# Homology Groups in Warped Product Submanifolds in Hyperbolic Spaces

Authors: Yanlin Li; Akram Ali; Fatemah Mofarreh; Nadia Alluhaibi
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this paper, we show that if the Laplacian and gradient of the warping function of a compact warped product submanifoldΩp+q in the hyperbolic space ℍm−1 satisfy various extrinsic restrictions, then Ωp+q has no stable integral currents, and its homology groups are trivial. Also, we prove that the fundamental group π1Ωp+q is trivial. The restrictions are also extended to the eigenvalues of the warped function, the integral Ricci curvature, and the Hessian tensor. The results obtained in the present paper can be considered as generalizations of the Fu–Xu theorem in the framework of the compact warped product submanifold which has the minimal base manifold in the corresponding ambient manifolds.


---

## Body

## 1. Introduction and Main Results

For any Riemannian manifoldΩn, it is well known that any integral homology class in HqΩn,ℤ which is nontrivial is correlated to integral stable currents. This result was initially proved by Federer and Fleming [1]. Utilizing the method of variational calculus for the geometric measure concept combining with the method of Federer and Fleming, Lawson and Simons [2] obtained the optimization for the second fundamental form, which leads to the vanishing homology in a range of intermediate dimensions and the nonexistence of stable currents in the submanifold in the simply connected space form, and obtained the key theorem of that paper.Theorem 1 (see [2–4]).
SupposeΩn is a compact submanifold of dimension n in the space form M˜c of constant curvature c. If the strict inequality for the second fundamental form Π(1)∑α=1p∑β=p+1n2Πeα,eβ2−gΠeα,eα,Πeβ,eβ<pn−sgncpcis satisfied, then Ωn has no stable p-currents with the vanished pth homology group, i.e.,(2)HpΩn,ℤ=HqΩn,ℤ=0,where p+q=n and sgnc is the signature of constant curvature c.The geometric structure and topological properties of submanifolds in different spaces have been studied on a large scale during the past few years [4–17]. Many results showed that there is a closed relationship between stable currents which are nonexistent and the vanished homology groups of submanifolds in a different class of the ambient manifold obtained by imposing conditions on the second fundamental form (1). For example, as an application of the Ricci curvature and the ambient manifold is an Euclidean space, Vlachoas [18] proved that a compact oriented submanifold Ωn of dimension n in Euclidean space Em of dimension m=n+k satisfies the pinching condition RicX>δ1ngAHX,X, in which X is any unit vector, AH is the shape operator regarding the mean curvature H, and δ1n is a constant such that δ1n=n−2 if n is even and δ1n=nn−3/n−1 if n is odd. Therefore, Ωn has no stable currents. Moreover, Ωn is homeomorphic to Sn. Moreover, using Theorem 1 in [19], it was found that if a compact oriented submanifold of dimension nΩn in space form Fcm satisfied the second fundamental form pinching condition S<an,k,H,c, for any integer k in which 0<k<n and c is a constant sectional curvature, then the pth homology groups are vanishing, HpΩn,ℤ=0, for all p∈k,…,n−k, and if the fundamental group π1Ωn is finite and simply connected, then Ωn is homeomorphic to Sn. Using Theorem 1, Xu and Gu [16] extended the pinching condition in terms of the Ricci curvature and showed that Ωn in Fcm satisfied Ric>n−2c+H2; then, Ωn is homeomorphic to Sn. Motivated by the nonexistence of stable submanifolds or stable currents, a number of topological properties have been studied by many authors [3, 4, 14, 15, 17, 19–24] inspired by Theorem 1.Inspired by the aforementioned results, we want to obtain some similar results of warped product submanifold theory where the second fundamental form pinching condition shall be replaced by the warping function. Using Theorem1, we now give the first main result of this note.Theorem 2.
LetΨ:Ωp+q=N1p×hN2q⟶ℍm−1 be an isometric embedding from a compact warped product submanifold Ωp+q into an m-dimensional hyperbolic space ℍm−1 in which the base manifold N1p is minimal in ℍm−1 and the following inequality(3)3hΔh<2q∇h2+pq2p+qh2,hold, where ∇h and Δh are the gradient and Laplacian of the function h, respectively. Then, we have the following:(a)
The warped product submanifoldΩp+q does not have any stable integral p-currents.(b)
Thei th integral homology groups of Ωp+q with integer coefficients are vanished, which means(4)ℍpΩp+q,ℤ=ℍqΩp+q,ℤ=0.(c)
Ifp=1, then the fundamental group π1Ω is null, i.e., π1Ω=0. Moreover, Ωp+q is a simply connected warped product manifold.Motivated by the geometric rigidity (Theorem2), the second goal of this approach is to prove a new vanishing theorem for compact warped product submanifolds in terms of the Ricci curvature and using the eigenvalue of Laplacian of the warping function. In particular, we can give the following theorem.Theorem 3.
Under the assumption of Theorem2 and if the warping function h is an eigenfunction of Laplacian of Ωp+q corresponding to the first positive eigenvalue λ1 and satisfying the strict inequality(5)∇2h2+Ric∇h,∇hλ12q4p2q+2p+3λ1h2>0for integral Hessian tensor ∇2h of the warping function h and the integral Ricci curvature along the gradient ∇h, then we have the following:(a)
The warped product submanifoldΩp+q does not have any stable integral p-currents.(b)
Theithe integral homology groups of Ωp+q with integer coefficients are vanished; that is,(6)ℍpΩp+q,ℤ=ℍqΩp+q,ℤ=0.(c)
The fundamental groupπ1Ω is null, i.e., π1Ω=0. Moreover, Ωp+q is a simply connected warped product manifold.Now, we give a direct application of Theorem3.Theorem 4.
Assume thatΨ:Ωp+q=N1p×hN2q⟶ℍm−1 is an isometric embedding from a compact warped product submanifold Ωp+q into an m-dimensional hyperbolic space ℍm−1 satisfying the following inequality:(7)∫Ωp+q∇2h2dV<∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1−λ12q4p2q+2p+3λ1∫Ωp+qh2dV.
Then, statements (a), (b), and (c) in Theorem2 hold.Another interesting result obtained from Theorem4 is the following:Corollary 1.
Under the same assumption as Theorem4 and if ∇h∈KerΠ with the following holds,(8)∫Ωp+q‖∇2h‖2dV<p−1−λ12q4p2q+2p+3λ1∫Ωp+qh2dV,then, statements (a), (b), and (c) in Theorem 2 are satisfied.Remark 1.
Theorem2 is the main vanishing homology theorem for a compact warped product submanifold with no need for Ωp+q to be simply connected. Moreover, our result is of significance due to involving the new pinching conditions in terms of the warping function, the integral of the squared norm of the Hessian tensor, the integral Ricci curvature, and the first nontrivial eigenvalue of the warped function.


## 2. Preliminaries

Letℍmc denote the hyperbolic space with dimension m and constant sectional curvature c=−1. We use the fact that ℍmc has a canonical isometric embedding in Lorentz–Minkowski space L1m+1 which is the vector space ℝm+1 with the metric(9)<,>=dx12+⋯dxn2−dxn+12,where x1⋯xn+1 are canonical coordinates in L1m+1. Therefore, the hyperbolic space ℍm with negative constant curvature c is(10)ℍmc=X∈L1m+1:X2=1c,xn+1>0.Thus,(11)R˜X,Y,Z,W=gY,WgX,Z−gX,WgY,Z,∀X,Y,Z,W∈ΓTM˜, where TM˜ is a tangent bundle of ℍm and R˜ is the Riemannian curvature tensor of ℍm. This means that ℍm is a manifold of constant sectional curvature −1.Assume thatΩn admits isometric immersion into a Riemannian manifold M˜m with induced metric g. Then, the formula(12)R˜X,Y,Z,W=RX,Y,Z,W+gΠX,Z,ΠY,W−gΠX,W,ΠY,Zis the Gauss equation for Ωn in which X,Y,Z,W∈XTΩ, and R and R˜ are the curvature tensors on Ωn and M˜, respectively. The mean curvature vector H for a local orthonormal basis e1,e2,…,en on Ω is defined by(13)H2=1n2∑r=n+1m∑α=1nΠαα2.In addition, we set(14)Παβr=gΠeα,eβ,er,andΠeα,eβ2=∑α,β=1ngΠeα,eβ,Πeα,eβ2.In this connection, we shall define the scalar curvatureτTxΩn of Riemannian submanifold Ωn, which is considered as a Riemannian intrinsic invariant, at some x in Ωn as follows:(15)τTxΩn=∑1≤β<≤αnKβα,where Kβα=Keβ∧eα. The first equality (15) is equivalent to the following equation which will be frequently used in subsequent proofs:(16)2τTxΩn=∑1≤β<α≤nKβα,1≤β,α≤n.In a similar manner, the scalar curvatureτLx of an L−plane is defined as(17)τLx=∑1≤β<α≤nKβα.If the plane sections are spanned byeβ and eα at x, we give denotations K˜βα and Kβα, respectively, for the sectional curvature of the Riemannian manifold M˜m and submanifold Ωn, which are considered as the extrinsic and intrinsic sectional curvature of the span eβ,eα at x. Using Gauss (12) and (15), we have(18)∑1≤β<α≤n+kKβα=∑1≤β<α≤n+kK˜βα+∑r=n+1mΠββrΠααr−Πβαr2.On the contrary, the conception of warped product manifolds was originally introduced by Bishop and O’Neill [25] for manifolds of negative curvature. Assume that N1p and N2q are two Riemannian manifolds with Riemannian metrics g1 and g2, respectively. Assume that h is a differentiable function on N1p. A warped product manifold is Ωp+q=N1p×hN2q with n=p+q and the Riemannian metric g=g1+h2g2. Assume that Ωp+q=N1p×hN2q is a warped product manifold. Hence, ∀W1∈ΓTN1 and W2∈ΓTN2, we attain that(19)∇W2W1=∇W1W2=W1lnhW2.Furthermore,∇lnh is the gradient of lnh, given by(20)g∇lnh,W1=W1lnh.Thus, from [26], we have(21)RW1,W2Y=HhW1,YhW2,where Hh is a Hessian tensor of h. Let e1,…,en be a local orthonormal basis of vector field Ωp+q; thus, the squared norm of the gradient of the positive differential function h for an orthonormal basis e1,…,en is(22)∇h2=∑i=1neih2.Now, if we replacep+q in (20) by n, we get(23)g∇⟶h,W1=W1h,and∇⟶h=∑β=1neβheβ,and Laplacian Δh of h is defined as(24)Δh=−∑β=1n∇eβeβh−eβeβh=∑β=1ng∇eβgradh,eβ=trHessh.Remark 2.
It should be noted that we consider the opposite sign of Chen [27] of Laplacian of h; that is, Δh=div∇h.Moreover, because unit vector fieldsW1 and W2 are tangent to N1p and N2q, respectively, we get(25)KW1∧W2=gRW1,W2W1,W2=∇W1W1lnhgW2,W2−g∇W1W1lnhW2,W2,=∇W1W1lnhgW2,W2−g∇W1W1lnhW2+W2lnh∇W1W2,W2,=∇W1W1lnhgW2,W2−W1lnh2−W1W1lnh.Let us sum up over the vector fields such that(26)∑β=1p∑α=1qKeβ∧eα=∑β=1p∑α=1q∇eβeβlnh−eβeβlnh−eβlnh2,which leads to(27)∑β=1p∑α=1qKeβ∧eα=−qΔhh.


## 3. Proof of Main Results

We also need to use the following method which is an important tool in proving our result. In the first case, we assume that the warped product submanifold is isometrically immersed in a hyperbolic spaceℍm−1; we will give the proof of our main results.


### 3.1. Proof of Theorem2

LetdimN1=p and dimN2=q, and consider e1,e2,…,ep and ep+1∗,…,en∗ to be orthonormal frames of TN1 and TN2, respectively. Then, from Gauss equation (12) for the standard hyperbolic space ℍm−1, we have(28)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=∑β=1p∑α=1qgReβ,eαeβ,eα−∑β=1p∑α=1qgR˜eβ,eαeβ,eα+∑r=1m∑β=1p∑α=1qΠβαr2.FromReβ,eαeβ=ℋheβ,eβ/heα in (21), we derive(29)∑β=1p∑α=1qgReβ,eαeβ,eα=qh∑β=1pg∇eβ∇h,eβ.Thus, from (28) and (29), we derive(30)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=qh∑β=1pg∇eβ∇h,eβ+−∑β=1p∑α=1qgR˜eβ,eαeβ,eα∑r=1m∑β=1p∑α=1qΠβαr2.Computing the LaplacianΔh on Ωp+q, one obtains(31)Δh=∑β=1pg∇eβgradh,eβ+∑i=1ng∇eigradh,ei=∑α=1qg∇eαgradh,eα.SinceN1p is totally geodesic in Ωp+q, gradh∈XTN1, and by utilizing the definition of the warped product, we obtain(32)Δh=1h∑α=1qgeα,eα∇h2+∑β=1pg∇eβgradh,eβ.Multiplying the above equation by1/h, we get(33)Δhh=1h∑β=1pg∇eβgradh,eβ+q∇lnh2.After some computations, we find that(34)qh∑β=1pg∇eβgradh,eβ=qΔhh−q2∇lnh2.Thus, from (30) and (34), one obtains(35)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=∑r=n+1m∑β=1p∑α=1qΠβαr2+qΔhh−q2∇lnh2−∑β=1p∑α=1qgR˜eβ,eαeβ,eα.Next, using Gauss equation (12) for hyperbolic space ℍm−1, we find that(36)n2H2−nn−1=Π2+∑1≤A<B≤nKeA∧eB.By rewriting the above equation forΩp+q and utilizing (13) and (21), we attain(37)∑t=n+1m∑A=1nΠAAt2∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2+∑β=1p∑α=1qKeβ∧eα+∑1≤s<l≤pKes∧el+∑1≤a<b≤qKea∧eb.Using (18) and (27) in the above equation, we derive(38)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh+nn−1+∑1≤s<l≤pK˜es∧el+∑1≤a<b≤qK˜ea∧eb+∑t=n+1m∑1≤s<l≤pΠsstΠllt−Πslt2+∑t=n+1m∑1≤a<b≤qΠaatΠbbt−Πabt2.Thus, using the curvature equation and the sphereℍm−1 and rearranging the last equation, we attain(39)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+nn−1−pp−1−qq−1+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh−∑t=n+1m∑1≤s<l≤pΠslt2+∑t=n+1m∑1≤s<l≤pΠsstΠllt+∑t=n+1mΠ11t2+⋯+Πpp2−∑t=n+1mΠ11t2+⋯+Πpp2+∑t=n+1m∑1≤a<b≤qΠaatζbbt−∑t=n+1m∑1≤a<b≤qΠabt2+∑t=n+1mΠp+1p+1t2+⋯+Πnn2−∑t=n+1mΠp+1p+1t2+⋯+Πnn2.After some rearrangements of the above equation, we get(40)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh−2pq+∑t=n+1m∑1≤s<l≤pΠsstΠllt+Π11t2+⋯+Πpp2−∑t=n+1m∑1≤s<l≤pΠslt2+Π11t2+⋯+Πpp2+∑t=n+1m∑1≤a<b≤qΠaatΠbbt+Πp+1p+1t2+⋯+Πnn2−∑t=n+1m∑1≤a<b≤qΠabt2+Πp+1p+1t2+⋯+Πnn2.Utilizing the binomial theorem and the fact that the baseN1p of a warped product manifold N1p×hN2q is minimal, we can readily obtain that(41)∑t=n+1m∑A=p+1nΠAAt2=−2pq+∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh+∑t=n+1mΠ11t2+⋯+Πpp2−∑t=n+1m∑s,l=1pΠslt2−∑t=n+1m∑a,b=1qΠabt2+∑t=n+1mΠp+1p+1t2+⋯+Πnn2.Utilizing the assumption of the theorem and sinceN1p is minimal, the fifth term of the right-hand side (RHS) in equation (41) is identically zero, and the first term of the left-hand side is equal to the seventh term of the RHS. Hence, we get(42)2∑t=n+1m∑β=1p∑α=1qΠβαt2=qΔhh+2pq.From (36) and (42), we get(43)∑β=1p∑α=1q2Πeβ,eα2−gΠ,Πeα,eαeβ,eβ=qΔhh−q2∇lnh2+qΔh2h+pq−∑β=1p∑α=1qgR˜eβ,eαeβ,eα.Then, from equation (11), one obtains(44)∑β=1p∑α=1qgR˜eβ,eαeβ,eα=pq.This follows that(45)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=3qΔh2h−q2h2∇h2.From our assumption (3) and (45), we obtain(46)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ<p2p+q=pn+p.Applying Theorem1 for constant holomorphic sectional curvature c=1, we obtain that there are no stable p-currents in Ωp+q and HpΩp+q,ℤ=HqΩp+q,ℤ=0, which completes the proof of (a) and (b) of the theorem. In the other part, from (45), substituting p=1, we have(47)∑α=2n2Πe1,eα2−gΠeα,eα,Πe1,e1=3qΔh2h−qh2∇h2.If the pinching condition (22) for p=1 and q=n−1 holds, then we get(48)∑α=2n2Πe1,eα2−gΠeα,eα,Πe1,e1<n+1.Then, there are no stable 1-currents inΩ1+q and H1Ω1+q,ℤ=Hn−1Ω1+q,ℤ=0. Let us assume that π1Ω≠0. The compactness property of Ω1+q; it follows from the classical theorem, using the results of Cartan and Hadamard, which states that there is a minimal closed geodesic in any nontrivial homotopy class in π1Ω, and this leads to a contradiction. Therefore, π1Ω = 0. This is the third part of the theorem. If the finite fundamental group is null for any Riemannian manifold, this Riemannian manifold is simply connected. As a result, Ωp+q is simply connected.


### 3.2. Proof of Theorem3

IfΔh=−λ1h, that is, h is the first eigenfunction of Laplacian Δh=div∇h of Ωp+q corresponding to the first nonzero eigenvalue λ1 of h. Recall the Bochner formula for a differentiable function h defined on a Riemannian manifold (see, e.g., [28]) which states that the following formula(49)12Δ∇h2=∇2h2+Ric∇h,∇h+g∇h,∇Δh,holds. Then, by integrating the above equation and using Stokes’ theorem, we get(50)∫Ωp+q∇2h2dV+∫Ωp+qRic∇h,∇hdV+∫Ωp+qg∇h,∇ΔhdV=0.Now, usingΔh=−λ1h and making some rearrangements in equation (50), we derive(51)∫Ωp+q∇h2dV=1λ1∫Ωp+q∇2h2dV+∫Ωp+qRic∇h,∇hdV.By assumption of the theorem, (5) holds, then we have(52)∫Ωp+q∇2h2+Ric∇h,∇hdV+λ12q4p2q+2p+3λ1∫Ωp+qh2dV>0.Then, substituting equation (52) in (51), we get(53)−λ12q4p2q+2p+3λ1∫Ωp+qh2dV<λ1∫Ωp+q∇h2dV,which implies that(54)−3λ1∫Ωp+qh2dV<2pq2p+q∫Ωp+qh2dV+2q∫Ωp+q∇h2dV.Now, usingΔ=−λ1h in the left-hand side of equation (54), we arrive at(55)∫Ωp+q3hΔh−2q∇h2−2pq2p+qh2dV<0.Then, the above equation gives us(56)3hΔh<2q∇h2+2pq2p+qh2.Then, using the above equation and also Theorem2, we get the conclusion of our theorem.


### 3.3. Proof of Theorem4

As we know thatΩp+q is N1P-minimal compact warped product submanifold, from the Gauss equation, one obtains(57)Rjkli=−δikδjl−δilδjk+∑r=1mΠikrΠjlr−ΠilrΠjkr,which implies the following:(58)Rjiji=−δiiδjj−δijδji+∑r=1mΠiirΠjjr−ΠijrΠjir.Taking into account thatN1p is a minimal submanifold and using the argument of the Ricci curvature for the hyperbolic space, we get(59)Ricei,ej=−p−1δij−∑r=1m∑l=1pΠilrΠjlr.The above equation yields that(60)Richiei,ejhj=−p−1δijhihj−∑r=1m∑l=1pΠilrΠjlrhihj.Using equation (60), we get(61)Ric∇h,∇h=−p−1∇h2−∑i=1pΠ∇h,ei2.Plugging the above equation in (51), we obtain(62)∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1+λ1∫Ωp+q∇h2dV=∫Ωp+q∇2h2dV.If our assumption (1.4) is satisfied, then(63)∫Ωp+q∇2h2dV<∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1−λ12q4p2q+2p+3λ1∫Ωp+qh2dV.Rewrite the above equation usingΔh=−λ1h:(64)3p−1−λ12q∫Ωp+qhΔhdV+∫Ωp+q∇2h2dV<∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1−λ12q4p2q+2p∫Ωp+qh2dV.Inserting the above equation in (62), we derive that(65)3p−1−λ12q∫Ωp+qhΔhdV<p−1−λ1∫Ωp+q∇h2dV+p−1−λ12q4p2q+2p∫Ωp+qh2dV,which implies the following:(66)3hΔh<2q∇h2+4p2q+2ph2.Thus, we get the conclusion of our theorem.


### 3.4. Proof of Corollary1

Using the hypothesis of the corollary,∇h∈KerΠ, we get Π∇h,ei=0. Using this condition in (7), we can easily obtain the required result.


## 3.1. Proof of Theorem2

LetdimN1=p and dimN2=q, and consider e1,e2,…,ep and ep+1∗,…,en∗ to be orthonormal frames of TN1 and TN2, respectively. Then, from Gauss equation (12) for the standard hyperbolic space ℍm−1, we have(28)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=∑β=1p∑α=1qgReβ,eαeβ,eα−∑β=1p∑α=1qgR˜eβ,eαeβ,eα+∑r=1m∑β=1p∑α=1qΠβαr2.FromReβ,eαeβ=ℋheβ,eβ/heα in (21), we derive(29)∑β=1p∑α=1qgReβ,eαeβ,eα=qh∑β=1pg∇eβ∇h,eβ.Thus, from (28) and (29), we derive(30)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=qh∑β=1pg∇eβ∇h,eβ+−∑β=1p∑α=1qgR˜eβ,eαeβ,eα∑r=1m∑β=1p∑α=1qΠβαr2.Computing the LaplacianΔh on Ωp+q, one obtains(31)Δh=∑β=1pg∇eβgradh,eβ+∑i=1ng∇eigradh,ei=∑α=1qg∇eαgradh,eα.SinceN1p is totally geodesic in Ωp+q, gradh∈XTN1, and by utilizing the definition of the warped product, we obtain(32)Δh=1h∑α=1qgeα,eα∇h2+∑β=1pg∇eβgradh,eβ.Multiplying the above equation by1/h, we get(33)Δhh=1h∑β=1pg∇eβgradh,eβ+q∇lnh2.After some computations, we find that(34)qh∑β=1pg∇eβgradh,eβ=qΔhh−q2∇lnh2.Thus, from (30) and (34), one obtains(35)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=∑r=n+1m∑β=1p∑α=1qΠβαr2+qΔhh−q2∇lnh2−∑β=1p∑α=1qgR˜eβ,eαeβ,eα.Next, using Gauss equation (12) for hyperbolic space ℍm−1, we find that(36)n2H2−nn−1=Π2+∑1≤A<B≤nKeA∧eB.By rewriting the above equation forΩp+q and utilizing (13) and (21), we attain(37)∑t=n+1m∑A=1nΠAAt2∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2+∑β=1p∑α=1qKeβ∧eα+∑1≤s<l≤pKes∧el+∑1≤a<b≤qKea∧eb.Using (18) and (27) in the above equation, we derive(38)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh+nn−1+∑1≤s<l≤pK˜es∧el+∑1≤a<b≤qK˜ea∧eb+∑t=n+1m∑1≤s<l≤pΠsstΠllt−Πslt2+∑t=n+1m∑1≤a<b≤qΠaatΠbbt−Πabt2.Thus, using the curvature equation and the sphereℍm−1 and rearranging the last equation, we attain(39)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+nn−1−pp−1−qq−1+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh−∑t=n+1m∑1≤s<l≤pΠslt2+∑t=n+1m∑1≤s<l≤pΠsstΠllt+∑t=n+1mΠ11t2+⋯+Πpp2−∑t=n+1mΠ11t2+⋯+Πpp2+∑t=n+1m∑1≤a<b≤qΠaatζbbt−∑t=n+1m∑1≤a<b≤qΠabt2+∑t=n+1mΠp+1p+1t2+⋯+Πnn2−∑t=n+1mΠp+1p+1t2+⋯+Πnn2.After some rearrangements of the above equation, we get(40)∑t=n+1m∑A=1nΠAAt2=∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh−2pq+∑t=n+1m∑1≤s<l≤pΠsstΠllt+Π11t2+⋯+Πpp2−∑t=n+1m∑1≤s<l≤pΠslt2+Π11t2+⋯+Πpp2+∑t=n+1m∑1≤a<b≤qΠaatΠbbt+Πp+1p+1t2+⋯+Πnn2−∑t=n+1m∑1≤a<b≤qΠabt2+Πp+1p+1t2+⋯+Πnn2.Utilizing the binomial theorem and the fact that the baseN1p of a warped product manifold N1p×hN2q is minimal, we can readily obtain that(41)∑t=n+1m∑A=p+1nΠAAt2=−2pq+∑t=n+1m∑s,l=1pΠslt2+∑t=n+1m∑a,b=1qΠabt2+2∑t=n+1m∑β=1p∑α=1qΠβαt2−qΔhh+∑t=n+1mΠ11t2+⋯+Πpp2−∑t=n+1m∑s,l=1pΠslt2−∑t=n+1m∑a,b=1qΠabt2+∑t=n+1mΠp+1p+1t2+⋯+Πnn2.Utilizing the assumption of the theorem and sinceN1p is minimal, the fifth term of the right-hand side (RHS) in equation (41) is identically zero, and the first term of the left-hand side is equal to the seventh term of the RHS. Hence, we get(42)2∑t=n+1m∑β=1p∑α=1qΠβαt2=qΔhh+2pq.From (36) and (42), we get(43)∑β=1p∑α=1q2Πeβ,eα2−gΠ,Πeα,eαeβ,eβ=qΔhh−q2∇lnh2+qΔh2h+pq−∑β=1p∑α=1qgR˜eβ,eαeβ,eα.Then, from equation (11), one obtains(44)∑β=1p∑α=1qgR˜eβ,eαeβ,eα=pq.This follows that(45)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ=3qΔh2h−q2h2∇h2.From our assumption (3) and (45), we obtain(46)∑β=1p∑α=1q2Πeβ,eα2−gΠeα,eα,Πeβ,eβ<p2p+q=pn+p.Applying Theorem1 for constant holomorphic sectional curvature c=1, we obtain that there are no stable p-currents in Ωp+q and HpΩp+q,ℤ=HqΩp+q,ℤ=0, which completes the proof of (a) and (b) of the theorem. In the other part, from (45), substituting p=1, we have(47)∑α=2n2Πe1,eα2−gΠeα,eα,Πe1,e1=3qΔh2h−qh2∇h2.If the pinching condition (22) for p=1 and q=n−1 holds, then we get(48)∑α=2n2Πe1,eα2−gΠeα,eα,Πe1,e1<n+1.Then, there are no stable 1-currents inΩ1+q and H1Ω1+q,ℤ=Hn−1Ω1+q,ℤ=0. Let us assume that π1Ω≠0. The compactness property of Ω1+q; it follows from the classical theorem, using the results of Cartan and Hadamard, which states that there is a minimal closed geodesic in any nontrivial homotopy class in π1Ω, and this leads to a contradiction. Therefore, π1Ω = 0. This is the third part of the theorem. If the finite fundamental group is null for any Riemannian manifold, this Riemannian manifold is simply connected. As a result, Ωp+q is simply connected.


## 3.2. Proof of Theorem3

IfΔh=−λ1h, that is, h is the first eigenfunction of Laplacian Δh=div∇h of Ωp+q corresponding to the first nonzero eigenvalue λ1 of h. Recall the Bochner formula for a differentiable function h defined on a Riemannian manifold (see, e.g., [28]) which states that the following formula(49)12Δ∇h2=∇2h2+Ric∇h,∇h+g∇h,∇Δh,holds. Then, by integrating the above equation and using Stokes’ theorem, we get(50)∫Ωp+q∇2h2dV+∫Ωp+qRic∇h,∇hdV+∫Ωp+qg∇h,∇ΔhdV=0.Now, usingΔh=−λ1h and making some rearrangements in equation (50), we derive(51)∫Ωp+q∇h2dV=1λ1∫Ωp+q∇2h2dV+∫Ωp+qRic∇h,∇hdV.By assumption of the theorem, (5) holds, then we have(52)∫Ωp+q∇2h2+Ric∇h,∇hdV+λ12q4p2q+2p+3λ1∫Ωp+qh2dV>0.Then, substituting equation (52) in (51), we get(53)−λ12q4p2q+2p+3λ1∫Ωp+qh2dV<λ1∫Ωp+q∇h2dV,which implies that(54)−3λ1∫Ωp+qh2dV<2pq2p+q∫Ωp+qh2dV+2q∫Ωp+q∇h2dV.Now, usingΔ=−λ1h in the left-hand side of equation (54), we arrive at(55)∫Ωp+q3hΔh−2q∇h2−2pq2p+qh2dV<0.Then, the above equation gives us(56)3hΔh<2q∇h2+2pq2p+qh2.Then, using the above equation and also Theorem2, we get the conclusion of our theorem.


## 3.3. Proof of Theorem4

As we know thatΩp+q is N1P-minimal compact warped product submanifold, from the Gauss equation, one obtains(57)Rjkli=−δikδjl−δilδjk+∑r=1mΠikrΠjlr−ΠilrΠjkr,which implies the following:(58)Rjiji=−δiiδjj−δijδji+∑r=1mΠiirΠjjr−ΠijrΠjir.Taking into account thatN1p is a minimal submanifold and using the argument of the Ricci curvature for the hyperbolic space, we get(59)Ricei,ej=−p−1δij−∑r=1m∑l=1pΠilrΠjlr.The above equation yields that(60)Richiei,ejhj=−p−1δijhihj−∑r=1m∑l=1pΠilrΠjlrhihj.Using equation (60), we get(61)Ric∇h,∇h=−p−1∇h2−∑i=1pΠ∇h,ei2.Plugging the above equation in (51), we obtain(62)∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1+λ1∫Ωp+q∇h2dV=∫Ωp+q∇2h2dV.If our assumption (1.4) is satisfied, then(63)∫Ωp+q∇2h2dV<∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1−λ12q4p2q+2p+3λ1∫Ωp+qh2dV.Rewrite the above equation usingΔh=−λ1h:(64)3p−1−λ12q∫Ωp+qhΔhdV+∫Ωp+q∇2h2dV<∫Ωp+q∑i=1pΠ∇h,ei2dV+p−1−λ12q4p2q+2p∫Ωp+qh2dV.Inserting the above equation in (62), we derive that(65)3p−1−λ12q∫Ωp+qhΔhdV<p−1−λ1∫Ωp+q∇h2dV+p−1−λ12q4p2q+2p∫Ωp+qh2dV,which implies the following:(66)3hΔh<2q∇h2+4p2q+2ph2.Thus, we get the conclusion of our theorem.


## 3.4. Proof of Corollary1

Using the hypothesis of the corollary,∇h∈KerΠ, we get Π∇h,ei=0. Using this condition in (7), we can easily obtain the required result.



---

*Source: 8554738-2021-09-30.xml*