Almost Cosymplectic and Almost Kenmotsu (κ, μ, ν)-Spaces

Abstract

We study the Riemann curvature tensor of (κ, μ, ν)-spaces when they have almost cosymplectic and almost Kenmotsu structures, giving its writing explicitly. This leads to the definition and study of a natural generalisation of the contact metric (κ, μ, ν)-spaces. We present examples or obstruction results of these spaces in all possible cases.Mathematics Subject Classification (2010). Primary 53C15, Secondary 53C25.

A Generalization of the Goldberg Conjecture for CoKähler Manifolds

“…In the present paper, we shall call a (κ, μ)-almost coKähler manifolds with κ < 0 a proper (κ, μ)-almost coKähler manifold or a non-coKähler (κ, μ)-almost coKähler manifold. Such manifolds with both κ and μ being constants were first introduced by Endo [11] and were generalized to (κ, μ, ν)-spaces by Dacko and Olszak in [10] (see also Carriazo and Martín-Molina [6] and [21]). Using (4.1) we have l = −κφ 2 + μκ and putting this into (2.7) gives that…”

“…First, κ being a constant and μ satisfying dμ ∧ η = 0 follows directly from [21,Proposition 9] and [21,Theorem 6] (see also [6]). Moreover, applying [6,Theorem 3.7], the Riemannian curvature tensor R is given by…”

Invariant Pseudoparallel Submanifolds of an Almost $\alpha$-Cosymplectic $(\kappa,\mu,\nu)$-Space

“…(2) An almost contact manifold M 2n+1 (φ, ξ, η) is said to be normal if the tensor field N = [φ, φ] + 2dη ⊗ ξ = 0, where [φ, φ] denote the Nijenhuis tensor field of φ. It is well known that any almost contact manifold M 2n+1 (φ, ξ, η) has a Riemannian metric such that g(φX, φY ) = g(X, Y ) − η(X)η(Y ), (3) for any vector fields X, Y on M [5]. Such metric g is called compatible metric and manifold M 2n+1 together with the structure (φ, η, ξ, g) is called an almost contact metric manifold and denoted by M 2n+1 (φ, η, ξ, g).…”

“…∇ X ξ = −φX − φhX, hφ + φh = 0, trh = trφh = 0, hξ = 0, (5) where ∇ is the Levi-Civita connection on M 2n+1 [3].…”

Characteristic Jacobi Operator on Almost Cosymplectic 3-Manifolds

“…Let us take a local orthonormal field {e 1 , e 2 , e 3 } on a 3-dimensional generalized almost cosymplectic (κ, µ, ν)space as in Lemma 4.1, then we have Remark 7.1. Carriazo and Martín-Molina [9] obtained the following curvature formula for 3-dimensional generalized almost cosymplectic (κ, µ, ν)-space satisfying (3.2):…”