Inequalities on Generalized Sasakian Space Forms

In this paper, we ﬁ nd 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.


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][4][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 S 2n+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 Section 2, 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.

Harmonic Maps on Riemannian Manifolds
We can view harmonic maps on Riemannian manifolds as the generalization of geodesics that is the case of a onedimensional 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 : ðS, gÞ ⟶ ðQ, hÞ. Let the dimension of the Riemannian manifold ðS, gÞ be s and the dimension of ðQ, hÞ be q. The function eðψÞ: S ⟶ ½0,∞Þ that is smooth can be considered as the energy density of ψ and is expressed as at a point p ∈ S and for any orthonormal basis fu 1 , ⋯, u s g of T p S. Considering the compact domain of a Riemannian manifold S, we take the energy density integral as the energy EðψÞ of ψ; that is, we have where the volume measure is represented by υ g that is related to the metric g on manifold S. In the set C ∞ ðS, QÞ of all smooth maps from ðS, gÞ to ðQ, hÞ, a critical point of the energy E is named as a harmonic map. That is, for any smooth variation ψ t ∈ C ∞ ðS, QÞ of ψðt ∈ ð−ε, εÞÞ with ψ 0 = ψ, we can take Now, we consider ðS, gÞ as a compact Riemannian man-ifold and take a map ψ : ðS, gÞ ⟶ ðQ, hÞ that is harmonic. We consider smooth variation ψ r,t through constraints r, t ∈ ð−ε, εÞ satisfying ψ 0,0 = ψ. Respective variational vector fields are represented through W and Z. Therefore, we can define Hessian H ψ for a harmonic map ψ through the following relation: The expression regarding the second variation of E is as follows ( [6,16]): where J ψ is the second order operator that is self-adjoint upon the space Γðψ −1 ðTQÞÞ of variation vector fields and is represented as for U ∈ Γðψ −1 ðTQÞÞ and any local orthonormal frame fu 1 , ⋯, u s g on S. Here, R Q shows the curvature tensor of ðQ, hÞ , and∇ illustrates the pull-back connection of ψ along with the Levi-Civita connection of Q.
We compute the dimension of the biggest subspace of Γðψ −1 ðTQÞÞ where the Hessian H ψ has values that are negative definite known as the index of a harmonic map ψ : ðS , gÞ ⟶ ðQ, hÞ. Therefore, if the index of harmonic map ψ is zero, then it is stable; otherwise, it is unstable.
An operator Δ ψ is represented by It is named the rough Laplacian. We consider the spectra of J ψ ; 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.

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].

Journal of Function Spaces
A Riemannian manifold P 2n+1 with odd dimensions is known as an almost contact manifold if a ð1, 1Þ-tensor field φ exists on P and ξ and a vector field η and a 1-form exist so that Further, φ and η satisfy φðξÞ = 0 and ηoφ = 0. A compatible metric g on any almost contact manifold is defined as for any vector fields W 1 , W 2 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 ΩðW 1 , W 2 Þ = gð W 1 , φW 2 Þ for W 1 , W 2 ∈ ΓðTPÞ. Like the parallel condition of integrability for almost complex manifolds, the almost contact metric structure on P becomes normal when The Nijenhuis torsion of φ is represented by ½φ, φ and is defined as A Sasakian manifold is a normal contact metric manifold, and if dη = 0, a normal almost contact metric manifold is known as the Kenmotsu manifold with 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: where Y 1 , Y 2 , and Y 3 are vector fields on P. An almost contact metric manifold Pðφ, ξ, η, gÞ can be identified as a generalized Sasakian space form provided that there are three functions f 1 , f 2 , f 3 upon P so as the curvature tensor on P is represented with the following relation: provided that vector fields V 1 , V 2 , and V 3 are on P, see [10].
In particular, if f 1 = ðc + 3Þ/4 and f 2 = f 3 = ðc − 1Þ/4, then P can be identified as a Sasakian space form. f 1 = ðc − 3 Þ/4 and f 2 = f 3 = ðc + 1Þ/4 can lead to a Kenmotsu-space form [10,18]. The semisymmetric metric connection ∇ ′ and the Levi Civita connection ∇ defined on contact metric manifold ð P 2m+1 , gÞ are related by the following expression that is obtained by Yano [19] and is represented as where W 1 and W 2 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.

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 form Mðφ, ξ, η, gÞ that is ðϕ = 1 M Þ. Then, the second variation formula is ([2]) as follows: where V ∈ ΓðTMÞ and fu 1 , ⋯, u 2n+1 g represents the local orthonormal frame on TM. The rough Laplacian defined by (7) upon a generalized Sasakian manifold M 2n+1 admitting a semisymmetric metric connection can be computed by the following lemma.  Journal of Function Spaces the adopted frame field fe 1 , ⋯, e n , ϕe 1 , ⋯, ϕe n , ξg is given bý where B Y ðV, WÞ = ηð∇ V YÞW.
Proof. Let ∇ and ∇ represent the semisymmetric connection and the Levi Civita connection on the generalized Sasakian space form, respectively. Therefore, it can be computed aś We have ∇ V ðηðYÞVÞ = ∇ V ðgðξ, YÞVÞ. Then, from equation (19), we havé