Harmonic Maps and Stability on f-Kenmotsu Manifolds

In Section 2, we give preliminaries on f-Kenmotsu manifolds. The concept of f-Kenmotsu manifold, where f is a real constant, appears for the first time in the paper of Jannsens and Vanhecke 1 . More recently, Olszak and Roşca 2 defined and studied the f-Kenmotsu manifold by the formula 2.3 , where f is a function on M such that df ∧ η 0. Here, η is the dual 1-form corresponding to the characteristic vector field ξ of an almost contact metric structure on M. The condition df ∧ η 0 follows in fact from 2.3 if dimM ≥ 5. This does not hold in general if dimM 3. A 1-Kenmotsu manifold is a Kenmotsu manifold see Kenmotsu 3, 4 . Theorem 2.1 provides a geometric interpretation of an f-Kenmotsu structure. In Section 3, we initiate a study of harmonic maps when the domain is a compact fKenmotsu manifold and the target is a Kähler manifold. Ianus and Pastore 5, 6 defined a φ, J -holomorphic map between an almost contact metric manifold M φ, η, ξ, g and an almost Hermitian manifold N J, h as a smooth map F : M→N such that the condition F ◦ φ J ◦ F is satisfied. Then, the formula J τ F F divφ −Trgβ holds, where τ F is the tension field of F and β X,Y ̃ ∇XJ F Y ,


Introduction
In Section 2, we give preliminaries on f-Kenmotsu manifolds. The concept of f-Kenmotsu manifold, where f is a real constant, appears for the first time in the paper of Jannsens and Vanhecke 1 . More recently, Olszak and Roşca 2 defined and studied the f-Kenmotsu manifold by the formula 2.3 , where f is a function on M such that df ∧ η 0. Here, η is the dual 1-form corresponding to the characteristic vector field ξ of an almost contact metric structure on M. The condition df ∧ η 0 follows in fact from 2.3 if dim M ≥ 5. This does not hold in general if dim M 3.
A 1-Kenmotsu manifold is a Kenmotsu manifold see Kenmotsu 3,4 . Theorem 2.1 provides a geometric interpretation of an f-Kenmotsu structure.
In Section 3, we initiate a study of harmonic maps when the domain is a compact f-Kenmotsu manifold and the target is a Kähler manifold.
Ianus and Pastore 5, 6 defined a ϕ, J -holomorphic map between an almost contact metric manifold M ϕ, η, ξ, g and an almost Hermitian manifold N J, h as a smooth map F : M→N such that the condition F • ϕ J • F is satisfied. Then, the formula J τ F F div ϕ −Tr g β holds, where τ F is the tension field of F and β X, Y ∇ X J F Y , ∇ being the connection induced in the pull-back bundle F TN see 7 . It is easy to see that in our assumptions div ϕ 0 and Tr g β 0 so that a ϕ, J -holomorphic map between an 2 International Journal of Mathematics and Mathematical Sciences f-Kenmotsu manifold M and a Kähler manifold N is a harmonic map. If M is a compact manifold, a second-order elliptic operator J F , called the Jacobi operator, is associated to the harmonic map F. It is well known that the spectrum of J F consists only of a discrete set of an infinite number of eigenvalues with finite multiplicities, bounded by the first one. We define the Morse index of the harmonic map F as the sum of multiplicities of negative eigenvalues of the Jacobi operator J F 8, 9 . A harmonic map is called stable if the Morse index is zero. We have proven that any ϕ, J -holomorphic map from a compact f-Kenmotsu manifold to a Kähler manifold is a stable harmonic map see 10 .

f-Kenmotsu manifolds
A differentiable 2n 1 -dimensional manifold M is said to have a ϕ, ξ, η -structure or an almost contact structure if there exist a tensor field ϕ of type 1, 1 , a vector field ξ, and a 1-form η on M satisfying where I denotes the identity transformation. It seems natural to include also ϕξ 0 and η • ϕ 0; both can be derived from 2.1 . Let g be an associated Riemannian metric on M such that Putting Y ξ in 2.2 and using 2.1 , we get η X g X, ξ , for any vector field X on M. In this paper, we denote by C ∞ M and Γ E the algebra of smooth functions on M and the C ∞ M -module of smooth sections of a vector bundle E, respectively. All manifolds are assumed to be connected and of class C ∞ . Tensors fields, distribution, and so on are assumed to be of class C ∞ if not stated otherwise.
We say that M is an f-Kenmotsu manifold if there exists an almost contact metric structure ϕ, ξ, η, g on M satisfying The following theorem provides a geometric interpretation of any f-Kenmotsu structure.

Theorem 2.1 Olszak-Roşca . Let M be an almost contact metric manifold. Then, M is f-Kenmotsu if and only if it satisfies the following conditions:
a the distribution D Ker η is integrable and any leaf of the foliation F corresponding to D is a totally umbilical hypersurface with constant mean curvature; b the almost Hermitian structure J, g induced on an arbitrary leaf is Kähler; c ∇ ξ ξ 0 and L ξ ϕ 0.

Vittorio Mangione 3
Moreover, we have which gives div ξ 2nf. The characteristic vector field of an f-Kenmotsu manifold also satisfies Levy proven that a second-order symmetric parallel nonsingular tensor on a space of constant curvature is a constant multiple of the metric tensor 11 . On the other hand, Sharma proven that there is no nonzero skew-symmetric second-order parallel tensor on a Sasakian manifold 12 . For an f-Kenmotsu manifold we have the following theorem.
for any X, Y, Z ∈ Γ T M . Now, let M J, g be a 2m-dimensional almost Hermitian manifold. A surjective map π : M→M is called a contact-complex Riemannian submersion if it is a Riemannian submersion and satisfies 10 In 13 , we have proven the following theorem.

Harmonic maps and stability
Let M, g and N, h be two Riemannian manifolds and F : M→N a differentiable map. Then, the second fundamental form α F of F is defined by

International Journal of Mathematics and Mathematical Sciences
where ∇ is the Levi-Civita connection on M and ∇ is the connection induced by F on the bundle F −1 TN , which is the pull-back of the Levi-Civita connection ∇ on N, and satisfies the following formula see 8 : The tension field τ F of F is defined as the trace of the second fundamental form α F , that is τ F x α F e i , e i x , where e 1 , . . . , e m is an orthonormal basis for T x M at x ∈ M. In what follows, we will use Einstein summation convention, so we will omit the sigma symbol.
We say that a map F : M→N is a harmonic map τ F x ∈ M.
Examples. Now let us consider a variation F s,t ∈ C ∞ M, N , with s, t ∈ −ε, ε and F 0,0 F. If the corresponding variation vector fields are denoted by V and W, the Hessian of F is given by where V g is the canonical measure associated to the Riemannian metric g and J F V is a second-order self-adjoint operator acting on Γ F −1 TN by where R is the curvature operator on N, h . We say that a map f : M, ϕ, ξ, η, g → N, J, h from an almost contact metric manifold to an almost Hermitian manifold is a ϕ, J -holomorphic map if and only if F • ϕ J • F .
If M ϕ, ξ, η, g is a Sasaki manifold and N J, h is a Kähler manifold, then any ϕ, Jholomorphic map from M to N is a harmonic map 14 .
Then, we can prove the same result for any ϕ, J -holomorphic map from an f-Kenmotsu manifold to a Kähler manifold see also 15 . Our main result is the following. If M is compact, the spectrum of J F consists only of a discrete set of an infinite number of eigenvalues with finite multiplicities, bounded below by the first one. We define the Morse index of the harmonic map F : M→N as the sum of multiplicities of negative eigenvalues of the Jacobi operator J F . Equivalently, the Morse index of F equals the dimension of the largest subspace of Γ f −1 TN on which the Hessian H F is negative definite see 8, 9 . Vittorio Mangione 5 We recall the following formula see 5, 9 : where we omitted the summation symbol for repeated indices a 1, . . . , n, n dim M 5 . Now, let e 1 , . . . , e m ; f 1 , . . . , f m , ξ be a local orthonormal ϕ-basis on M ϕ, ξ, η, g such that f i ϕe i , i 1, . . . , m.
From the ϕ, J -holomorphicity of F and by ϕξ 0, we have F ξ 0. Thus, from 3.5 , we obtain the following.

Lemma 3.3. Let T be a vector field on M such that
for any X ∈ Γ D , where D Ker η and g T, ξ 0. Then,

3.9
By using 3.7 and 2.3 , we obtain 3.14 For any V ∈ Γ F −1 TN , we define the operator for any X ∈ Γ TM see 5 .
Using Lemmas 3.2, 3.3, and 3.14 , by a straightforward calculation, we obtain because M div T V g 0. Thus, we have H F V, V ≥ 0 for any V ∈ Γ F −1 TN , so that F is a stable harmonic map.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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