
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12803-4
10.1016/j.heliyon.2024.e36772
e36772
Research Article
Homology groups in CR-warped products of complex space forms
Li Yanlin a
Ali Akram akramali133@gmail.com
akali@kku.edu.sa
b⁎
Alluhaibi Nadia c
Mofarreh Fatemah d
Ozel Cenap e
a School of Mathematics, Hangzhou Normal University, Hangzhou 311121, China
b Department of Mathematics, College of Sciences, King Khalid University, Abha, Saudi Arabia
c Department of Mathematics, Science and Arts College, Rabigh Campus, King Abdulaziz University, Jeddah, Saudi Arabia
d Mathematical Science Department, Faculty of Science, Princess Nourah bint Abdulrahman University, Riyadh 11546, Saudi Arabia
e Department of Mathematics, College of Science, King Abdulaziz University, Jeddah, Saudi Arabia
⁎ Corresponding author. akramali133@gmail.comakali@kku.edu.sa
26 8 2024
15 9 2024
26 8 2024
10 17 e367725 5 2023
21 8 2024
22 8 2024
© 2024 The Authors
2024
https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
In the present paper, by using the result of Lawson-Simons [15], we adopt a different technique to show that there are no stable integral q-currents and a non-trivial homology group in a compact oriented CR-warped product submanifold Nn in a complex space forms space Qcm(4c) by imposing some restrictions on the squared norm of gradient and the Laplacian of warped function. Finally, we show that the same approach can be given by using Dirichlet energy, positive eigenvalue, and Hamiltonian.

MSC

primary, 53C40
secondary, 58C35, 53C55, 58Z05, 58J60
Keywords

Homology groups
Stable currents
Eigenvalue
Compact CR-warped products
Homotopic
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pmc1 Main results with their motivations

The classification of topological obstructions of submanifolds has an important character in global Riemannian geometry. An interesting fact is that the essential curvature invariants of a Riemannian manifold affect the topology of a warped product submanifold. In this regard, the classical object in geometry and physics depends on the behavior of the homology groups called topological invariants. They provide important knowledge about the topological character of defined manifolds. This type of connection between stable currents and integral homology classes in Hq(N,G) is provided by Federer-Fleming's in [10]. If the pinching condition on the second fundamental form of a submanifold Nn is satisfied then a submanifold Nn of Sn does not possess stable currents or homology groups see [15]. The present article aims to construct the vanishing homology theorems of warped product submanifolds with holomorphic constant sectional curvature at most one and zero. In this background, the following theorem obtained previously in [15], [27], will be useful Theorem 1.1 [15], [17], [27]LetNnbe a compact submanifold of dimension n in a space formQ˜m(c)of curvaturec≥0. If σ denotes the second fundamental form ofNn, and satisfies the relation(1.1) ∑r=1q∑β=q+1n(2||σ(vr,vs)||2−g(σ(vr,vr),σ(vs,vs)))<qpc,

for any integersq,p>0such thatq+p=n, in this case, there isn't a stable integral q-current inNnandHq(Nn,G)=Hn−q(Nn,G)=0, where{vj}1≤j≤nis an orthonormal frame for the tangent spaceTxNn,

The importance of homotopy theory accentuates its applications of low-dimension statistical-mechanical systems, singularities in liquid crystal, and phase transition in physics [14] while studying warped products and differential topology approaches in mathematical physics effectively relevant in general relativity [12], [20], [21]. Particularly, space-time homology is one of the main apparatuses for quantum gravity [11], [19], [26]. Some new and interesting results for trivial homology groups and stable currents on submanifolds have been gotten by putting some restrictions on the second fundamental form in several structures such as the Euclidean spaces [16], complex projective spaces [18], CR-warped product submanifolds in Sasakian space forms [23], CR-warped product submanifold in the Euclidean spaces [24], CR-warped product in nearly Kaehler manifold [25] ad many others (see [1], [2], [3], [4], [27], [29], [24]). This has motivated the authors of the study to link the concept of warped product manifold and homotopy-homology theory. In [6], [7], Chen started the investigation of warped product CR-submanifolds of Kahler manifolds and provided the characterization theorem for CR-warped product which isometrically immersed in complex space form Qcm, complex hyperbolic space CHm(−4), and complex projective space CPm(4) [see in [6]]. Such work of Chen [6] and the above-mentioned studies have pushed us to extend the study of Lawson and Simon [15] to CR-warped product submanifolds in a complex space form Qcm(4c). Theorem 1.2 Assume thatNn=NTq×fN⊥pis a compact CR-warped product submanifold in a complex space formQcm(4c). If the inequality is satisfied(1.2) (1−p)‖∇f‖2+‖σμ‖2f2<3qcf2+fΔf.

In this case, there isn't a stable integral q-current inNnandHq(Nn,G)=Hp(Nn,G)=0.

Motivated by Lawson and Simon [17, p. 441 Theorem 4], the second goal of the study is to find some new results on the topology for warped product submanifolds. If we replace condition (1.1) with one of the warping functions. Particularly, we prove the following theorem.

Theorem 1.3 Assume thatNn=NTq×fN⊥pis a compact CR-warped product submanifold inQcm(4c). Then, if the following condition is satisfied(1−p)‖∇h‖2+‖σμ‖2h2<3qch2+hΔh,

we have:• Ifq+p≠3, the submanifoldNq+pis homeomorphic to sphereSq+p,

• Ifq+p=3,Nq+pis homotopic to a sphereSq+p.

Let a compact Riemannian manifold N and ϕ be a positive smooth function on Nn, in which ϕ∈F(Nn), then the Lagrangian and the Dirichlet energy ϕ are defined by (see [9]):(1.3) Lϕ=12||∇ϕ||2.

(1.4) E(ϕ)=12∫Nn||∇ϕ||2dV,0<E(φ)<∞

Furthermore, the Euler-Lagrange equation for (1.3) is that(1.5) Δϕ=0.

If manifold Nn without boundary is compact oriented, i.e., ∂Nn=∅. Then, we provide a strong result using (1.4), that is Theorem 1.4 LetNn=NTn×fN⊥pbe compact oriented CR-warped product submanifolds of complex space formQcm(4c). Assume that the following relation is satisfied(1.6) ∫Nn‖σμ‖2f2dV<3q∫Nncf2dV+2pE(f),

whereE(f)is the energy defined in(1.4). Therefore, integral q-current is not stable inNnandHq(Nn,G)=Hp(Nn,G)=0.

A 2th order Laplacian operator on compact manifold Nn is presented byΔϕ=−div(∇ϕ).

An eigenvalue problem of Δ can be used as an example of how such a Laplacian has found many applications in math as well as physics. If there exists a non-zero function ϕ satisfying the following equationΔϕ=λϕ,onNn,

then a real number λ is called eigenvalue depending on boundary conditions. The first nonzero eigenvalue of Δ, denoted by λ1, has a variational characterization (cf. [6]):(1.7) λ1=inf⁡{∫N||∇ϕ||2dV∫N|ϕ|2dV|ϕ∈W1,2(Nn)﹨{0},∫NϕdV=0},

with manifold Nn has empty boundary. Inspired by the above characterization, we give result as follows; Theorem 1.5 LetQcm(4c)be a complex space form andNn=NTn×fN⊥pbe compact oriented CR-warped product ofQcm(4c)satisfies the following inequality(1.8) ‖σμ‖2<(3qc+λ1p),

whereλ1is a positive eigenvalue endowed with Laplacian operator Δ, then integral q-current is not stable in Nn as well as homology groups are trivial, i.e., Hq(Nn,G)=Hp(Nn,G)=0. In addition, Nq+p is homeomorphic to sphere Sq+p when q+p≠3, and Nq+p is homotopic to a sphere Sq+p if q+p=3

The significance of our results comes from the new conditions on the Laplacian of the warped function. Such results can be considered as a vanishing homology theorem as warped product submanifolds have sectional curvature at most one and zero.

2 Preliminaries

A Riemannian manifold Q˜ of dimension m with an almost complex structure J and a Riemannian metric g is called an almost hermitian manifold if the following conditions are satisfiedJ2=−I,andg(JF1,JF2)=g(F1,F2),

∀,F1,F2∈X(TQ˜), where the tangent bundle on Q˜ is denoted by TQ˜. If the almost complex structure J holds the parallel condition, i.e., (∇˜F1J)F2=0, for any F1,F2∈X(TQ˜), then Q˜ is called a Kahler manifold according to Yano and Kon [28]. In such a case, the almost complex structure J is integrable and the fundamental 2 form is closed. On the other hand, a Kahler manifold Q˜m with holomorphic constant sectional curvature c is represented to the complex space form Qcm(4c). In the whole paper, we consider the complex space form Qcm(4c) with curvature tensor R˜ given by;(2.1) R˜(F1,F2,F3,F4)=c{g(F2,F3)g(F1,F4)−g(F2,F4)g(F1,F3)+g(F1,JF3)g(JF2,F4)−g(F2,JF3)g(JF1,F4)+2g(F1,JF2)g(JF3,F4)},

for any F1,F2,F3,F4∈X(Qcm(4c)).

The Gauss and Weingarten formulas for a submanifold Nn of Kaehler manifold Q˜ with induced connections ∇ and ∇⊥ on the tangent bundle TNn and the normal bundle T⊥Nn of Nn, respectively, are defined as;∇˜F1F2=∇F1F2+σ(F1,F2),∇˜F1ξ=−AξF1+∇F1⊥ξ,

∀ F1,F2∈X(TNn) and ξ∈X(T⊥Nn). The notation is used in the above formulas, i.e., A and σ are denoted as shape operators, and the fundamental form, respectively, with the relation holds g(σ(F1,F2),ξ)=g(AξF1,F2). Now, as usual notations, we have(i)JF1=PF1+FF1,(ii)Jξ=tξ+fξ.

The notations PF1(tξ) and FF1(fξ) are referred as tangential and normal parts of JF1(Jξ), respectively. If P=0, then Nn is known as a totally real submanifold. A holomorphic submanifold Nn is defined as J(TxNn)⊆TxNn, for each x∈Nn. Similarly, Nn is totally real if J(TxNn)⊆T⊥Nn, for each x∈Nn.

Definition 2.1 [7] If the pair of orthogonal distributions DT and D⊥ exists in which TNn=DT⊕D⊥, where holomorphic DT and totally real distributions are defined as J(DT)⊆DT, and D⊥ is JD⊥⊆(T⊥Nn), respectively, a Riemannian submanifold Nn of Kahler manifold Q˜m is referred to as a CR-submanifold.

If dim(DT)=q and dim(D⊥)=p in a CR-submanifold Nn of Q˜m, then Nn is• totally real if q=0,

• holomorphic if p=0,

• a proper CR-submanifold if neither q=0 nor p=0.

The normal bundle T⊥Nn can be split asT⊥Nn=JD⊥⊕μ,

where, μ is a holomorphic subspace along J of T⊥Nn. The equation Gauss for a submanifold Nn is presented as;(2.2) R˜(F1,F2,F3,F4)=R(F1,F2,F3,F4)+g(σ(F1,F3),σ(F2,F4))−g(σ(F1,F4),σ(F2,F3)),

∀ F1,F2,F3,F4∈X(TNn), where R˜ and R are presented for scalar curvatures of Q˜m and Nn, respectively. The mean curvature H on Nn is defined byH=1ntrace(σ)=1n∑j=1nσ(vj,vj),

where {v1,v2,⋯vn} is a orthonormal basis for the tangent space TNn and n=dim⁡Nn. In addition, we set(2.3) σjir=g(σ(vj,vi),vr),and||σ||2=∑j,i=1ng(σ(vj,vi),σ(vj,vi)).

According to Bishop and O'Neill [5], the product manifold Nn=N1q×N2p is the warped product manifold N1q×fN2p associated with the Riemannian metric g=g1+f2g2 such that f is classified on the domain in N1q. In this case, f is referred to as a warping function on Nn. Consequently, the given result is very useful. Lemma 2.1 [5]LetNn=N1q×fN2pbe a warped product manifold, then the following properties are well-defined(i) ∇Z2F1=∇F1Z2=(F1f)fZ2,

(ii) ∇Z1Z2=∇Z1′Z2−g(Z1,Z2)∇ln⁡f,

∀ F1∈X(TN1) and Z1,Z2∈X(TN2), where ∇ and ∇′ stand for the Levi-Civitas connections on Nn and N2, respectively. Further, the gradient of ln⁡f is denoted by ∇ln⁡f and presented as;g(∇ln⁡f,F1)=F1(ln⁡f).

Therefore, utilizing the first property of the above lemma, we get(2.4) R(F1,F2)Z1=Hf(F1,Z1)fF2,

where Hf is a Hessian tensor of f and R is the scalar curvature of warped product manifold [5].

Remark 2.1 Trivial or simply a Riemannian product manifold is just a warped product manifold Nn=N1q×fN2p with a constant warping function f along N1q.

Remark 2.2 Moreover, N1q is totally geodesic and N2p is totally umbilical submanifold of Nn in Nn=N1q×fN2p.

One of the interesting results for warped product submanifold proved by Chen in [8] is the following.(2.5) ∑i=1q∑j=1pK(vi∧vj)=pΔff.

From the above, we get(2.6) Δff=Δ(ln⁡f)−||∇(ln⁡f)||2.

As long as the submanifolds of the warped product are completely real and holomorphic, it is referred to as CR-warped. In the setting of Kahler manifold according to Chen in [6], we have two cases of CR-warped products,(i)N⊥p×fNTq,and(ii)NTq×fN⊥p.

For case (i), Chen [6] proved that no warped product CR-submanifold Nn=N⊥p×fNTq exists that is non-trivial. In the same paper, he proved that there are many CR-warped products of the Nn=NTq×fN⊥p and proved the following results(2.7) g(σ(Z1,JF1),JZ2)=(F1ln⁡f)g(Z1,Z2),g(σ(F1,F2),JZ1)=0

∀ F1,F2∈X(TNT) and Z1,Z2∈X(TN⊥).

Proof of Theorem 1.2 Let Nn=NTq×fN⊥p be an n=q+p-dimensional CR-warped product submanifold with dim(NTq)=q=2r and dim(N⊥p)=p=s such that N⊥p and NTq are integral manifolds of D⊥ and D, respectively. Thus, we consider the {v1,v2,⋯vr,vr+1=Jv1,⋯v2r=Jvr} and {v2r+1=e1⁎,⋯v2r+s=ep⁎} to be orthonormal basis of TNT and TN⊥, respectively. Thus the orthonormal basis of the normal subbundles JD⊥ and μ are {vn+1=e¯1,⋯vn+s=e¯β,} and {vn+s+1,⋯vm}, respectively. Then, using (2.2), we get∑r=1q∑s=1pg(R(vr,vs)vr,vs)=∑r=1q∑s=1pg(R˜(vr,vs)vr,vs)+||σ(vr,vs)||2−∑r=1q∑s=1pg(σ(vs,vs),σ(vr,vr).

Add ||σ(vr,vs)||2 to both sides, one obtains(2.8) ∑r=1q∑s=1pg(R(vr,vs)vr,vs)+||σ(vr,vs)||2=∑r=1q∑s=1pg(R˜(vr,vs)vr,vs)−∑r=1q∑s=1pg(σ(vs,vs),σ(vr,vr)+2||σ(vr,vs)||2.

The orthonormal frames {vr}1≤r≤q and {vs}1≤s≤p of NTq and N⊥p, respectively in (2.4), we deriveR(vr,vs)vr=vsfHf(vr,vr).

For the basis {vs}1≤s≤p of tangent space TN which is orthonormal, we derive(2.9) ∑r=1q∑s=1pg(R(vr,vs)vr,vs)=pf∑r=1qg(∇vr∇f,vr).

Thus from Eq (2.8) and (2.9), we derive(2.10) ∑r=1q∑s=1p(2||σ(vr,vs)||2−g(σ(vs,vs),σ(vr,vr)))+∑r=1q∑s=1pg(R˜(vr,vs)vr,vs)=pf∑r=1q∑s=1pg(∇vr∇f,vr)+||σ(vr,vs)||2.

Now, we compute the Laplacian of fΔh=−∑i=1ng(∇vigradf,vi)=−∑r=1qg(∇vrgradf,vr)−∑s=1pg(∇vsgradf,vs).

Using Eq (2.6), we find that(2.11) 1f∑r=1qg(∇vrgradf,vr)=−Δ(ln⁡f)+(1−p)||∇ln⁡f||2.

Therefore, utilizing (2.10) and (2.11), we get(2.12) ∑r=1q∑s=1p(2||σ(vr,vs)||2−g(σ(vs,vs),σ(vr,vr)))+∑r=1q∑s=1pg(R˜(vr,vs)vr,vs)=p(1−p)||∇(ln⁡f)||2−pΔ(ln⁡f)+∑r=1q∑s=1p||σ(vr,vs)||2.

Now, set X=vr and Z=vs for 1≤r≤q and 1≤s≤p, respectively. Follows the (2.3), we defined as∑r=1q∑s=1p||σ(vr,vs)||2=∑r=1q∑s=1pg(σ(vr,vs⁎),vs)2.

In the equation above, there are two components on the right-hand side: the μ-component and the JD⊥-component. Taking the summation over NTq and N⊥p, using the adopted frame for orthonormal vector fields and then utilizing (2.7), we obtain(2.13) ∑r=1q∑s=1p||σ(vr,vs)||2=p||∇ln⁡f||2+‖σμ‖2.

Using (2.12) and (2.13), we find that(2.14) −pΔ(ln⁡f)+p(1−p)||∇(ln⁡f)||2+p||∇(ln⁡f)||2+‖σμ‖2=∑r=1q∑s=1p(2||σ(vr,vs)||2−g(σ(vs,vs),σ(vr,vr)))+∑r=1q∑s=1pg(R˜(vr,vs)vr,vs).

By symmetry of curvature tensor R˜, we get(2.15) ∑r=1q∑s=1pK˜(vr∧vs)=∑r=1q∑s=1pg(R˜(vr,vs)vr,vs).

Hence from (2.1), we get∑r=1q∑s=1pK˜(vr∧vs)=∑r=1q∑s=1p{g(vs,vr)g(vr,vs)−g(vr,vr)g(vs,vs)−g(Jvr,vs)g(Jvs,vs)+g(Jvs,vr)g(Jvr,vs)+2g(vr,Jvs)g(Jvr,vs)},

which implies that(2.16) ∑r=1q∑s=1pK˜(vα∧vβ)=−qpc.

Therefore, from (2.14), (2.15) and (2.16), we get(2.17) ∑r=1q∑s=1p(2||σ(vr,vs)||2−g(σ(vs,vs),σ(vr,vr)))=−pΔ(ln⁡f)+p(1−p)||∇(ln⁡f)||2+p||∇(ln⁡f)||2+‖σμ‖2+qpc.

As our supposition (1.2) holds if and only if the inequality holds from (2.17)∑r=1q∑s=1p(2||σ(vr,vs)||2−g(σ(vs,vs),σ(vr,vr)))<qpc′.

Applying Theorem 1.1 for complex space form Qm(4c) of constant holomorphic sectional curvature c′=4c (1.1), would complete the proof of Theorem 1.2. □

Proof of Theorem 1.4 Using the divergence theorem ([28]) on a compact manifold possibly without boundary, we get that ∫Nn(Δf)dV=0. Therefore, we have:0=∫NnΔ(f22)dV=−∫Nndiv(∇(f22))dV=−∫Nndiv(f∇f)dV=−∫Nng(∇f,∇f)dV+∫NnfΔfdV

which implies that(2.18) ∫NnfΔfdV=∫Nn‖∇f‖2dV.

If the inequality (1.6) holds and from (1.3), we get∫Nn‖σμ‖2f2dV<3qc∫Nnf2dV+p∫Nn‖∇f‖2dV.

Adding and subtracting the Dirichlet energy terms, we get(1−p)∫Nn‖∇f‖2dV+∫Nn‖σμ‖2f2dV<3q∫Nncf2dV+∫Nn‖∇f‖2dV.

Utilizing the (2.18) into above equation, we find that(1−p)∫Nn‖∇f‖2dV+∫Nn‖σμ‖2f2dV<3q∫Nncf2dV+∫NnfΔfdV,

which implies the following(1−p)‖∇f‖2+‖σμ‖2f2<3qcf2dV+fΔf.

By combining Theorem 1.2 with inequality (1.2), we obtain (1.6). As a result, the proof has been completed. □

Proof of Theorem 1.5 From the hypothesis of theorem, the inequality (1.8) is satisfied, then‖σμ‖2<3qc+pλ1

Assuming the above equation is multiplied by f2 on both sides, and integration along the volume element, we have∫Nn‖σμ‖2f2dV<3qc∫Nnf2dV+pλ1∫Nnf2dV.

Using the property (1.7) for the eigenvalue λ1 in proceeding equation, we arrive at(2.19) ∫Nn‖σμ‖2f2dV<3qc∫Nnf2dV+p∫Nn‖∇f‖2dV.

Adding the term ∫Nn‖∇f‖2dV and using (2.18), we reached(1−p)∫Nn‖∇f‖2dV+∫Nn‖σμ‖2f2dV<3qc∫Nnf2dV+∫NnfΔfdV,

this corresponds to the following:(1−p)‖∇f‖2+‖σμ‖2f2<3qcf2+fΔf.

This leads to the result from Theorem 1.2. This is the end of the proof. □

3 Classifications

For any smooth function φ∈C∞(Nn) with a positive tensor is defined asΔφ=−traceHφ.

Then H is called the Hessian tensor. Based on the above relation and the Theorem 1.2, Theorem 1.3, we can verify that the warping function's Hessian tensor has been pinching in the following way. Corollary 3.1 Suppose thatNn=NTq×fN⊥pis a compact oriented CR-warped product submanifold of a complex space formQcm(4c). If the relation(1−p)‖∇f‖2+‖σμ‖2f2+ftraceHf<3qcf2.

In this case, there isn't a stable integral q-current inNnand homology groups are as;Hq(Nn,G)=Hp(Nn,G)=0.

As an analogy, the Dirichlet energy function is used to prove the topological sphere theorem Corollary 3.2 LetQcm(4c)be a complex space form andNn=NTq×fN⊥pbe compact oriented CR-warped product ofQcm(4c)satisfies the following∫Nn‖σμ‖2f2dV<3q∫Nncf2dV+2pE(f),

holds, then• ifq+p≠3, thenNq+pis homeomorphic to sphereSq+p,

• ifq+p=3,Nq+pis homotopic to a sphereSq+p.

Similarly, for Lagrangian satisfied the Euler-Lagrange equation, we have

Corollary 3.3 LetQcm(4c)be a complex space form andNn=NTq×fN⊥pbe compact oriented CR-warped product ofQcm(4c)with inequalityLf<12(1−p){f2(3qc−‖σμ‖2)},

whereLfis the Lagrangian defined in the equation(1.4), thenNnhas no stable integral q-currents and it is homology groups are trivial, i.e.,Hq(Nn,G)=Hp(Nn,G)=0. In addition, statement ofCorollary 3.2holds.

Proof We can deduce the result by using (1.2), (1.4) and utilizing Euler-Lagrange equation condition (1.5). □

The Hamiltonian at any point x∈Nn in the local orthonormal basis can be given as (see [9]):H(p,x)=12∑j=1np(vj)2.

Put p=dφ in the above equation, where d is the differential operator, then it leads to:(3.1) H(dφ,x)=12∑j=1ndφ(vj)2=12∑j=1nvj(φ)2=12||∇φ||2.

From (3.1) and (1.2), we derive a new result as follows Corollary 3.4 LetQcm(4c)be a complex space form andNn=NTq×fN⊥pbe a compact CR-warped product ofQcm(4c)the following relation holdsH(df,x)<f2(1−p){Δf+f(3qc−‖σμ‖2)},

whereH(df,x)stand for the Hamiltonian of f, then there isn't a stable integral q-current inNnand homology groups are as;Hq(Nn,G)=Hp(Nn,G)=0.

Proof Using (3.1) in (1.2), we get desired result. □

Corollary 3.5 Let Qcm(4c) be a complex space form and Nn=NTq×fN⊥p be a compact oriented CR-warped product of Qcm(4c) , and the following condition hold H(df,x)<f2(1−p){Δf+f(3qc−‖σμ‖2)}.

Then Nn is homeomorphic to sphere Sn when q+p≠3 and if q+p=3 , Nq+p is homotopic to a sphere Sn .

4 Conclusion remark

The singularity structure has important applications in liquid crystals and statistical mechanics with low dimensional as well as physical phase transitions (referring to [14]). Moreover, space-time is modeled by warped product manifolds in General Relativity. A warped product space can be divided into two categories. There are two types of static space-times: Robertson-Walker generalizations and standard static space-times [26], [23]. The differential topological methods of mathematical physics are crucial to the theory of general relativity. Quantum gravity particularly uses space-time homology ([13], [22], [23]). Its results can be applied as physical applications due to the connection between warped product manifolds and homotopy-homology theory.

CRediT authorship contribution statement

Yanlin Li: Writing – review & editing, Writing – original draft, Funding acquisition. Akram Ali: Writing – original draft, Conceptualization. Nadia Alluhaibi: Writing – original draft, Supervision. Fatemah Mofarreh: Writing – review & editing, Writing – original draft, Project administration, Conceptualization. Cenap Ozel: Writing – review & editing, Writing – original draft, Formal analysis.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability

No data was used for this manuscript.

Acknowledgement

The authors thank the referees for their invaluable comments and suggestions. The authors extend their appreciation to the Deanship of Scientific Research and Graduate Studies at King Khalid University for this work through a Large Research Project under grant number R.G.P.2/453/45 . Also, the authors express their gratitude to 10.13039/501100004242 Princess Nourah Bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R27 ), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
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