A Remark on Four-dimensional Almost Kähler-einstein Manifolds with Negative Scalar Curvature

Concerning the Goldberg conjecture, we will prove a result obtained by applying the result of Iton in terms of L 2-norm of the scalar curvature.

As a corollary, he also proved the following.
In this paper, concerning the Goldberg conjecture, we will prove a result obtained by using Corollary 1.3 (see Theorem 2.2).

Preliminaries and the result.
Let M = (M,J,g) be a four-dimensional almost Kähler-Einstein manifold with the almost-complex structure J and the Hermitian metric g.We denote by Ω the Kähler form of M defined by Ω(X, Y ) = g(X, JY ) for X, Y ∈ X(M), the set of all smooth vector fields on M. We assume that M is oriented by the volume form dV = Ω 2 /2.We denote by ∇, R, ρ, and τ the Riemannian connection, the curvature tensor, the Ricci tensor, and the scalar curvature of M, respectively.We assume that the curvature tensor is defined by We denote by ρ * the Ricci * -tensor of M defined by for x, y ∈ T p M, p ∈ M. Now, let ∧ 2 M be the vector bundle of all real 2-forms on M. The bundle ∧ 2 M inherits a natural inner product g and we have an orthogonal decomposition where LM (resp., ∧ where ι is the duality between the tangent bundle and the cotangent bundle of M by means of the metric g.Let {e 1 ,e 2 = Je 1 , e 3 ,e 4 = Je 3 } be a (local) unitary frame field and put e i = ι(e i ).Then, the Kähler form is represented by Ω = −e 1 ∧ e 2 − e 3 ∧ e 4 .Further, we see that and so on, where the Latin indices run over the range 1, 2, 3, 4. We define functions A, B, C, D, G, and K on M by where u = g((Φ), Φ), v = g((JΦ), JΦ), and w = g((Φ), JΦ).First, we will prove the following.
Next, we recall the following equalities established in [6]: where LM is the restriction of to LM and LM = P LM • LM , the composition of LM and the natural projection P LM : ∧ 2 M → LM.We define a vector field η = (η a ) on M by η a = 4 i,j=1 (∇ a J ij )ρ * i j , then we obtain the following (see [6, (2.23)]): Further, from (2.12) and the curvature identity by Gray [3] for almost Kähler manifold, we have (2.15) Thus, from (2.12) and this equality, we obtain

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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First
Round of ReviewsMay 1, 2009 We note that if M is Kählerian, the Ricci tensor and the Ricci * -tensor coincide on M. The * -scalar curvature τ * of M is the trace of the linear endomorphism Q for x, y, z ∈ T p M, the tangent space of M at p ∈ M. The Ricci * -tensor satisfies ρ * (x, y) = ρ * (Jy, Jx) for any x, y∈ T p M, p ∈ M. * defined by g(Q * x, y) = ρ * (x, y) for x, y ∈ T p M, p ∈ M. Since ∇J 2 = 2(τ * −τ), Mis a Kähler manifold if and only if τ * −τ = 0 on M.An almost Hermitian manifold M is called a weakly * -Einstein manifold if ρ * = λ * g (λ * = τ * /4) and a * -Einstein if M is weakly * -Einstein with constant * -scalar curvature.