# Inequalities on Generalized Sasakian Space Forms

Authors: Najma Abdul Rehman; Abdul Ghaffar; Esmaeil Abedi; Mustafa Inc; Mohammed K. A. Kaabar
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this paper, we find the second variational formula for a generalized Sasakian space form admitting a semisymmetric metric connection. Inequalities regarding the stability criteria of a compact generalized Sasakian space form admitting a semisymmetric metric connection are established.


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## Body

## 1. Introduction

The harmonic maps have aspects from both Riemannian’s geometry and analysis. Harmonic mappings are considered a vast field, and because of the minimization of energy due to its dual nature, it has many applications in the field of mathematics, physics, relativity, engineering, geometry, crystal liquid, surface matching, and animation. Some particular examples of harmonic maps are geodesics, immersion, and solution of the Laplace equation. In physics,p-harmonic maps were studied in image processing. Exponential harmonic maps were discussed in the field of gravity. Due to generalized properties, F-harmonic maps have many applications in cosmology. Harmonic maps have played a significant role in Finsler’s geometry. On complex manifolds, we have interesting and useful outcomes of harmonic maps (for details, see [1, 2]).During the past years, harmonicity on almost contact metric manifolds has been considered a parallel to complex manifolds ([3–5]). The identity map on a Riemannian manifold with a compact domain becomes a trivial case of the harmonicity. However, the stability and second variation theory are complex and remarkable here. In [6], a Laplacian upon functions with its first eigenvalue is used to explain stability on Einstein’s manifolds. From [7, 8], we know about the stability-based classification of a Riemannian that simply connected irreducible spaces with a compact domain.From [6], we know a well-known result about the stability of S2n+1. Further in [5], identity map stability upon a compact domain of the Sasakian space form was explained by Gherge et al. (see also [9]). Considering the generalization of Sasakian space forms, Alegre et al. presented the generalized Sasakian space forms [10]. Therefore, we naturally study the identity map stability upon a compact domain of generalized Sasakian space forms, as discussed in some results in [11]. One of the most important terms in differential geometry is connection. Research on manifolds is incomplete without the notion of connection. In manifold theory, from the relation of metric and connection, we have a very important notion known as curvature tensor. The concept of a semisymmetric metric connection was initiated by Friedmann and Schouten in 1932 [12, 13]. Semisymmetric metric connections have many applications in the field of Riemannian manifolds and are useful to study many physical problems. In the current paper, we compute the stability criteria of a generalized Sasakian space form admitting a semisymmetric metric connection.After recollecting the essential facts about harmonic maps between Riemannian manifolds in Section2, we explain generalized Sasakian space forms throughout Section 3. In Section 4, we give the main results for a second variational formula and establish the inequalities for the identity map stability criteria upon a compact domain generalized Sasakian space form admitting a semisymmetric metric connection.


## 2. Harmonic Maps on Riemannian Manifolds

We can view harmonic maps on Riemannian manifolds as the generalization of geodesics that is the case of a one-dimensional domain and range as Euclidean space. In common, a map is known as harmonic if its Laplacian becomes zero and is known as totally geodesic if its Hessian becomes zero. In this present section, the basic facts of the harmonic maps theory [14, 15] are provided. Consider a smooth map ψ:S,g⟶Q,h. Let the dimension of the Riemannian manifold S,g be s and the dimension of Q,h be q. The function eψ:S⟶0,∞ that is smooth can be considered as the energy density of ψ and is expressed as(1)eψp=12Trgψ∗hp=12∑i=1shψ∗pui,ψ∗pui,at a point p∈S and for any orthonormal basis u1,⋯,us of TpS. Considering the compact domain of a Riemannian manifold S, we take the energy density integral as the energy Eψ of ψ; that is, we have(2)Eψ=∫Seψυg,where the volume measure is represented by υg that is related to the metric g on manifold S. In the set C∞S,Q of all smooth maps from S,g to Q,h, a critical point of the energy E is named as a harmonic map. That is, for any smooth variation ψt∈C∞S,Q of ψt∈−ε,ε with ψ0=ψ, we can take(3)ddtEψtt=0=0.Now, we considerS,g as a compact Riemannian manifold and take a map ψ:S,g⟶Q,h that is harmonic. We consider smooth variation ψr,t through constraints r,t∈−ε,ε satisfying ψ0,0=ψ. Respective variational vector fields are represented through W and Z. Therefore, we can define Hessian Hψ for a harmonic map ψ through the following relation:(4)HψW,Z=∂2∂r∂tEψr,tr,t=0,0.The expression regarding the second variation ofE is as follows ([6, 16]):(5)HψW,Z=∫PhJψW,Zυg,whereJψ is the second order operator that is self-adjoint upon the space Γψ−1TQ of variation vector fields and is represented as(6)JψU=−∑i=1s∇~ui∇~ui−∇~∇uiuiU−∑i=1sRQU,dψuidψui,for U∈Γψ−1TQ and any local orthonormal frame u1,⋯,us on S. Here, RQ shows the curvature tensor of Q,h, and ∇~ illustrates the pull-back connection of ψ along with the Levi-Civita connection of Q.We compute the dimension of the biggest subspace ofΓψ−1TQ where the Hessian Hψ has values that are negative definite known as the index of a harmonic map ψ:S,g⟶Q,h. Therefore, if the index of harmonic map ψ is zero, then it is stable; otherwise, it is unstable.An operatorΔ¯ψ is represented by(7)Δ¯ψU=−∑i=1s∇~ui∇~ui−∇~∇uiuiU,V∈Γϕ−1TQ. It is named the rough Laplacian. We consider the spectra of Jψ; because of the Hodge de Rham Kodaira theory, this spectra is constructed as a discrete set of infinite number of eigenvalues with finite multiplicities with no accumulation points.


## 3. Generalized Sasakian Space Forms

Generalized Sasakian space forms have the generalized curvature expression that combines the curvature expessions of Sasakian, Kenmotsu, and Cosymplectic space forms. Due to a generalized curvature expression, generalized Sasakian space forms have very useful and interesting properties. The current unit presents basics of almost contact metric manifolds particularly of generalized Sasakian space forms [17].A Riemannian manifoldP2n+1 with odd dimensions is known as an almost contact manifold if a 1,1-tensor field φ exists on P and ξ and a vector field η and a 1-form exist so that(8)φ2=−I+η⊗ξ,ηξ=1.Further,φ and η satisfy φξ=0 and ηoφ=0. A compatible metric g on any almost contact manifold is defined as(9)gφW1,φW2=gW1,W2−ηW1ηW2,for any vector fields W1, W2 on manifold P known as an almost contact metric manifold. An almost contact metric manifold becomes a contact metric manifold if for a fundamental 2-form Ω, we have dη=Ω, and ΩW1,W2=gW1,φW2 for W1, W2∈ΓTP. Like the parallel condition of integrability for almost complex manifolds, the almost contact metric structure on P becomes normal when(10)φ,φ+2dη⊗ξ=0.The Nijenhuis torsion ofφ is represented by φ,φ and is defined as(11)φ,φY1,Y2=φ2Y1,Y2+φY1,φY2−φφY1,Y2−φY1,φY2.A Sasakian manifold is a normal contact metric manifold, and ifdη=0, a normal almost contact metric manifold is known as the Kenmotsu manifold with(12)dΩY1,Y2,Y3=23σY1,Y2,Y3ηY1ϕY2,Y3,Y1,Y2,Y3∈ΓTP,where the cyclic sum is represented by σ. A real space form is a Riemannian manifold with a constant sectional curvature c, and its curvature tensor is represented by the following relation:(13)RY1,Y2Y3=cgY2,Y3Y1−gY1,Y3Y2,where Y1, Y2, and Y3 are vector fields on P. An almost contact metric manifold Pφ,ξ,η,g can be identified as a generalized Sasakian space form provided that there are three functions f1, f2, f3 upon P so as the curvature tensor on P is represented with the following relation:(14)RV1,V2V3=f1gV2,V3V1−gV1,V3V2+f2gV1,ϕV3ϕV2−gV2,ϕV3ϕV1+2gV1,ϕV2ϕV3+f3ηV1ηV3V2−ηV2ηV3V1+gV1,V3ηV2ξ−gV2,V3ηV1ξ,provided that vector fields V1, V2, and V3 are on P, see [10]. In particular, if f1=c+3/4 and f2=f3=c−1/4, then P can be identified as a Sasakian space form. f1=c−3/4 and f2=f3=c+1/4 can lead to a Kenmotsu-space form [10, 18].The semisymmetric metric connection∇′ and the Levi Civita connection ∇ defined on contact metric manifold P2m+1,g are related by the following expression that is obtained by Yano [19] and is represented as(15)∇W1′W2=∇W1W2+ηW2W1−gW1,W2ξ,where W1 and W2 are vector fields on P. As mentioned in [20], we have the following relation of the curvature tensor R with respect to the Levi-Civita connection ∇ and the curvature tensor R′ regarding the semisymmetric metric connection ∇′ of the generalized Sasakian space form.(16)R′V1,V2V3=RV1,V2V3+gϕV2,V3V1−gϕV1,V3V2+gV2,V3ϕV1−gV1,V3ϕV2+ηV2V1−ηV1V2ηV3+gV2,V3ηV1−gV1,V3ηV2ξ,taking vector fields V1, V2, V3, on P.


## 4. Stability on Generalized Sasakian Space Forms with Semisymmetric Metric Connection

Identity maps are always harmonic maps, but here, the second variational formula is not a trivial case. In this section, with the help of the second variational formula, we derive the inequalities for the stability criteria on the generalized Sasakian space forms with a semisymmetric metric connection. Consider the identity map on a compact generalized Sasakian space formMφ,ξ,η,g that is ϕ=1M. Then, the second variation formula is ([2]) as follows:(17)H1MV,V=∫MhΔ¯V,Vυg−∑i=12n+1∫MhRV,uiui,Vυg,where V∈ΓTM and u1,⋯,u2n+1 represents the local orthonormal frame on TM.The rough Laplacian defined by (7) upon a generalized Sasakian manifold M2n+1 admitting a semisymmetric metric connection can be computed by the following lemma.Lemma 1.
For a generalized Sasakian space form admitting semisymmetric metric connection, the rough Laplacian in the adopted frame fielde1,⋯,en,ϕe1,⋯,ϕen,ξ is given by(18)Δ¯´Y=Δ¯Y+2trBY−gtr∇,Yξ−2divYξ+2ηYξ−2Y+ϕY+∑gei,Yϕei+∑gei,ϕYei,where BYV,W=η∇VYW.Proof.
Let∇` and ∇ represent the semisymmetric connection and the Levi Civita connection on the generalized Sasakian space form, respectively. Therefore, it can be computed as(19)∇´V∇´VY=∇V∇´VY+η∇´VYV−gV,∇´VYξ=∇V∇VY+∇VηYV−∇VgV,Yξ+η∇VYV+ηVηYV−gV,YV−gY,∇VVξ−gV,∇VYξ−gV,Y∇Vξ.
We have∇VηYV=∇Vgξ,YV. Then, from equation (19), we have(20)∇´V∇´VY=∇V∇VY+g∇Vξ,YV−∇VgV,Yξ+ηY∇VV+2η∇VYV+ηVηYV−gV,YV−g∇VV,Yξ−gV,∇VYξ−gV,Y∇Vξ=∇V∇VY+g∇Vξ,YV+ηY∇VV+2η∇VYV+ηVηYV−gV,YV−2g∇VV,Yξ−2gV,∇VYξ−2gV,Y∇Vξ=∇V∇VY+gV,ϕYV+ηY∇VV+2η∇VYV−2g∇VV,Yξ−2gV,∇VYξ−2gV,Y∇Vξ=∇V∇VY+gV,ϕYV+ηY∇VV+2η∇VYV−2g∇VV,Yξ−2gV,∇VYξ−2gV,Y∇Vξ=∇V∇VY+gV,ϕYV+ηY∇VV+2η∇VYV−2g∇VV,Yξ−2gV,∇VYξ+2gV,YϕV−2gV,YV+2ηVgV,Yξ.
Also, we have(21)∇´V∇´VY−∇´∇VVY=∇V∇VY−∇∇VVY+gV,ϕYV+2η∇VYV−g∇VV,Yξ−2gV,∇VYξ+2gV,YϕV−2gV,YV+2ηVgV,Yξ.
Take into account thatBYV,W=η∇VYW. Then, in an adopted frame field e1,⋯,en,ϕe1,⋯,ϕen,ξ, we arrived at(22)Δ¯´Y=Δ¯Y+2trBY−gtr∇,Yξ−2divYξ+2ηYξ−2Y+ϕY+∑gei,Yϕei+∑gei,ϕYei.Theorem 2.
The second variation formula for the identity map on the generalized Sasakian space form admitting a semisymmetric connection is expressed as(23)H1MY,Y=∫MhΔ¯Y,Yυg−3f2+2nf1−f3−2n+3∫MhY,Yυg+3f2+2n−1f3−2n+3∫MηYηYυg.Proof.
(24)H1MY,Y=∫MhΔ′¯Y,Yυg−∑i=12n+1∫MhR′Y,uiui,Yυg,hΔ´¯Y,Y=hΔY¯,Y+2htrBY,Y−htr∇,Yhξ,Y−2divYhξ,Y+2ηYhξ,Y−2hY,Y+hϕY,Y+∑hei,Yhϕei,Y+∑hei,ϕYhei,Y,Δ′¯Y,Y=hΔY¯,Y+2htrBY,Y−htr∇,Yhξ,Y−2divYhξ,Y+2ηYhξ,Y−2hY,Y+hϕY,Y+∑hei,Yhϕei,Y+∑hei,ϕYhei,Y,since ∫MdivY=0, over a compact domain M, by Green’s formula and η∇eiY=h∇eiY,ξ=eihY,ξ−hY,∇eiξ=0, similarly, htr∇,Yhξ,Y=0. Therefore, we have (25)∫MhΔ′¯Y,Yυg=∫MhΔ¯Y,Yυg+2∫Mη2Yυg−2∫MhY,Yυg.
Now, we consider aφ-adapted orthonormal local frame ei,φei,ξ. After that, we have(26)∑i=12n+1hRei,Yei,Y=f1−3f2∑i=1nhY,ei2+hY,φei2−2n+1f1−f3hY,Y+2n−1f3+f1hY,ξ2,and thus, we have(27)∑i=12n+1hRei,Yei,Y=−3f2+2nf1−f3hY,Y+3f2+2n−1f3hY,ξ2,and with semisymmetric metric connection, it can be written as(28)∑i=12n+1hR′Y,eiei,Y=∑i=12n+1hRY,eiei,Y−2n−1hY,Y+2n−1η2Y=3f2+2nf1−f3hY,Y−3f2+2n−1f3hY,ξ2−2n−1hY,Y+2n−1η2Y.
From (24) and (28), we have acquired the result of ((24)).Proposition 3.
Consider a compact generalized Sasakian space formM admitting a semisymmetric metric connection. The identity map 1M is weakly stable, if 3f2+2nf1−f3−2n+3≤0 and 3f2+2n−1f3−2n+3≥0.Proof.
We can easily prove that(29)∫MhΔ¯V,Vυg=∫Mh∇~V,∇~Vυg,V∈ΓTM.☐Now, the second variation formula with respect to a semisymmetric connection becomes(30)H1MY,Y=∫Mh∇~Y,∇~Y−3f2+2nf1−f3−2n+3∫MhY,Yυg+3f2+2n−1f3−2n+3∫MηYηYυg.Therefore, for the inequalities3f2+2nf1−f3−2n+3≤0 and 3f2+2n−1f3−2n+3≥0, the identity map is weakly stable.Corollary 4.
LetM be the Kenmotsu space form admitting a semisymmetric metric connection; then, the identity map on its compact domain is stable if 3n−7/n+1≤c≤7n−1/n+1.
On the Kenmotsu space formM, f1=c−3/4, f2=f3=c+1/4 [10]. And 3f2+2nf1−f3−2n+3≤0 implies c≤7n−1/n+1, and 3f2+2n−1f3−2n+3≥0 implies c≥3n−7/n+1. Then, by the above results, the identity of the 1M map becomes stable for the values of c∈3n−7/n+1,7n−1/n+1.


## 5. Conclusion

The2nd variational formula for a generalized Sasakian space form admitting a semisymmetric metric connection has been successfully obtained in this work. All results in this work are novel where inequalities concerning the stability criteria of a compact generalized Sasakian space form admitting a semisymmetric metric connection have been established. Further research works can be conducted depending on all our obtained results in this paper.



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