, we have [26]
(33)∫MΔΦdV=0,where Φ is a function on M and dV is the volume element of M.The Ricci soliton idea was given by Hamilton [26]. It is regarded as the natural generality of Einstein metrics, and it is the self-similar solution of the Ricci flow ∂/∂tgt=−2Rict. If ∃ a smooth vector field Y such that the Ricci tensor meets the following condition:
(34)Ric+12LYg=α,for any constant α, where LY is the Lie derivative, then the metric g on a complete Riemannian manifold M¯ is named as Ricci soliton. If α<0,α=0, and α>0, then the Ricci soliton is called expanding, steady, and shrinking, respectively. If we specify Y=∇Φ for a smooth function Φ on M¯, then g admits gradient Ricci soliton with the potential function Φ. For this case, (34) takes the form
(35)Ric+∇2Φ=αg.Since the LaplacianΔ and the gradient ∇2 are related as Δ=∇2, in terms of Hessian, (35) can be expressed as
(36)Ric=αg+Hessψ.Note 1.
M,g,∇Φ,α is an Einstein manifold if the potential function Φ is constant on a gradient Ricci soliton.


## 3. Warped Product Pointwise Semi Slant Submanifolds

In [12], Park proved that the WP-PSS submanifolds of the type NT×ΦNθ of the contact metric manifolds exist and achieved the result of the following lemma.Lemma 5.
Suppose that M=NT×ΦNθ is a WP-PSS of a Sasakian manifold M¯ such that η∈TNT, where NT and Nθ are invariant and pointwise slant submanifolds of M¯, respectively. Then, we have
(37)gσY,W,FPZ=cos2θYlnΦgW,Z−ΨYlnΦgW,PZ−ζYgW,PZ,
∀ Y∈TNT and W,Z∈TNθ.In particular, if we replaceZ by PW in the above equation, then one can conclude the following:
(38)gσY,W,FW=−ΨYlnΦW2−ζYW2.Now, letM=NT×ΦNθ be a WP-PSS of a Sasakian manifold M¯ and we consider η∈TNT. If D is invariant distribution and Dθ is proper pointwise slant distribution with θ as a slant function, then TM and T⊥M are decomposed (resp.) as follows:
(39)TM=D⊕Dθ⊕η,T⊥M=FDθ⊕μ,where μ is the orthogonal complementary distribution of FDθ in T⊥M. One can easily check that μ is an invariant subbundle of T⊥M regarding Ψ.In view of the above direct decomposition,σ can be written as
(40)σU1,U2=σFDϑU1,U2+σμU1,U2,for U1,U2∈TM, where σFDθU1,U2 and σμU1,U2 are the components of σU1,U2 in the normal subbundles FDθ and μ, respectively. Moreover, if V1,V2,⋯,Vq is a local orthonormal basis of vector fields of Dθ, we have
(41)σFDθU1,U2=∑r=1qσrU1,U2FVr,where
(42)σrU,V=csc2θgσU,V,FVr.To ensure the existence, we construct an example of a WP-PSS of the typeM=NT×ΦNθ in the Sasakian manifold with η tangent to NT.Example 1.
We know thatR2m+1,Ψ0,ζ,η,g denotes a Sasakian manifold with the structure as follows:
(43)ζ=12dz−∑i=1myidxi,η=2∂∂z,g=ζ⊗ζ+14∑i=1mdxi⊗dxi+dyi⊗dyi,Ψ0∑i=1mXi∂∂xi+Yi∂∂yi+Z∂∂z=∑i=1mYi∂∂xi−Xi∂∂yi+∑i=1mYiyi∂∂z,where xi,yi,z,i=1,⋯,m, are the Cartesian coordinates. It is well known that 2∂/∂yi,2∂/∂xi+yi∂/∂z,η is an orthonormal basis of TR2m+1.Now, consider the 9-dimensional Sasakian manifoldR9,Ψ0,ζ,η,g and a submanifold isometrically immersed in R9 as follows:
(44)M=2u,0,wet,0,v,setcosθ,setsinθ,0,f∈R9,and consider a basis e1,e2,e3,e4,e5 of orthogonal vector fields on M as
(45)e1=2∂∂x1+y1∂∂z,e2=2∂∂y1,e3=2et∂∂x3,e4=2etcosθ∂∂y3+2etsinθ∂∂y4,e5=2∂∂z=η.Hence, it can be seen thatD=spane1,e2,e5 and Dθ=spane3,e4 define the invariant and pointwise slant distributions with θ∈0,π/2 as a slant function on the Sasakian manifold R9. If NT and Nθ stand for the integral manifold of D and Dθ, respectively, the metric g on M is defined by
(46)g=gNT+e2tgNθ.Then,M=NT×ΦNθ is a WP-PSS with the Φt=et.In this section, for convention, we denote byY,X∈TNT and V,U∈TNθ as the vector fields of respective tangent bundles of NT and Nθ. At first, some initial results need to be proved.Lemma 6.
Suppose that NT×ΦNθ is a WP-PSS of a Sasakian manifold M¯. Then, we have
(i)
ηlnΦ=0
(ii)
gσΨY,V,FV=YlnΦV2
(iii)
gσΨY,V,ΨσY,V=σμY,V2+cos2θYlnΦ2V2
∀ Y∈TNT and V∈TNθ, where σμ is the μ component of σ.Proof.
From (5), (8), and (26), it is easy to see that ηlnΦ=0. Moreover, replacing Y by ΨY in (38), using (2) and part (i), we get part (ii). To prove part (iii), on making use of (8) and (4), we get
(47)σΨY,V=−ζUV+ΨσY,V+Ψ∇VY−∇VΨY.By utilizing (26), the form of the above equation can be changed to as follows:
(48)σΨY,V=−ζYV+ΨσY,V+YlnΦΨV−ΨYlnΦV.By comparing the normal parts, we get(49)σΨY,V=ΨσμY,V+YlnΦFV,taking the inner product with ΨσY,V, we obtain
(50)gσΨY,V,ΨσY,V=σμY,V2+YlnΦgΨσY,V,FV.Calculating the last term of the above equation by using (8), (4), and (22) as follows:
(51)gΨσY,V,FV=gσΨY,V,FV−sin2θYlnΦV2.Utilizing part (ii), we get(52)gΨσY,V,FV=cos2θYlnΦV2,using (50), we obtain the required result.Lemma 7.
Suppose that M=NT×ΦNθ is a WP-PSS submanifold of a Sasakian manifold M¯. Then,
(53)gσY,PV,FV=−gσY,V,FPV=−cos2θYlnΦV2,
for allY∈TNT, V∈TNθ.Proof.
From part (ii) of Lemma6, the following is attained:
(54)gσPY,V,FU+gσPY,U,FV=2YlnΦgV,U,∀ Y∈TNT and V,U∈TNθ. Replacing U by PV∈Dθ and using the fact that V and PV are perpendicular, the following is obtained:
(55)gσY,PV,FV=−gσY,V,FPV.
By using (15), (16), and (26), we get
(56)PYlnΦV−YlnΦPV=tσY,V−ζYV.
Now, taking the inner product withU∈TNθ in the above equation, we have
(57)PYlnΦgV,U−YlnΦgPV,U=−gσY,V,FU−ζXgV,U.
InterchangingV and U and subtracting the resultant from equation (57) lead to
(58)−gσY,V,FU+gσY,U,FV=2YlnΦgV,PU.
In particular, replacingU by PV∈Dθ, we get
(59)gσY,V,FPV−gσY,PV,FV=−2cos2θYlnΦV2.
Using (55) yields
(60)gσY,PV,FV=−gσY,V,FPV=−cos2θYlnΦV2.Lemma 8.
On a WP-PSS submanifold M=NT×ΦNθ of a Sasakian manifold M¯, we obtain
(61)∑i=1q∑j,k=12pgσΨei,ek,Fejgσei,Pek,Fej−gσei,ek,FejgσΨei,Pek,Fej=−4pcos2θ∇lnΦ2,where e0=η,e1,e2,⋯,eq,Ψe1,Ψe2,⋯,Ψeq and e1,e2,⋯,ep,ep+1=secθPe1,ep+2=secθPe2,⋯,e2p=secθPep are the frames of the orthonormal vector fields on TNT and TNθ, respectively.Proof.
First, we expand the left hand term in the following way:(62)∑i=1q∑j,k=12pgσΨei,ek,Fejgσei,Pek,Fej=∑i=1q∑j=12pgσΨei,ej,Fejgσei,Pej,Fej+∑j≠k=12pgσΨei,ek,Fejgσei,Pek,Fej=∑i=1q∑j=12pgσΨei,ej,Fejgσei,Pej,Fej+∑j=1pgσΨei,ej,Fej+pgσei,Pej,Fej+p+∑j=1pgσΨei,ej+p,Fejgσei,Pej+p,Fej=∑i=1q∑j=12pgσΨei,ej,Fejgσei,Pej,Fej+sec2θ∑j=1pgσΨei,ej,FPejgσei,Pej,FPej−∑j=1pgσΨei,Pej,Fejgσei,ej,Fej.Using part (ii) of Lemma6 and Lemma 7 and utilizing (28), we get
(63)∑i=1q∑j,k=12pgσΨei,ek,Fejgσei,Pek,Fej=∑i=1q−2pcos2θeilnΦ2−2pcos2θΨeilnΦ2=−2pcos2θ∇lnΦ2.Replacingei by Ψei in the above equation, we get
(64)∑i=1q∑j,k=12pgσei,ek,FejgσΨei,Pek,Fej=2pcos2θ∇lnΦ2.By subtracting the above two findings, the required result gets attained.The next theorem provides the following characterization.Theorem 9.
Suppose that M=NT×ψNθ is a WP-PSS submanifold of a Sasakian space form M¯c such that NT is a compact submanifold. Then, M is a Riemannian product submanifold if the following inequalities hold
(65)∑i=12q∑j=12pσμei,ej2≤qpc−1sin2θ−2pcos2θ+2cot2θ∇lnΦ2,∑i=1q∑j=12pgσμΨei,ej,σμei,Pej≥0,where σμ stands for the component of σ in μ and 2q+1 and 2p are the dimensions of NT and Nθ, respectively.Proof.
LetY∈TNT and V∈TNθ be unit vector fields. Then, by making use of (2), (6), and (22), we have
(66)R¯Y,ΨY,V,FV=−c−12sin2θY2V2.
However, the Codazzi equation gives(67)R¯Y,ΨY,V,FV=g∇Y⊥σΨY,V,FV−g∇ΨY⊥σY,V,FV+gσY,∇ΨYV,FV−gσΨY,∇YV,FV−gσ∇YΨY,V,FV+gσ∇ΨYY,V,FV.Now, we compute the values of the terms involved in (67). First, we get
(68)g∇Y⊥σΨY,V,FV=YgσΨY,V,FV−gσΨY,V,∇Y⊥FV.Using part (ii) of Lemma6 leads to
(69)g∇Y⊥σΨY,V,FV=Y2lnΦV2+2YlnΦ2V2−gσΨY,V,∇Y⊥FV.Now, we compute the last term of (69) using (11) as follows:
(70)gσΨY,V,∇Y⊥FV=gσΨY,V,∇¯YΨV−PV.By the use of (8) and (13), the above equation becomes
(71)gσΨY,V,∇Y⊥FV=gσΨY,V,∇¯YΨV+Ψ∇¯YV−gσΨY,V,σY,PV.Using (4), (8), (26), and parts (ii) and (iii) of Lemma 6, we get
(72)gσΨY,V,∇Y⊥FV=YlnΦ21+cos2θV2+σμY,V2−gσΨY,V,σY,PV.Utilizing (72) in (69), we obtain
(73)g∇Y⊥σΨY,V,FV=Y2lnΦV2+YlnΦ2sin2θV2−σμY,V2+gσΨY,V,σY,PV.Similarly, we can write(74)g∇ΨY⊥σY,V,FV=−ΨY2lnΦV2−ΨYlnΦ2sin2θV2+σμΨY,V2+gσY,V,σΨY,PV.From the second part of Lemma6, we get
(75)gAFVV,ΨY=YlnΦV2.ReplacingY by ∇YY (using the totally geodesicness of NT, ∇YY∈TNT) in the above equation, we have
(76)gAFVV,Ψ∇YY=∇YYlnΦV2.By using (8), we obtain
(77)gAFVV,Ψ∇¯YY−ΨσY,Y=∇YYlnΦV2.SinceNT is totally geodesic in M, it can be ready to verify that σY,X∈μ, for all Y,X∈TNT. Using this and (13) in equation (77), we have
(78)gσ∇YΨY,V,FV=∇YYlnΦV2.Similarly, we can write(79)gσ∇ΨYY,V,FV=−∇ΨYΨYlnΦV2.By use of (26) and the second part of Lemma 6, we attain
(80)gσΨY,∇YV,FV=YlnΦ2V2,(81)gσY,∇ΨYV,FV=−ΨYlnΦ2V2.Substituting values from (66), (73), (74), (78), (79), (80), and (81) in (67) yields
(82)−c−12sin2θY2V2=Y2lnΦV2+ΨY2lnΦV2−YlnΦ2cos2θV2−ΨYlnΦ2cos2θV2−σμY,V2−σμΨY,V2−∇YYlnΦV2−∇ΨYΨYlnΦV2+gσΨY,V,σY,PV−gσY,V,σΨY,PV.Lete0=η,e1,e2,⋯,eq,eq+1=Ψe1,eq+2=Ψe2,⋯,e2q=Ψeq be the orthonormal frame on TNT and e1,e2,⋯,ep,secθPe1,secθPe2,⋯,secθPep be an orthonormal frame on TNθ. By using the decomposition (40) and formulas (42), (8), and (4) in the last term of the above equation using the orthonormal frame of TNθ, we get
(83)−c−12sin2θY2V2=Y2lnΦV2+ΨY2lnΦV2−YlnΦ2cos2θV2−ΨYlnΦ2cos2θV2−σμY,V2−σμΨY,V2−∇YYlnΦV2−∇ΨYΨYlnΦV2+csc2θ∑j=12pgσΨY,V,FejgσY,PV,Fej−gσΨY,V,FejgσΨY,PV,Fejej2+2gσμΨY,V,σμY,PV.Now, taking the sum of the above equation overi=1,2,⋯,q and j=1,2,⋯2p, using (28) and (29) and part (iii) of Lemma 6, one can get
(84)2pΔlnΦ=qpc−1sin2θ−2pcos2θ∇lnΦ2−∑i=12q∑j=12pσμei,ej2−4pcot2θ∇lnΦ2+2∑i=1q∑j=12pgσμΨei,ej,σμei,Pej.From (84), if
(85)∑i=12q∑j=12pσμei,ej2≤qpc+1sin2θ−2pcos2θ+2cot2θ∇lnΦ2,∑i=1q∑j=12pgσμΨei,ej,σμei,Pej≥0,then ΔlnΦ≥0; hence, from Hopf’s Lemma, lnΦ is constant which means that Φ is constant, which proves the theorem.The squared norm ofσ with reference to the warping function and slant function is obtained in the next theorem as follows.Theorem 10.
Let M¯c be a 2m+1-dimensional Sasakian space form and M=NT×ΦNθ be an n‐dimensional WP-PSS submanifold such that NT is a 2q+1-dimensional invariant submanifold and Nθ is a 2p‐dimensional proper pointwise slant submanifold of M¯c. If
(86)∑i=1q∑j=12pgσΨei,ej,σei,Pej≥0,then
(87)iσ2≥qpc−1sin2θ+2psin2θ∇lnΦ2−2pΔlnΦ.
(ii)
the necessary and sufficient conditions for the equality sign of (87) to be held identically are
(a)
NT is the totally geodesic invariant in M¯c. Furthermore, it is a Sasakian space form
(b)
Nθ is totally umbilical in M¯c
(c)
∑i=1q∑j=12pgσΨei,ej,σei,Pej=0Proof.
From (84), we have
(88)∑i=12q∑j=12pσμei,ej2≥qpc−1sin2θ−2pcos2θ+2cot2θ∇lnΦ2−2pΔlnΦ.For the orthonormal framese0=η,e1,e2,⋯,eq,eq+1=Ψe1,eq+2=Ψe2,⋯,e2q=Ψeq and e1,e2,⋯,ep,secθPe1,secθPe2,⋯,secθPep, in view of formulas (40) and (41) and part (ii) of Lemma 6, we get
(89)∑i=02q∑j=12pσFDθei,ej2=∑i=02q∑j,k=12pcsc2θgσei,ej,Fek2=csc2θ∑i=02q∑j=12pgσei,ej,Fej2+∑j≠k=12pgσei,ej,Fek2=csc2θ∑i=02q2peilnΦ2+sec2θ∑j=1pgσei,ej,FPej2+gσei,Pej,Fej2.Further using Lemma7 and (28), the above equation is reduced to
(90)∑i=02q∑j=12pσFDθei,ej2=2pcsc2θ∇lnΦ2+2pcot2θ∇lnΦ2.From (88) and (90), we attain the desired inequality.To prove part (ii), suppose thatσ′ is the second fundamental form for the immersion of Nθ in M. Then, ∀ V,U∈TNθ and Y∈TNT, the Gauss formula gives
(91)gσ′V,U,Y=g∇VU,Y=−YlnΦgV,U.Using (27), we attain
(92)gσ′V,U,Y=−gV,Ug∇lnΦ,Y,(93)orσ′V,U=−gV,U∇lnΦ.For the equality case of (87), we get
(94)σD,D=0,σDθ,Dθ=0,(95)gσμΨD,Dθ,σμD,PDθ=0.From the first equation of (94), we go to the conclusion that NT is totally geodesic in M. Furthermore, one can readily check that gσY,ΨX,FV=0, ∀ Y,X∈TNT, V∈TNθ. It leads to the conclusion that NT is totally geodesic in M¯c and it is a Sasakian space form as well. Moreover, the second equation of (94) together with (93) yields that Nθ is totally umbilical. This proves the theorem.


## 4. Some Applications

Theorem10 motivates to compute the formula for Dirichlet energy for the warping function Φ. Here, the Dirichlet energy of Φ is denoted by EΦ and is calculated for a compact orientable WP-PSS submanifold M=NT×ΦNθ in a Sasakian space form M¯c as follows.Theorem 11.
Suppose that M=NT×ΦNθ is a compact orientable WP-PSS submanifold of a Sasakian space form M¯c, such that NT is a 2q+1‐dimensional invariant submanifold tangent to the structure vector field η and Nθ is a 2p‐ dimensional pointwise slant submanifold of M¯c. Then, for every y∈Nθ, the necessary and sufficient conditions for the Dirichlet energies of the warping functions to fulfil the following
(96)ElnΦ=14psin2θ∫NT×yσ2dV−pc−14VolNTare the following:
(i)
NT is the totally geodesic invariant in M¯c, and it is a Sasakian space form
(ii)
Nθ is totally umbilical in M¯c
(iii)∑i=1q∑j=12pgσΨei,ej,σei,Pej=0Proof.
By integrating the case of equality of (87) and using the definition of Dirichlet energy and (33), we get the desired result.
Ifθ=π/2, the compact orientable WP-PSS submanifolds become contact CR-warped product submanifolds. The above theorem can be used for deducing the following results.Corollary 12.
Suppose that M=NT×ΦN⊥ is a compact orientable contact CR-warped product submanifold of a Sasakian space form M¯c, such that NT is a 2q+1‐ dimensional invariant submanifold tangent to η and N⊥ is a 2p-dimensional anti-invariant submanifold of M¯c. Then, for every x∈N⊥, the necessary and sufficient conditions for the relation
(97)ElnΦ=14p∫NT×yσ2dV−pc−14VolNTto be held are the following:
(i)
NT is a Sasakian space form and totally geodesic invariant in M¯c
(ii)
N⊥ is a totally umbilical anti-invariant in M¯cIf equality sign of (87) holds, then
(98)∑i=12q∑j=12pσμei,ej2=qpc−1sin2θ−2pcsc2θ+2cot2θ∇lnΦ2−2pΔlnΦ.The Laplacian of a smooth functionΦ is the trace of the Hessian of the function. In terms of Hessian, (98) can be written as follows:
(99)∑i=12q∑j=12pσμei,ej2=qpc−1sin2θ−2pcsc2θ+2cot2θ∇lnΦ2+2p∑i=1qHesslnΦei,ei+HesslnΦΨei,Ψei.Now, the study comprises the following classification theorem for the WP-PSS submanifolds admitting the gradient Ricci soliton fulfilling the equality case of (87).Theorem 13.
Suppose that M¯c is a 2m+1-dimensional Sasakian space form and M=NT×ΦNθ is a WP-PSS submanifold admitting a shrinking gradient Ricci soliton. Then, if
(100)∑i=12q∑j=12pσμei,ej2+4qp=qpc−1sin2θ+2p∑i=12qRicei,ei,one among the following is true:
(i)
θ=π/2, i.e., M is a contact CR-warped product submanifold
(ii)
Φ is constant, i.e., M is trivial Riemannian product pointwise semi slant submanifoldProof.
Suppose that the WP-PSS submanifoldM=NT×ψNθ satisfies the basic equation of the Ricci soliton, such that the potential function τ=lnΦ, then
(101)RicY,X=αgY,X+HessτY,X,for all Y,X∈TNT. Considering that e1,e2,⋯,eq,eq+1=Ψe1,⋯,e2p=Ψeq is an orthonormal basis of the vector fields on TNT. Now, taking summation over i=1,2,⋯,q for Y=X in (101), we have
(102)∑i=1qRicei,ei=αq+∑i=1qHessτei,ei.Replacingei by Ψei in the above equation, we get
(103)∑i=1qRicΨei,Ψei=αq+∑i=1qHessτΨei,Ψei.From (102) and (103), we have
(104)∑i=12qRicei,ei=2αq++∑i=1qHessτei,ei+HessτΨei,Ψei.By the assumption that the equality in (87) holds, then by (99),
(105)12p∑i=12q∑j=12pσμei,ej2=qc−12sin2θ−cos2θ+2cot2θ∇lnΦ2+∑i=12qRicei,ei−2αq,or∑i=12q∑j=12pσμei,ej2=qpc−1sin2θ−2pcos2θ+2cot2θ∇lnΦ2+2p∑i=12qRicei,ei−4qpα.By assumption (100), we get
(106)cos2θ+2cot2θ∇lnΦ2=0.From the above equation, it is evident thatθ=π/2 or Φ is constant, which proves the theorem.If the submanifoldM=NT×ΦNθ admits the steady gradient Ricci soliton, then the following can be easily concluded from the above theorem.Theorem 14.
Suppose that M¯c is a 2m+1-dimensional Sasakian space form and M=NT×ΦNθ is a WP-PSS submanifold admitting a steady gradient Ricci soliton. If the relation
(107)∑i=12q∑j=12pσμei,ej2=qpc−1sin2θ+2p∑i=12qRicei,eiholds, then one can be true from the following:
(i)
θ=π/2, i.e., M is a contact CR-warped product submanifold
(ii)
Φ is constant, i.e., M is trivial Riemannian product pointwise semi slant submanifold
In terms of Ricci curvature, the study constitutes the following classification.Theorem 15.
Suppose that M¯c is a 2m+1-dimensional Sasakian space form and M=NT×ΦNθ is a WP-PSS submanifold with the equality in (87) holds. If the following holds
(108)2p∫MRic∇lnΦ,−dV=qpc−1sin2θ−∑i=12q∑j=12pσμei,ej2,then one among the listed statements must be correct:
(i)
θ=π/2, i.e., M is a contact CR-warped product submanifold
(ii)
Φ is constant, i.e., M is trivial Riemannian product pointwise semi slant submanifoldProof.
For a connection∇ on a smooth manifold M, there exists a tensor R of type (1, 3) called the curvature tensor of the connection ∇ given by
(109)∇U∇VZ−∇V∇UZ−∇V,UZ=RV,UZ,∀ V,U,Z∈TM.For a warping functionτ=lnΦ, from (109), we obtain
(110)∇2dτU,V,Z−∇2dτV,U,Z=dτRV,UZ.By the smoothness property ofΦ on NT and ∇VU2=∇V∇U−∇∇VU, then RV,UZ behaves like a derivative. Since dτ is closed, then it is ready to check that ∇2dτV,U,Z=∇2dτU,V,Z, ∀ V,V,Z∈TNT. Now, for a local orthonormal basis e1,e2,⋯,e2q on NT and for a fixed point t∈NT such that ∇eiejt=0, for 1≤i,j≤2q+1. If we specify ∇eiVt=0, for any V∈TNT, and taking trace with respect to U and Z in the following equation:
(111)∇2dτU,V,Z=∇2dτV,U,Z,and utilizing (110), we have
(112)∑i=1q∇2dτei,ej,V=−dΔτV+Ric∇,V.Further solving the left hand side of (112), it becomes
(113)divHessτV+dΔτV=Ric∇τ,V,the above equation can be written in the form of an operator as follows:
(114)divHessτ+dΔτ=Ric∇τ,⋯.SinceM=NT×ΦNθ is a compact orientable WP submanifold, then on integrating
(115)Δτ+∫MdivHessτdV=∫MRic∇τ,⋯dV,where dV is the volume element.SinceΔΦ=−div∇Φ [26] and ∫MdivVdV=0 for any V∈TNT. So, it is easy to conclude that ∫MdivHessτdV=0. Then,
(116)Δτ=∫MRic∇τ,⋯dV.Utilizing the above equation in (98), we have
(117)∑i=12q∑j=12pσμei,ej2=qpc−1sin2θ−2pcsc2θ+2cot2θ∇lnΦ2−2p∫MRic∇lnΦ,⋯dV.By assumption (108), we get
(118)cos2θ+2cot2θ∇lnΦ2=0.From the above equation, it is evident thatθ=π/2 or Φ is constant, which proves the theorem.


## 5. Conclusion

We obtain the inequalities characterizing the existence of WP-PSS submanifolds of Sasakian space forms via Hopf’s Lemma. Furthermore, we gave an estimation of the squared norm of the second fundamental form with regard to the slant function and warping function. To strengthen our results, we provided an example of a WP-PSS submanifold in a Sasakian manifold. Moreover, some applications in the form of the Dirichlet energy of the warping function are derived. The results obtained may be helpful in further studies on the Dirichlet energy of smooth functions.



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