1. Introduction

The symplectic structures of symplectic manifolds ðM, ω, JÞ are, by Darboux's theorem 2.1, locally equivalent to the standard symplectic structure on Euclidean space.

In Section 2, we introduce basic definitions on symplectic manifolds [1–5, 10–13] and flux homomorphism. In Section 2.1, we recall J-holomorphic curve, moduli space of J-holomorphic curves, Gromov-Witten invariant and Gromov-Witten potential, quantum product and quantum cohomology, and in Section 2.2, symplectic action functional and its gradient flow line, Maslov type index of critical loop, Floer cochain complex and Floer cohomology, and theorem of Arnold conjecture.

In Section 3, we introduce almost contact metric manifolds ðM, g,ϕ, η, ξ, φÞ with a closed fundamental 2-form φ and their product [4, 7, 8]. In Section 3.1, we study ϕ-coholomorphic

distribution, and eproduction in any medium, provided the original work is properly cited.

map, moduli space of ϕ-coholomorphic maps which represent a homology class of dimension two, Gromov-Witten type cohomology, quantum type product and quantum type cohomology, Gromov-Witten type potentials on the product of a symplectic manifold, and an almost contact metric manifold [5, 6, 13]. In Section 3.2, we investigate the symplectic type action functional on the universal covering space of the contractible loops, its gradient flow line, the moduli space of the connecting flow orbits between critical loops, Floer type cochain complex, and Floer type cohomology with coefficients in a Novikov ring [7, 9, 13].

In Section 4, as conclusions we show that the Floer type cohomology and the quantum type cohomology of an almost contact metric manifold with a closed fundamental 2-form are isomorphic [7, 13], and present some examples of almost contact metric manifolds with a closed fundamental 2-form.
