Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds

Abstract

In this chapter, we introduce Gromov-Witten invariant, quantum cohomology, Gromov-Witten potential, and Floer cohomology on symplectic manifolds, and in connection with these, we describe Gromov-Witten type invariant, quantum type cohomology, Gromov-Witten type potential and Floer type cohomology on almost contact metric manifolds. On the product of a symplectic manifold and an almost contact metric manifold, we induce some relations between Gromov-Witten type invariant and quantum cohomology and quantum type invariant. We show that the quantum type cohomology is isomorphic to the Floer type cohomology.

1. Introduction

The symplectic structures of symplectic manifolds $(M,\omega ,J)$ 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 $(M,g,\phi ,\eta ,\xi ,\varphi )$ with a closed fundamental 2-form $\varphi$ and their product [4, 7, 8]. In Section 3.1, we study $\phi$ -coholomorphic map, moduli space of $\phi$ -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.

2. Symplectic manifolds

By a symplectic manifold, we mean an even dimensional smooth manifold $M^{2n}$ together with a global 2-form $\omega$ which is closed and nondegenerate, that is, the exterior derivative $d\omega =0$ and the n-fold wedge product ${\omega }^n$ never vanishes.

Examples: (1) The $2n$ -dimensional Euclidean space ${\mathbb{R}}^{2n}$ with coordinates $(x_1,\dots ,x_n,y_1,\dots ,y_n)$ admits symplectic form ${\omega }_0={\displaystyle {\displaystyle \sum }_{i=1}^n}d{x}_i\wedge d{y}_i$ .

(2) Let $M$ be a smooth manifold. Then its cotangent bundle $T^{*}M$ has a natural symplectic form as follows. Let $\pi :T^{*}M\to M$ be the projection map and $x_1,\dots ,x_n$ are local coordinates of $M$ . Then $q_i=x_i\circ \pi$ , $i=1,2,\dots ,n$ together with fiber coordinates $p_1,\dots ,p_n$ give local coordinates of $T^{*}M$ . The natural symplectic form on $T^{*}M$ is given by

$\omega ={\displaystyle {\displaystyle \sum }_{i=1}^n}d{q}_i\wedge d{q}_j.$ E1

(3) Every Kähler manifold is symplectic.

Darboux’s Theorem 2.1 ([6]). Every symplectic form $\omega$ on $M$ is locally diffeomorphic to the standard form ${\omega }_0$ on ${\mathbb{R}}^{2n}$ .

A symplectomorphism of $(M,\omega )$ is a diffeomorphism $\varphi \in Diff(M)$ which preserves the symplectic form ${\varphi }^{*}\omega =\omega$ . Denote by $\text{Sym}(M)$ the group of symplectomorphims of $M$ . Since $\omega$ is nondegenerate, there is a bijection between the vector fields $X\in \text{Γ}(TM)$ and 1-forms $\omega \left(X,\cdot \right)\in {\text{Ω}}^1(M)$ . A vector field $X\in \text{Γ}(TM)$ is called symplectic if $\omega (X,\cdot )$ is closed.

Let $M$ be closed, i.e., compact and without boundary. Let $\varphi :\mathbb{R}\to \text{Diff}(M)$ , $\text{t}\mapsto {\varphi }_t$ be a smooth family of diffeomorphisms generated by a family of vector fields $X_t\in \text{Γ}(TM)$ via,

$\frac{d}{dt}{\varphi }_t=X_t\circ {\varphi }_t,{\varphi }_0=\text{id}.$ E2

Then ${\varphi }_t\in \text{Symp}(M)$ if and only if $X_t\in \text{Γ}(TM,\omega )$ the space of symplectic vector fields on $M$ . Moreover, if $X$ , $Y\in \text{Γ}(TM,\omega )$ , then $\left[\text{X},\text{Y}\right]\in \Gamma (TM,\omega )$ and $\omega \left(\left[X,Y\right],\cdot \right)=dH$ , where $H=\omega \left(X,Y\right):M\to R$ . Let $H:M\to R$ be a smooth function. Then the vector field $X_H$ on $M$ determined by $\omega \left(X_H,\cdot \right)=dH$ is called the Hamiltonian vector field associated with $H$ . If $M$ is closed, then $X_H$ generates a smooth 1-parameter group of diffeomorphisms ${\varphi }_H^t\in \text{Diff}(M)$ such that

$\frac{d}{dt}{\varphi }_H^t=X_H\circ {\varphi }_H^t,{\varphi }_H^0=\text{id}.$ E3

This $\left\{{\varphi }_H^t\right\}$ is called the Hamiltonian flow associated with $H$ . The flux homomorphism $\text{Flux}$ is defined by

$\text{Flux}\left\{{\varphi }_H^t\right\}={\displaystyle \underset{0}{\overset{1}{{\displaystyle \int }}}}\omega \left(X_t,\cdot \right)dt.$ E4

Theorem 2.2 ([6]). $\varphi \in Sym(M)$ is a Hamiltonian symplectomorphism if and only if there is a homotopy $\left[0,1\right]\to \text{Sym}\left(M\right)$ , $t\mapsto {\varphi }_t$ such that ${\varphi }_0=\text{id}$ , ${\varphi }_1=\varphi$ , and $\text{Flux}\left(\left\{{\varphi }_t\right\}\right)=0$ .

2.1. Quantum cohomology

Let $(M,\omega )$ be a compact symplectic manifold. An almost complex structure is an automorphism of $TM$ such that $J^2=-I$ . The form $\omega$ is said to tame $J$ if $\omega \left(v,Jv\right)>0$ for every $v\ne 0$ . The set ${\mathfrak{I}}_{\tau }(M,\omega )$ of almost complex structures tamed by $\omega$ is nonempty and contractible. Thus the Chern classes of $TM$ are independent of the choice $J\in {\mathfrak{I}}_{\tau }(M,\omega )$ . A smooth map $\varphi :(M_1,J_1)\to (M_2,J_2)$ from $M_1$ to $M_2$ is $(J_1,J_2)$ -holomorphic if and only if

$d{\varphi }_x\circ J_1=J_2\circ d{\varphi }_x$ E5

Hereafter, we denote by $H_2(M)$ the image of Hurewicz homomorphism ${\pi }_{2}M\to H_2(M,\mathbb{Z})$ . A $(i,J)$ -holomorphic map $u:(\text{Σ},z_1,\dots ,z_k)\to M$ from a reduced Riemann surface $(\text{Σ},j)$ of genus $g$ with $k$ marked points to $\left(M,J\right)$ is said to be stable if every component of $\text{Σ}$ of genus 0 (resp. 1), which is contracted by $u$ , has at least 3 (resp. 1) marked or singular points on its component, and the $k$ marked points are distinct and nonsingular on $\text{Σ}$ . For a two-dimensional homology class $A\in H_2(M)$ let ${\mathcal{M}}_{g,k}(M,A;J)$ be the moduli space of $(j,J)$ -holomorphic stable maps which represent $A$ .

Let $B:=C^{\infty }(\text{Σ},M;A)$ be the space of smooth maps

$u:\text{Σ}\to M$ E6

which represent $A\in H_2(M)$ .

Let us consider infinite dimensional vector bundle $E\to B$ whose fiber at $u$ is the space $E_u={\text{Ω}}^{0,1}(\text{Σ},u^{*}TM)$ of smooth $J$ -antilinear 1-forms on $\text{Σ}$ with values in $u^{*}TM$ . The map ${{\displaystyle \overline{\partial }}}_J:B\to E$ given by

${{\displaystyle \overline{\partial }}}_J\left(u\right)=\frac{1}{2}(du+J\circ du\circ j)$ E7

is a section of the bundle. The zero set of the section ${{\displaystyle \overline{\partial }}}_I$ is the moduli space ${\mathcal{M}}_{g,k}(M,A;J)$ .

For an element $u\in {\mathcal{M}}_{g,k}(M,A;J)$ we denote by

$D_u:{\text{Ω}}^0\left(\text{Σ},u^{*}TM\right)=T_{u}B\to {\text{Ω}}^{0,1}(\text{Σ},u^{*}TM)$ E8

the composition of the derivative

$d{({{\displaystyle \overline{\partial }}}_J)}_u:T_{u}B\to T_{(u,0)}E$ E9

with the projection to fiber $T_{(u,0)}E\to {\text{Ω}}^{0,1}(\text{Σ},u^{*}TM)$ . Then the virtual dimension of ${\mathcal{M}}_{g,k}(M,A;J)$ is

$\begin{array}{c}\dim {\mathcal{M}}_{g,k}\left(M,A;J\right)=\text{index}{D}_u:{\text{Ω}}^0\left(\text{Σ},u^{*}TM\right)\to {\text{Ω}}^{0,1}\left(\text{Σ},u^{*}TM\right)\\ =2c_1\left(TM\right)A+n\left(2-2g\right)+\left(6g-6\right)+2k.\end{array}$ E10

Theorem 2.1.1. For a generic almost complex structure $J\in {\mathfrak{I}}_{\tau }(M,\omega )$ the moduli space ${\mathcal{M}}_{g,k}(M,A;J)$ is a compact stratified manifold of virtual dimension,

$\text{dim}{\mathcal{M}}_{g,k}\left(M,A;J\right)=2c_1\left(TM\right)A+n\left(2-2g\right)+\left(6g-g\right)+2k.$ E11

For some technical reasons, we assume that $c_1(A)\ge 0$ if $\omega \left(A\right)>0$ and $A$ is represented by some $J$ -holomorphic curves. In this case, we call the symplectic manifold $M$ semipositive. We define the evaluation map by

$\text{ev}:{\mathcal{M}}_{g,k}(M,A;J)\to M^k,\text{ev}\left(\left[u;z_1,\dots ,z_k\right]\right)=(u\left(z_1\right),\dots ,u\left(z_k\right)).$ E12

Then the image $\text{Im}(\text{ev})$ is well defined, up to cobordism on $J$ , as a $\text{dim}{\mathcal{M}}_{g,k}\left(M,A;J\right):\equiv m$ -dimensional homology class in $M^k$ .

Definition. The Gromov-Witten invariant ${\text{Φ}}_{g,k}^{M,A}$ is defined by

${\text{Φ}}_{g,k}^{M,A}:H^m\left(M^k\right)\to Q,{\text{Φ}}_{g,k}^{M,A}\left(a\right)={\displaystyle \underset{{\mathcal{M}}_{g,k}\left(M,A;J\right)}{\overset{}{{\displaystyle \int }}}}{\text{ev}}^{*}•\alpha$ E13

where $\alpha =PD\left(a\right)\in H_{2nk-m}(M^k)$ and • is the intersection number of $\text{ev}$ and $\alpha$ in $M^k$ .

The minimal Chern number N of $(M,\omega )$ is the integer $N:=\text{min {}\lambda \text{|}{c}_1\left(A\right)=\lambda \ge 0,A\in H_2(M)\}$ . We define the quantum product $a*b$ of $a\in H^k(M)$ and $b\in H^{\mathcal{l}}(M)$ as the formal sum

$a*b={\displaystyle {\displaystyle \sum }_{A\in H_2(M)}}{(a*b)}_A{q}^{c_1(A)/N}$ E14

where $q$ is an auxiliary variable of degree $2N$ and ${(a*b)}_A\in H^{k+l-2c_1\left(A\right)}(M)$ is defined by

${\displaystyle \underset{C}{\overset{}{{\displaystyle \int }}}}{\left(a*b\right)}_A={\text{Φ}}_{0,3}^{M,A}(a\otimes b\otimes r)$ E15

for $C\in H_{k+l-2c_1\left(A\right)}(M)$ , $r=PD(C)$ . Hereafter, we use the Gromov-Witten invariants of $g=0$ and $k=3$ . Then the quantum product $a*b$ is an element of

$Q{H}^{*}:=H^{*}(M)\otimes \mathbb{Q}[q]$ E16

where $\mathbb{Q}[q]$ is the ring of Laureut polynomials of the auxiliary variable $q$ .

Extending $*$ by linearity, we get a product called quantum product

$*:Q{H}^{*}(M)\otimes Q{H}^{*}(M)\to Q{H}^{*}(M).$ E17

It turns out that $\text{*}$ is distributive over addition, skew-commutative, and associative.

Theorem 2.1.2. Let $(M,\omega )$ be a compact semipositive symplectic manifold. Then the quantum cohomology $(Q{H}^{*}\left(M\right),+,*)$ is a ring.

Remark. For $A=0\in H_2(M)$ , the all $J$ -holomorphic maps in the class $A$ are constant. Thus ${(a*b)}_0=a{{\displaystyle \cup }}^{\text{}}b$ . The constant term of $a*b$ is the usual cup product $a{{\displaystyle \cup }}^{\text{}}b$ .

We defined the Novikov ring ${\Lambda }_{\text{ω}}$ by the set of functions $\lambda :H_2(M)\to Q$ that satisfy the finiteness condition

$\#\{A\in H_2\left(M\right)\text{|}\lambda \left(A\right)\ne 0,\omega \left(A\right)<c\}<\infty$ E18

for every $c\in R$ . The grading is given by $\text{deg}\left(A\right)=2c_1(A)$ .

Examples ([5]). (1) Let $p\in H^2(\mathbb{C}{P}^n)$ and $A\in H_2(\mathbb{C}{P}^n)$ be the standard generators. There is a unique complex line through two distinct points in $\mathbb{C}{P}^n$ and so $p*p^n=q$ . The quantum cohomology of $\mathbb{C}{P}^n$ is

$Q{H}^{*}\left(\mathbb{C}{P}^n;\mathbb{Q}\left[q\right]\right)=\frac{\mathbb{Q}\left[p,q\right]}{<p^{n+1}=q>}.$ E19

(2) Let $G(k,n)$ be the Grassmannian of complex $k$ -planes in ${\mathbb{C}}^n$ . There are two natural complex vector bundles ${\mathbb{C}}^k\to E\to G(k,n)$ and ${\mathbb{C}}^{n-k}\to F\to G(k,n)$ . Let $x_i=c_i(E^{*})$ and $y_i=c_i(F^{*})$ be Chern classes of the dual bundles $E^{*}$ and $F^{*}$ , respectively. Since $E\oplus F$ is trivial, ${\displaystyle {{\displaystyle \sum }}_{i=0}^j}{x}_i{y}_{j-i}=0$ , $j=1,\dots ,n$ . By computation $x_k*y_{n-k}={(-1)}^{n-k}q$ . The quantum cohomology of $G(k,n)$ is

$Q{H}^{*}\left(G\left(k,n\right);\mathbb{Q}\left[q\right]\right)=\frac{\mathbb{Q}[x_1,\dots ,x_k,q]}{<y_{n-k+1},\dots ,y_{n-1},y_n+{\left(-1\right)}^{n-k}q>}.$ E20

Let $\{e_0,\dots ,e_n\}$ be an integral basis of $H^{*}(M)$ such that $e_0=1\in H^0(M)$ and each $e_i$ has pure degree. We introduce $n+1$ formal variables $t_0,\dots ,t_n$ and the linear polynomial $a_t$ in $t_0,\dots ,t_n$ with coefficients in $H^{*}(M)$ by $a_t=t_0{e}_0+\cdots +t_n{e}_n$ . The Gromov-Witten potential of $(M,\omega )$ is a formal power series in variables $t_0,\dots ,t_n$ with coefficients in the Novikov ring ${\Lambda }_{\omega }$

$\begin{array}{c}{\text{Φ}}^M\left(t\right)={\displaystyle {\displaystyle \sum }_{k\ge 3}^{}}{\displaystyle {\displaystyle \sum }_A^{}}\frac{1}{k!}{\text{Φ}}_{0,k}^{M,A}(a_t,\dots ,a_t)q^{\frac{c_1\left(A\right)}{N}}\\ ={\displaystyle {\displaystyle \sum }_{k_0+\cdots +k_n\ge 3}^{}}{\displaystyle {\displaystyle \sum }_A^{}}\frac{\epsilon \left(k_0,\dots ,k_n\right)}{k_0!\dots k_n!}{\text{Φ}}_{0,k}^{M,A}(e_0^{k_0}\otimes \cdots \otimes e_n^{k_n})\cdot (t^0)k_0\dots {(}^{t^n}{q}^{c_1(A)/N}.\end{array}$ E21

Examples ([4]). (1) ${\text{Φ}}^{\mathbb{C}{P}^1}\left(t\right)=\frac{1}{2}{t}_0{}^2{t}_1+\left(e^{t_1}-1-t_1-\frac{t_1{}^2}{2}\right).$

(2) ${\text{Φ}}^{\mathbb{C}{P}^n}\left(t\right)=\frac{1}{6}{\displaystyle {\displaystyle \sum }_{i+j+k=n}^{}}{t}_i{t}_j{t}_k+{\displaystyle {\displaystyle \sum }_{d>0}^{}}{\displaystyle {\displaystyle \sum }_{k_2\dots k_n}^{}}\begin{array}{c}{N}_d\left(k_2\dots k_n\right)\cdot \frac{t_2^{k_2}\dots t_n^{k_n}}{k_2!\dots k_n!}{e}^{d{t}_1}{q}^d,\\ \end{array}$

where $N_d\left(k_2\dots k_n\right)={\text{Φ}}_{0,k}^{\mathbb{C}{P}^{n,},dA}(p^2\dots p^2,\dots ,p^n\dots p^n)$ .

We define a nonsingular matrix $(g_{ij})$ by $g_{ij}={\displaystyle \underset{M}{\overset{}{{\displaystyle \int }}}}{e}_i{{\displaystyle \cup }}^{\text{}}{e}_j$ and denote by $(g^{ij})$ its inverse matrix.

Theorem 2.1.3 ([4, 5]). The Gromov-Witten potential ${\text{Φ}}^M$ of $(M,\omega )$ satisfies the WDVV-equations:

${\displaystyle {\displaystyle \sum }_{\upsilon ,\mu }^{}}{\partial }_{t_i}{\partial }_{t_j}{\partial }_{t_{\upsilon }}{\text{Φ}}^M\left(t\right)g^{\upsilon \mu }{\partial }_{t_{\mu }}{\partial }_{t_k}{\partial }_{t_{\mathcal{l}}}{\text{Φ}}^M\left(t\right)={\epsilon }_{ijk}\cdot {\displaystyle {\displaystyle \sum }_{\upsilon ,\mu }^{}}{\partial }_{t_j}{\partial }_{t_k}{\partial }_{t_{\upsilon }}{\text{Φ}}^M\left(t\right)g^{\upsilon \mu }{\partial }_{t_{\mu }}{\partial }_{t_i}{\partial }_{t_{\mathcal{l}}}{\text{Φ}}^M\left(t\right),$ E22

where ${\epsilon }_{ijk}={(-1)}^{\text{deg}(e_1)(\mathrm{deg}\left(e_j\right)+\mathrm{deg}\left(e_k\right))}$ .

2.2. Floer cohomology

Let a compact symplectic manifold $(M,\omega )$ be semipositive. Let $H_{t+1}:M\to R$ be a smooth 1-periodic family of Hamiltonian functions. The Hamiltonian vector field $X_t$ is defined by $\omega \left(X_t,\cdot \right)=d{H}_t$ . The solutions of the Hamiltonian differential equation ${\displaystyle \dot{x}}\left(t\right)=X_t(x\left(t\right))$ generate a family of Hamiltonian symplectomorphisms ${\varphi }_t:M\to M$ satisfying $\frac{d}{dt}{\varphi }_t=X_t\circ {\varphi }_t$ and ${\varphi }_0=\text{id}$ . For every contractible loop $x:\mathbb{R}/\mathbb{Z}\to M$ , there is a smooth map $u:D:=\text{{}z\in C\text{|}\left|z\right|\le 1\}\to M$ such that $u(e^{2\pi it})=x(t)$ . Two such maps $u_1$ and $u_2$ are called equivalent if their boundary $\text{sum}\left(u_1\right)\#(-u_2)$ is homologus to zero in $H_2(M)$ . Denote by $\left(x,\left[u_1\right]\right)=(x,\left[u_2\right])$ for equivalent pairs, $LM$ the space of contractible loops and ${\displaystyle \widetilde{LM}}$ the space of equivalence classes. Then ${\displaystyle \widetilde{LM}}\to LM$ is a covering space whose covering transformation group is $H_2(M)$ via, $A\left(x,\left[u\right]\right)=(x,\left[A\#u\right])$ for each $A\in H_2(M)$ and $(x,\left[u\right])\in {\displaystyle \widetilde{LM}}$ .

Definition. The symplectic action functional $a_H$ is defined by

$a_H:{\displaystyle \widetilde{LM}}\to R,a_H\left(x,\left[u\right]\right)=-{\displaystyle \underset{D}{\overset{}{{\displaystyle \int }}}}{u}^{*}\omega -{\displaystyle \underset{0}{\overset{1}{{\displaystyle \int }}}}{H}_t\left(x\left(t\right)\right)dt.$ E23

For each element ${\displaystyle \tilde{x}}:=(x,\left[u\right])\in {\displaystyle \widetilde{LM}}$ and $\xi \in T_{{\displaystyle \tilde{x}}}{\displaystyle \widetilde{LM}}$ , we have

$d{a}_H\left(x,\left[u\right]\right)\left(\xi \right)={\displaystyle \underset{0}{\overset{1}{{\displaystyle \int }}}}\omega \left({\displaystyle \dot{x}}\left(t\right)-X_t\left(x\left(t\right)\right),\xi \right)dt.$ E24

Thus the critical points of $a_H$ are one-to-one correspondence with the periodic solutions of ${\displaystyle \dot{x}}\left(t\right)-X_t\left(x\left(t\right)\right)=0$ . Denote by ${\displaystyle \widetilde{PH}}\subset {\displaystyle \widetilde{LM}}$ the critical points of $a_H$ and by $PH\subset LM$ the set of periodic solutions.

The gradient flow lines of $a_H$ are the solutions $u:{\mathbb{R}}^2\to M$ of the partial differential equation

${\partial }_u+J\left(u\right)\left({\partial }_{t}u-X_t\left(u\right)\right)=0$ E250

with conditions $u\left(s,t+1\right)=u(s,t)$ ,

${\displaystyle \underset{s\to \mathrm{\pm }\infty }{\lim }}u\left(s,t\right)=x^{\mathrm{\pm }}(t)$ E25

for some $x^{\pm }\in PH$ .

Let $\mathcal{M}({{\displaystyle \tilde{x}}}^{-},{{\displaystyle \tilde{x}}}^{+})$ be the space of such solutions $u$ with ${{\displaystyle \tilde{x}}}^{+}={{\displaystyle \tilde{x}}}^{-}\#u$ . This space is invariant under the shift $u(s,t)\mapsto u(s+s_0,t)$ for each $s_0\in R$ . For a generic Hamiltonian function, the space $\mathcal{M}({{\displaystyle \tilde{x}}}^{-},{{\displaystyle \tilde{x}}}^{+})$ is a manifold of dimension

$\dim \mathcal{M}\left({{\displaystyle \tilde{x}}}^{-},{{\displaystyle \tilde{x}}}^{+}\right)=\mu \left({{\displaystyle \tilde{x}}}^{-}\right)-\mu \left({{\displaystyle \tilde{x}}}^{+}\right).$ E26

Here $\mu :{\displaystyle \widetilde{PH}}\to Z$ is a version of Maslov index defined by the path of symplectic matrices generated by the linearized Hamiltonian flow along $x(t)$ .

Let $\mu \left({\displaystyle \tilde{x}}\right)-\mu \left({\displaystyle \tilde{y}}\right)=1$ . Then $\mathcal{M}({\displaystyle \tilde{x}},{\displaystyle \tilde{y}})$ is a one-dimensional manifold and the quotient by shift $\mathcal{M}({\displaystyle \tilde{x}},{\displaystyle \tilde{y}})/\mathbb{R}$ is finite. In this case, we denote by $n\left({\displaystyle \tilde{x}},{\displaystyle \tilde{y}}\right)=\#(\frac{\mathcal{M}\left({\displaystyle \tilde{x}},{\displaystyle \tilde{y}}\right)}{\mathbb{R}})$ the number of connecting orbits from ${\displaystyle \tilde{x}}$ to ${\displaystyle \tilde{y}}$ counted with appropriate signs.

We define the Floer cochain group $F{C}^{*}(M,H)$ as the set of all functions $\xi :{\displaystyle \widetilde{PH}}\to Q$ that satisfy the finiteness condition,

$\#\{{\displaystyle \tilde{x}}\in {\displaystyle \widetilde{PH}}\text{|}\xi ({\displaystyle \tilde{x}})\ne 0,a_H({\displaystyle \tilde{x}})\le c\}<\infty$ E27

for every $c\in R$ . The complex $F{C}^{*}(M,H)$ is a ${\Lambda }_{\text{ω}}$ -module with action

$\left(\lambda *\xi \right)\left({\displaystyle \tilde{x}}\right):={\displaystyle {\displaystyle \sum }_A^{}}\lambda \left(A\right)\xi \left(A\#{\displaystyle \tilde{x}}\right).$ E28

The degree $k$ part $F{C}^{*}(M,H)$ consists of all $\xi \in F{C}^{*}(M,H)$ that are nonzero only on elements ${\displaystyle \tilde{x}}\in {\displaystyle \widetilde{PH}}$ with $\mu \left({\displaystyle \tilde{x}}\right)=k$ . Thus the dimension of $F{C}^{*}(M,H)$ as a ${\Lambda }_{\omega }$ -module is the number $\#(PH)$ .

We define a coboundary operator $\delta :F{C}^{*}\left(M,H\right)\to F{C}^{k+1}(M,H)$ by

$\delta \left(\xi \right)\left({\displaystyle \tilde{x}}\right)={\displaystyle {\displaystyle \sum }_{\mu \left({\displaystyle \tilde{y}}\right)=k}^{}}\mathit{n}\left({\displaystyle \tilde{x}},{\displaystyle \tilde{y}}\right)\xi \left({\displaystyle \tilde{y}}\right).$ E29

The coefficients of $\delta (\delta (\xi )({\displaystyle \tilde{x}}))$ are given by counting the numbers of pairs of connecting orbits from ${\displaystyle \tilde{x}}$ to ${\displaystyle \tilde{y}}$ where $\mu \left({\displaystyle \tilde{x}}\right)-\mu \left({\displaystyle \tilde{y}}\right)=2=\dim \mathcal{M}({\displaystyle \tilde{x}},{\displaystyle \tilde{y}})$ . The quotient $\mathcal{M}({\displaystyle \tilde{\text{x}}},{\displaystyle \tilde{\text{y}}})/\mathbb{R}$ is a one-dimensional oriented manifold that consists of pairs counted by $\delta (\delta \left(\xi \right)\left({\displaystyle \tilde{x}}\right))$ . Thus the numbers cancel out in pairs, so that $\delta \left(\delta \left(\xi \right)\right)=0$ .

Definition. The cochain complex $(F{C}^{*}\left(M,H\right),\delta )$ induces its cohomology groups

$F{H}^k\left(M,H\right):=\frac{\text{Ker}\delta :F{C}^k\left(M,H\right)\to F{C}^{k+1}(M,H)}{\text{Im}\delta :F{C}^{k-1}\left(M,H\right)\to F{C}^k(M,H)}$ E30

which are called the Floer cohomology groups of $(M,\omega ,H,J)$ .

Remark. By the usual cobordism argument, the Floer cohomology groups $F{H}^{*}(M,H)$ are independent to the generic choices of $H$ and $J$ . Let $f:M\to R$ be a Morse function such that the negative gradient flow of $f$ with respect to the metric $g\left(\cdot ,\cdot \right)=\omega (\cdot ,J\cdot )$ is Morse-Smale. Let $H=-\epsilon f:M\to R$ be the time-independent Hamiltonian. If $\epsilon$ is small, then the 1-periodic solutions of ${\displaystyle \dot{x}}\left(t\right)-X_H\left(x\left(t\right)\right)=0$ are one-to-one correspondence with the critical points of $f$ . Thus we have $PH=\text{Crit}(f)$ and the Maslov type index can be normalized as

$\mu \left(x,\left[u\right]\right)={\text{ind}}_f\left(x\right)-n$ E31

where $u:D\to M$ is the constant map $u\left(D\right)=x$ .

We define a cochain complex $M{C}^{*}(M;{\Lambda }_{\omega })$ as the graded ${\Lambda }_{\omega }$ -module of all functions

$\text{ξ}:\text{Crit}(f)\times H_2(M)\to Q$ E32

that satisfy the finiteness condition

$\#\{\left(x,A\right)\text{|}\xi \left(x,A\right)\ne 0,\omega \left(A\right)\ge c\}<\infty$ E33

for every $c\in R$ . The ${\Lambda }_{\omega }$ -module structure is given by $\left(\lambda *\xi \right)\left(x,A\right)={{\displaystyle \sum }}^{\text{}}\lambda (B)\xi (x,A+B)$ and the grading $\mathrm{deg}\left(x,A\right)={\text{ind}}_f\left(x\right)-2c_1(A)$ . The gradient flow lines $u:\mathbb{R}\to M$ of $f$ are the solutions of ${\displaystyle \dot{u}}\left(s\right)=-\nabla f(u\left(s\right))$ . These solutions determine coboundary operator

$\delta :M{C}^k\left(M;{\Lambda }_{\omega }\right)\to M{C}^{k+1}(M;{\Lambda }_{\omega })$ E34

$\delta \left(\xi \right)\left(x,A\right)={\displaystyle {\displaystyle \sum }_y^{}}{n}_f(x,y)\xi (y,A)$ E35

where $n_f(x,y)$ is the number of connecting orbits $u$ from $x$ to $y$ satisfying ${\displaystyle \underset{s\to -\infty }{\lim }}u\left(s\right)=x$ , ${\displaystyle \underset{s\to +\infty }{\lim }}u\left(s\right)=y$ , counted with appropriate signs and ${\text{ind}}_f(x)-{\text{ind}}_f\left(y\right)=1$ .

Definition–Theorem 2.2.1. (1) The cochain complex $(M{C}^{*}\left(M;{\Lambda }_{\omega }\right),\delta )$ defines a cohomology group

$M{H}^{*}\left(M;{\Lambda }_{\omega }\right):=\frac{\text{Ker}\delta :M{C}^{*}\left(M;{\Lambda }_{\omega }\right)\to M{C}^{*+1}(M;{\Lambda }_{\omega })}{\text{Im}\delta :M{C}^{*-1}\left(M;{\Lambda }_{\omega }\right)\to M{C}^{*}(M;{\Lambda }_{\omega })}$ E36

which is called the Morse-Witten cohomology of $M$ .

(2) $M{H}^{*}(M;{\Lambda }_{\omega })$ is naturally isomorphic to the quantum cohomology $Q{H}^{*}(M;{\Lambda }_{\omega })$ .

Theorem 2.2.2 ([5]). Let a compact symplectic manifold $(M,\omega )$ be semipositive. There is an isomorphism

$\text{Φ}:F{H}^{*}(M,H)\to Q{H}^{*}(M;{\Lambda }_{\omega })$ E37

which is linear over the Novikov ring ${\Lambda }_{\omega }$ .

Let $H:M\to R$ be a generic Hamiltonian function and $\varphi :M\to M$ the Hamiltonian symplectomorphism of $H$ . By Theorems 2.2.1 and 2.2.2

$F{H}^{*}\left(M,H\right)\simeq Q{H}^{*}(M;{\Lambda }_{\omega })\simeq H^{*}(M)\otimes {\Lambda }_{\omega }$ E38

The rank of $F{C}^{*}(M,H)$ as a ${\Lambda }_{\omega }$ -module must be at least equal to the dimension of $H^{*}\left(M\right)$ . The rank is the number $\#(PH)$ which is the number of the fixed points of $\varphi$ .

Theorem 2.2.3 (Arnold conjecture). Let a compact symplectic manifold $(M,\omega )$ be semipositive. If a Hamiltonian symplectomorphism $\varphi :M\to M$ has only nondegenerate fixed points, then

$\#(\text{Fix}\left(\varphi \right))\ge {\displaystyle {\displaystyle \sum }_{j=0}^{2n}}{b}_j(M)$ E39

where $b_j(M)$ is the jth Betti number of $M$ .

3. Almost contact metric manifolds

Let be a real $\left(2n+1\right)$ -dimensional smooth manifold. An almost cocomplex structure on $M$ is defined by a smooth $\left(1,1\right)$ type tensor $\phi$ , a smooth vector field $\xi ,$ and a smooth 1-form $\eta$ on $M$ such that for each point $x\in M$ ,

${\phi }_x^2=-I+{\eta }_x\otimes {\xi }_x,{\eta }_x\left({\xi }_x\right)=1,$ E40

where $I:T_{x}M\to T_{x}M$ is the identity map of the tangent space $T_{x}M$ .

A Riemannian manifold $M$ with a metric tensor $g$ and with an almost co-complex structure $(\phi ,\xi ,\eta )$ such that

$g\left(X,Y\right)=g\left(\phi X,\phi Y\right)+\eta \left(X\right)\eta \left(Y\right),X,Y\in \text{Γ}\left(TM\right),$ E41

is called an almost contact metric manifold.

The fundamental 2-form $\varphi$ of an almost contact metric manifold $(M,g,\phi ,\xi ,\eta )$ is defined by

$\varphi \left(X,Y\right)=g(X,\phi Y)$ E42

for all $X,Y\in \text{Γ}(TM)$ . The $(2n+1)$ -form ${\varphi }^n\wedge \eta$ does not vanish on $M$ , and so $M$ is orientable. The Nijehuis tensor [8, 11] of the (1,1) type tensor $\phi$ is the (1,2) type tensor field $N_{\phi }$ defined by

$N_{\phi }\left(X,Y\right)=\left[\phi X,\phi Y\right]-\left[X,Y\right]-\phi \left[\phi X,Y\right]-\phi [X,\phi Y]$ E43

for all $X,Y\in \text{Γ}(TM)$ , where $[X,Y]$ is the Lie bracket of $X$ and $Y$ . An almost cocomplex structure $(\phi ,\xi ,\eta )$ on $M$ is said to be integrable if the tensor field $N_{\phi }=0$ , and is normal if $N_{\phi }+2d\eta \otimes \xi =0$ .

Definition. An almost contact metric manifold $(M,g,\phi ,\eta ,\xi ,\varphi )$ is said to be

1. almost cosymplectic (or almost co-Kähler) if $d\varphi =0$ and $d\eta =0$ ,

2. contact (or almost Sasakian) if $\varphi =d\eta$ ,

3. an almost $C$ -manifold if $d\varphi =0$ , $d\eta \ne 0$ , and $d\eta \ne \varphi$ ,

4. cosymplectic (co-Kähler) if $M$ is an integrable almost cosymplectic manifold,

5. Sasakian if $M$ is a normal almost Sasakian manifold,

6. a $C$ -manifold if $M$ is a normal almost $C$ -manifold.

An example of compact Sasakian manifolds is an odd-dimensional unit sphere $S^{2n+1}$ , and the one of the co-Kähler (almost cosymplectic) manifolds is a product $M\times S^1$ where $M$ is a compact Kähler (symplectic) manifold, respectively.

Let $(M_1^{2n_1+1},g_1,{\phi }_1,{\eta }_1,{\xi }_1)$ and $(M_2^{2n_2+1},g_2,{\phi }_2,{\eta }_2,{\xi }_2)$ be almost contact metric manifolds. For the product $M:=M_1\times M_2$ , Riemannian metric on $M$ is defined by

$g\left(\left(X_1,Y_1\right),\left(X_2,Y_2\right)\right)=g_1\left(X_1,X_2\right)+g_2\left(Y_1,Y_2\right).$ E44

An almost complex structure on $M$ is defined by

$J\left(X,Y\right)=\left({\phi }_1\left(X\right)+{\eta }_2\left(Y\right){\xi }_1,{\phi }_2\left(Y\right)-{\eta }_1\left(X\right){\xi }_2\right).$ E45

Then $J^2=-I$ and the fundamental 2-form $\varphi$ on $M$ is $\varphi ={\varphi }_1+{\varphi }_2+{\eta }_1\wedge {\eta }_2$ . If ${\varphi }_1$ , ${\varphi }_2$ and ${\eta }_1$ and ${\eta }_2$ are closed, then $\varphi$ is closed. Thus we have

Theorem 3.1. Let $(M_1^{2n_1+1},g_1,{\phi }_1,{\eta }_1,{\xi }_1)$ be almost contact metric manifolds, $j=1,2$ , and $(M,g,\varphi ,J)$ be the product constructed as above.

1. If ${\varphi }_i$ and ${\eta }_i$ , i = 1,2, are closed, then $\varphi$ is closed.

2. $J$ is an almost complex structure on $M$ .

3. If $M_i$ , $i=1,2$ , are cosymplectic, then $M$ is Kähler.

Let $(M_1^{2n_1},g_1,J_1)$ be a symplectic manifold, and $(M_2^{2n_2+1},g_2,{\phi }_2,{\eta }_2,{\xi }_2)$ be an almost contact metric manifold. Then ${\xi }_1={\eta }_1=0$ , and ${\omega }_1={\varphi }_1$ on $M_1$ .

Theorem 3.2. Let $(M,g,\phi ,\eta ,\xi )$ be the product constructed as above.

1. If $M_2$ is contact, then $M$ is an almost $C$ -manifold.

2. If $M_2$ is a $C$ -manifold, then $M$ is an almost $C$ -manifold.

3. If $M_2$ is almost cosymplectic, then $M$ is almost cosymplectic.

3.1. Quantum type cohomology

In [10, 11] we have studied the quantum type cohomology on contact manifolds. In this section, we want to introduce the quantum type cohomologies on almost cosymplectic, contact, and $C$ -manifolds.

Let $(M^{2n+1},g,\phi ,\eta ,\xi )$ be an almost contact metric manifold. Then the distribution $\mathfrak{H}=\{X\in TM\text{|}\eta \left(X\right)=0\}$ is an $n$ -dimensional complex vector bundle on $M$ .

Now fix the vector bundle $\mathfrak{H}\to M$ . As the symplectic manifolds, a (1,1) type tensor field $\phi :\mathfrak{H}\to H$ with ${\phi }^2=-I$ is said to be tamed by $\varphi$ if $\varphi \left(X,\phi X\right)>0$ for $X\in \mathfrak{H}\backslash \left\{0\right\}$ is said to be compatible if $\varphi \left(\phi X,\phi Y\right)=\varphi (X,Y)$ .

Assume that the almost contact metric manifold $M$ has a closed fundamental 2-form $\varphi$ , i.e., $d\varphi =0$ . An almost contact metric manifold $M$ with the $\varphi$ is called semipositive if for every $A\in {\pi }_2(M)$ , $\varphi \left(A\right)>0$ , $c_1\left(\mathfrak{H}\right)\left(A\right)\ge 3-n$ , then $c_1\left(\mathfrak{H}\right)\left(A\right)>0$ [13]. A smooth map $u:\left(\text{Σ},j\right)\to (M,\phi )$ from a Riemann surface $(\text{Σ},j)$ into $(M,\phi )$ is said to be $\phi$ -coholomorphic if $du\circ j=\phi \circ du$ .

Let $A\in H_2(M;\mathbb{Z})$ be a two-dimensional integral homology class in $M$ . Let ${\mathcal{M}}_{0,3}(M;A,\phi )$ be the moduli space of stable rational $\phi$ -coholomorphic maps with three marked points, which represent class $A$ .

Lemma 3.1.1. For a generic almost complex structure $\phi$ on the distribution, ${\mathbb{C}}^n\to H\to M$ , the moduli space ${\mathcal{M}}_{0,3}(M;A,\phi )$ is a compact stratified manifold with virtual dimension $2c_1\left(\mathfrak{H}\right)\left[A\right]+2n$ .

Consider the evaluation map given by

$\text{ev}:{\mathcal{M}}_{0,3}\left(M;A,\phi \right)\to M^3,$ E46

$\text{ev}\left(\text{Σ};z_1,z_2,z_3,u\right)=\left(u\left(z_1\right),u\left(z_2\right),u\left(z_3\right)\right).$ E47

We have a Gromov-Witten type invariant given by

${\text{Φ}}_{0,3}^{M,A,\phi }:H^{*}\left(M^3\right)\to Q$ E48

${\text{Φ}}_{0,3}^{M,A,\phi }\left(\alpha \right)={\displaystyle \underset{{\mathcal{M}}_{0,3}(M;A,\phi )}{\overset{}{{\displaystyle \int }}}}{\text{ev}}^{*}\left(\alpha \right)={\text{ev}}_{*}[{\mathcal{M}}_{0,3}\left(M;A,\phi \right)]\cdot PD(\alpha )$ E49

which is the number of these intersection points counted with signs according to their orientations.

We define a quantum type product $\text{*}$ on $H^{*}(M)$ , for $\alpha \in H^k(M)$ and $\beta \in H^{\mathcal{l}}(M)$ ,

$\alpha *\beta ={\displaystyle {\displaystyle \sum }_{A\in H_2(M)}^{}}{(\alpha *\beta )}_A{q}^{c_1\left(\mathfrak{H}\right)\left[A\right]/N},$ E50

where $N$ is called the minimal Chern number defined by

$<c_1\left(\mathfrak{H}\right),H_2\left(M\right)>=N\mathbb{Z}$ E51

The ${(\alpha *\beta )}_A\in H^{k+l-2c_1\left(\mathfrak{H}\right)\left[A\right]}(M)$ is defined for each $C\in H_{k+l-2c_1\left(\mathfrak{H}\right)\left[A\right]}(M)$ ,

${\displaystyle \underset{C}{\overset{}{{\displaystyle \int }}}}{(\alpha *\beta )}_A={\Phi }_{0,3}^{M,A,\phi }\left(\alpha \otimes \beta \otimes \gamma \right),\gamma =PD\left(C\right).$ E52

We denote a quantum type cohomology [11, 13] of $M$ by

$Q{H}^{*}\left(M\right):=H^{*}(M)\otimes Q[q]$ E53

where $\mathbb{Q}[q]$ is the ring of Laurent polynomials in $q$ of degree $2N$ with coefficients in the rational numbers $\mathbb{Q}$ . By linearly extending the product $\text{*}$ on $Q{H}^{*}(M)$ , we have

Theorem 3.1.2. The quantum type cohomology $Q{H}^{*}(M)$ of the manifold $M$ is an associative ring under the product $*$ .

Let $(M_1^{2n_1},g_1,J_1,{\omega }_1)$ be a symplectic manifold and $(M_2^{2n_2+1},g_2,{\phi }_2,{\eta }_2,{\xi }_{2,},{\varphi }_2)$ be an either almost cosymplectic or contact or $C$ -manifold.

Let the product $(M^{2n+1},g,\phi ,\eta ,\xi ,\varphi )$ be construct as Theorem 3.2 where $n=n_1+n_2$ . Now we will only consider the free parts of the cohomologies. By the Künneth formula, $H^{*}(M)\simeq H^{*}(M_1)\otimes H^{*}(M_2)$ in particular, $H_2(M)\simeq H_2(M_1\oplus \left(H_1\left(M_1\right)\otimes H_1\left(M_2\right)\right)\oplus H_2\left(M_2\right))$ .

Assume that a two-dimensional class $A=A_1+A_2\in H_2(M_1)\oplus H_2(2)\subset H_2(M)$ .

Lemma 3.1.3. Let $(M,g,\phi ,\eta ,\xi ,\varphi )$ be the product $M=M_1\times M_2$ constructed as above. For a generic almost cocomplex structure $\phi$ on $M$

(1) the moduli space ${\mathcal{M}}_{0,3}(M;A,\phi )$ is homeomorphic to the product

${\mathcal{M}}_{0,3}\left(M_1,A_1,J_1\right)\times {\mathcal{M}}_{0,3}\left(M_2,A_2,\phi \right),$ E54

$\dim {\mathcal{M}}_{0,3}\left(M,A,\phi \right)=2\left[c_1\left(T{M}_1\right)\left(A_1\right)+c_1\left({\mathfrak{H}}_2\right)\left(A_2\right)\right]+2\left(n_1+n_2\right).$ E55

Theorem 3.1.4. For the product $\left(M,g,\phi ,\eta ,\xi ,\varphi \right)=\left(M_1,g_1,J_1,{\omega }_1\right)\times \left(M_2,g_2,{\phi }_2,{\eta }_2,{\xi }_2,{\varphi }_2\right)$ , if $A=A_1+A_2\in H_2(M_1)\oplus H_2(M_2)\subset H_2(M)$ , then the Gromov-Witten type invariants satisfy the following equality

${\text{Φ}}_{0,3}^{M,A,\phi }={\text{Φ}}_{0,3}^{M_1,A_1,J_1}\cdot {\text{Φ}}_{0,3}^{M_2,A_2,{\phi }_2}.$ E56

The complex $(n_1+n_2)$ -dimensional vector bundle

$T{M}_1\oplus {\mathfrak{H}}_2\to M=M_1\times M_2$ E57

has the first Chern class $c_1\left(T{M}_1\oplus {\mathfrak{H}}_2\right)=c_1\left(T{M}_1\right)+c_1\left({\mathfrak{H}}_2\right)$ .

The minimal Chern numbers $N_1$ and $N_2$ are given by $N_1\mathbb{Z}=<c_1\left(T{M}_1\right),H_2\left(M_1\right)>$ and

$N_2\mathbb{Z}=<c_1\left({\mathfrak{H}}_2\right),H_2\left(M_2\right)>.$ E58

For cohomology classes

$\alpha ={\alpha }_1\otimes {\alpha }_2\in H^{k_1}\left(M_1\right)\otimes H^{k_2}\left(M_2\right)\subset H^k\left(M\right),$ E59

$\beta ={\beta }_1\otimes {\beta }_2\in H^{l_1}\left(M_1\right)\otimes H^{l_2}\left(M_2\right)\subset H^l\left(M\right),$ E60

$k_1+k_2=k$ , the quantum type product $\alpha *\beta$ is defined by

$\alpha *\beta ={\displaystyle {\displaystyle \sum }_{\begin{array}{c}{A}_1\in H_2\left(M_1\right)\\ A_2\in H_2\left(M_2\right)\end{array}}^{}}{({\alpha }_1*{\beta }_1)}_{A_1}{q}^{c_1(A_1)/N_1}\otimes {({\alpha }_2*{\beta }_2)}_{A_2}{q}^{c_1(A_2)/N_2}$ E61

where $q_i$ is a degree $2N_i$ auxiliary variable, $i=1,2$ , and the cohomology class ${({\alpha }_i*{\beta }_i)}_{A_i}\in H_{k_i+l_i-2c_1\left(A_i\right)}(M_i)$ is defined by the Gromov-Witten type invariants as follows:

${\displaystyle \underset{C_i}{\overset{}{{\displaystyle \int }}}}{({\alpha }_i*{\beta }_i)}_{A_i}={\text{Φ}}_{0,3}^{M_i,A_i,{\phi }_i}({\alpha }_i\otimes {\beta }_i\otimes {\gamma }_i)$ E62

where $C_i\in H_{k_i+l_i-2c_1\left(A_i\right)}(M_i)$ , ${\gamma }_i=PD(C_i)$ and ${\phi }_1:=J_1$ , $i=1,2$ , respectively.

The quantum type cohomology of $M$ is defined by the tensor product

$\text{Q}{H}^{*}\left(M\right)=H^{*}\left(M\right)\otimes Q\left[q_1,q_2\right],$ E63

where $\mathbb{Q}[q_1,q_2]$ is the ring of Laurent polynomials of variables $q_1$ and $q_2$ with coefficients in $\mathbb{Q}$ . Extend the product $\text{*}$ linearly on the quantum cohomology $\text{Q}{H}^{*}\left(M\right)$ ; similarly, we define the quantum cohomology rings

$\{\begin{array}{c}Q{H}^{*}\left(M_1\right)=H^{*}(M_1)\otimes Q\left[q_1\right],\\ \\ Q{H}^{*}\left(M_2\right)=H^{*}(M_2)\otimes Q\left[q_2\right].\end{array}$ E64

Theorem 3.1.5. There is a natural ring isomorphism between quantum type cohomology rings constructed as above,

$\text{Q}{H}^{*}\left(M\right)=Q{H}^{*}\left(M_1\right)\otimes Q{H}^{*}\left(M_2\right).$ E65

Let $(M,g,\phi ,\varphi )$ be the product of a compact symplectic manifold $(M_1^{2n_1},g_1,J_1,{\omega }_1)$ and an either almost cosymplectic or contact or $\text{C}$ -manifold $(M_2^{2n_2+1},g_2,{\phi }_2,{\eta }_2,{\xi }_2,{\varphi }_2)$ . We choose integral bases, $e_0,e_1,\dots ,e_{k_1}$ of $H^{*}(M_1)$ and $f_0,f_1,\dots ,f_{k_2}$ of $H^{*}(M_2)$ such that $e_0=1\in H^0(M_1)$ , $f_0=1\in H^0(M_2)$ and each basis element has a pure degree. We introduce a linear polynomial of $k_1+1$ variables $t_0,t_1,\dots ,t_{k_1}$ , with coefficients in $H^{*}(M_1)$

$a_t:=t_0{e}_0+t_1{e}_1+\cdots +t_{k_1}{e}_{k_1},$ E66

and a linear polynomial of $k_2+1$ variables $s_0,s_1,\cdots ,s_{k_2}$ with coefficients in $H^{*}(M_2)$

$a_s:=s_0{f}_0+s_1{f}_1+\cdots +s_{k_2}{f}_{k_2}.$ E67

By choosing the coefficients in $\mathbb{Q}$ , the cohomology of $M$ is

$H^{*}\left(M\right)\cong H^{*}\left(M_1\right)\otimes H^{*}\left(M_2\right).$ E68

Then, $H^{*}\left(M\right)$ has an integral basis $\{e_i\otimes f_i\text{|}i=0,\dots ,k_1,j=0,\dots ,k_2\}$ . The rational Gromov-Witten type potential of the product $(M,\omega )$ is a formal power series in the variables $\{t_i,s_j\text{|}i=0,\dots ,k_1,j=0,\dots ,k_2\}$ with coefficients in the Novikov ring ${\Lambda }_{\omega }$ as follows:

$\begin{array}{c}{\text{Ψ}}_0^M\left(t,s\right)={\displaystyle {\displaystyle \sum }_A}{\displaystyle {\displaystyle \sum }_m}\frac{1}{m!}{\Phi }_{0,m}^{M,A,\phi }\left(a_t\otimes a_s,\dots ,a_t\otimes a_s\right)e^{-{\displaystyle \underset{A}{\overset{}{{\displaystyle \int }}}}\varphi }\\ ={\displaystyle {\displaystyle \sum }_{A_1}}{\displaystyle {\displaystyle \sum }_{m_1}}\frac{1}{m_1!}{\text{Φ}}_{0,m_1}^{M_1,A_1,J_1}\left(a_t,\dots ,a_t\right)e^{-{\displaystyle \underset{A_1}{\overset{}{{\displaystyle \int }}}}{\omega }_1}\cdot {\displaystyle {\displaystyle \sum }_{A_2}}{\displaystyle {\displaystyle \sum }_{m_2}}\frac{1}{m_2!}{\text{Φ}}_{0,m_2}^{M_2,A_2,J_2}\left(a_s,\dots ,a_s\right)e^{-{\displaystyle \underset{A_2}{\overset{}{{\displaystyle \int }}}}{\varphi }_2}\\ ={\text{Ψ}}_0^{M_1}\left(t\right)\cdot {\text{Ψ}}_0^{M_2}\left(s\right).\end{array}$ E69

Theorem 3.1.6. The rational Gromov-Witten type potential of $(M,\phi )$ is the product of the rational Gromov-Witten potentials of $M_1$ and $M_2$ , that is,

${\text{Ψ}}_0^M\left(t,s\right)={\text{Ψ}}_0^{M_1}\left(t\right)\cdot {\text{Ψ}}_0^{M_2}\left(s\right).$ E70

3.2. Floer type cohomology

In this subsection, we assume that our manifold $(M_{}^{2n+1},g,\phi ,\eta ,\xi ,\varphi )$ is either a almost cosymplectic, contact, or $C$ -manifold.

Let $H_t=H_{t+1}:M\to R$ be a smooth 1-periodic family of Hamiltonian functions. Denoted by $X_t:M\to TM$ the Hamiltonian vector field of $H_t$ .

The vector fields $X_t$ generate a family of Hamiltonian contactomorphisms ${\psi }_t:M\to M$ satisfying $\frac{d}{dt}{\psi }_t=X_t\circ {\psi }_t$ and ${\psi }_0=\text{id}$ .

Let $a:\mathbb{R}/\mathbb{Z}\to M$ be a contractible loop, then there is a smooth map $u:D\to M$ , defined on the unit disk $D=\{z\in C\text{|}|z|\le 1\}$ , which satisfies $u(e^{2\pi it})=a(t)$ . Two such maps $u_1,u_2:D\to M$ are called equivalent if their boundary sum $u_1\#\left(-u_2\right):S^2\to M$ is homologus to zero in $H_2(M)$ .

Let ${\displaystyle \tilde{a}}:=(a,\left[u\right])$ be an equivalence class and denoted by ${\displaystyle \widetilde{LM}}$ the space of equivalence classes. The space ${\displaystyle \widetilde{LM}}$ is the universal covering space of the space $LM$ of contractible loops in $M$ whose group of deck transformation is $H_2(M)$ .

The symplectic type action functional $a_H:{\displaystyle \widetilde{LM}}\to R$ is defined by

$a_H\left(a,[u]\right)=-{\displaystyle \underset{D}{\overset{}{{\displaystyle \int }}}}{u}^{*}\varphi -{\displaystyle \underset{0}{\overset{1}{{\displaystyle \int }}}}{H}_t\left(a\left(t\right)\right)dt,$ E71

then satisfies $a_H\left(A\#{\displaystyle \tilde{a}}\right)=a_H\left({\displaystyle \tilde{a}}\right)-\varphi \left(A\right).$

Lemma 3.2.1. Let $(M,\varphi )$ the manifold with a closed fundamental 2-form $\varphi$ and fix a Hamiltonian function $H\in C^{\infty }(\mathbb{R}/\mathbb{Z}\times M)$ . Let $\left(a,\left[u\right]\right)\in LM$ and $V\in T_{a}LM=C^{\infty }(\mathbb{R}/\mathbb{Z},a^{*}TM)$ . Then

${\left(d{a}_H\right)}_{\left(a,\left[u\right]\right)}\left(V\right)={\displaystyle \underset{0}{\overset{1}{{\displaystyle \int }}}}\varphi \left({\displaystyle \dot{a}}-X_{H_t}\left(a\right),V\right)dt.$ E72

We denote by ${\displaystyle \widetilde{P\left(H\right)}}\subset {\displaystyle \widetilde{LM}}$ the set of critical points of $a_H$ and by $P(H)\subset LM$ the corresponding set of periodic solutions.

Consider the downward gradient flow lines of $a_H$ with respect to an $L^2$ -norm on $LM$ . The solutions are

$u:{\mathbb{R}}^2\to M,(s,t)\mapsto u(s,t)$ E73

of the partial differential equation

${\partial }_s\left(u\right)+\phi \left(u\right)\left({\partial }_{t}u-X_t\left(u\right)\right)=0$ E74

with periodicity condition

$u\left(s,t+1\right)=u(s,t)$ E75

and limit condition

${\displaystyle \underset{s\to -\infty }{\lim }}u\left(s,t\right)=a\left(t\right),{\displaystyle \underset{s\to +\infty }{\lim }}u\left(s,t\right)=b\left(t\right),$ E76

where $a,b\in P(H)$ .

Let $\mathcal{M}\left({\displaystyle \tilde{a}},{\displaystyle \tilde{b}}\right):=M({\displaystyle \tilde{a}},{\displaystyle \tilde{b}},H,\phi )$ be the space of all solutions $u(s,t)$ satisfying (74)–(76) with

${\displaystyle \tilde{a}}\#u={\displaystyle \tilde{b}}.$ E77

The solutions are invariant under the action $u(s,t)\mapsto u(s+r,t)$ of the time shift $r\in R$ . Equivalent classes of solutions are called Floer connecting orbits.

For a generic Hamiltonian function $H$ , the space $\mathcal{M}({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})$ is a finite dimensional manifold of dimension

$\dim \mathcal{M}({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})=\mu \left({\displaystyle \tilde{a}}\right)-\mu \left({\displaystyle \tilde{b}}\right),$ E78

where the function $\mu :{\displaystyle \widetilde{P(H)}}\to Z$ is a version of the Maslov index defined by the path of unitary matrices generated by the linealized Hamiltonian flow along $a(t)$ on $D$ .

If $H_t\equiv H$ is a $C^2$ -small Morse function, then a critical point $(a,\left[u\right])$ of $H_t$ is a constant map $u\left(D\right)=a$ with index ${\text{ind}}_H(a)$ .

If $\mu \left({\displaystyle \tilde{a}}\right)-\mu \left({\displaystyle \tilde{b}}\right)=1$ , then the space $\mathcal{M}({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})$ is a one-dimensional manifold with $\mathbb{R}$ action by time shift and the quotient $\mathcal{M}({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})/\mathbb{R}$ is a finite set. In fact, $\mu ({\displaystyle \tilde{a}})\in {\pi }_1(U\left(n\right))\simeq Z$ .

If $\mu \left({\displaystyle \tilde{a}}\right)-\mu \left({\displaystyle \tilde{b}}\right)=1$ , ${\displaystyle \tilde{a}},{\displaystyle \tilde{b}}\in {\displaystyle \widetilde{P(H)}}$ , then we denote

$\eta \left({\displaystyle \tilde{a}},{\displaystyle \tilde{b}}\right):=\#\left(\frac{\mathcal{M}\left({\displaystyle \tilde{a}},{\displaystyle \tilde{b}}\right)}{\mathbb{R}}\right),$ E79

where the connection orbits are to be counted with signs determined by a system of coherent orientation $s$ of the moduli space $\mathcal{M}({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})$ . These numbers give us a Floer type cochain complex.

Let $F{C}^{*}(M,H)$ be the set of functions

$\text{ξ}:{\displaystyle \widetilde{P(H)}}\to R$ E80

that satisfy the finiteness condition

$\#\{{\displaystyle \tilde{x}}\in {\displaystyle \widetilde{P\left(H\right)}}\text{|}\xi ({\displaystyle \tilde{x}})\ne 0,a_H({\displaystyle \tilde{x}})\le c\}<\infty$ E81

for all $\text{c}\in \mathbb{R}$ .

Now we define a coboundary operator

${\delta }^k:F{C}^k\left(M,H\right)\to F{C}^{k+1}\left(M,H\right),$ E82

$\left({\delta }^k\xi \right)\left({\displaystyle \tilde{a}}\right)={\displaystyle {\displaystyle \sum }_{\mu \left({\displaystyle \tilde{a}}\right)=\mu \left({\displaystyle \tilde{b}}\right)+1}}\eta ({\displaystyle \tilde{a}},{\displaystyle \tilde{b}})\xi ({\displaystyle \tilde{b}})$ E83

where $\xi \in F{C}^k(M,H)$ , $\mu \left({\displaystyle \tilde{a}}\right)=k+1$ and $\mu \left({\displaystyle \tilde{b}}\right)=k$ .

Lemma 3.2.2. Let $(M,\phi )$ be a semipositive almost contact metric manifold with a closed functional 2-forms. The coboundary operators satisfy ${\delta }^{k+1}\circ {\delta }^k=0$ , for all $k$ .

Definition - Theorem 3.2.3. (1) For a generic pair $\left(H,\phi \right)$ on $M$ , the cochain complex $\left(F{C}^{*},\delta \right)$ defines cohomology groups

$F{H}^{*}\left(M,\varphi ,H,\phi \right):=\frac{\text{Ker}\delta }{\text{Im}\delta }$ E84

which are called the Floer type cohomology groups of the $\left(M,\varphi ,H,\phi \right)$ .

(2) The Floer type cohomology group $F{H}^{*}(M,\varphi ,H,\phi )$ is a module over Novikov ring ${\Lambda }_{\varphi }$ and is independent of the generic choices of $H$ and $\phi$ .

4. Quantum and Floer type cohomologies

In this section, we assume that our manifold $M$ is a compact either almost cosymplectic or contact or $C$ -manifold. In Section 3.1, we study quantum type cohomology of $M$ and in Section 3.2 Floer type cohomology of $M$ . Consequently, we have:

Theorem 4.1. Let $(M,g,\phi ,\eta ,\xi ,\varphi )$ be a compact semipositive almost contact metric manifold with a closed fundamental 2-form $\varphi$ . Then, for every regular pair $\left(H,\phi \right)$ , there is an isomorphism between Floer type cohomology and quantum type cohomology

$\text{Φ}:F{H}^{*}\left(M,\varphi ,H,\phi \right){\displaystyle \stackrel{\sim }{\to }}Q{H}^{*}\left(M,{\Lambda }_{\varphi }\right).$ E85

Proof. Let $h:M\to R$ be a Morse function such that the negative gradient flow of $h$ with respect to the metric $\varphi \left(\cdot ,\phi \left(\cdot \right)\right)+\eta \otimes \eta$ is Morse-Smale and consider the time-independent Hamiltonian

$H_t:=-\epsilon h,t\in R.$ E86

If $\epsilon$ is sufficiently small, then the 1-periodic solutions of

${\displaystyle \dot{a}}\left(t\right)=X_t(a\left(t\right))$ E87

are precisely the critical point of $h$ . The index is

$\mu \left(a,u_a\right)=n-{\text{ind}}_h\left(a\right)={\text{ind}}_{-h}\left(a\right)-n$ E88

where $u_a:D\to M$ is the constant map $u_a\left(z\right)=a$ .

The downward gradient flow lines $u:\mathbb{R}\to M$ of $h$ are solutions of the ordinary differential equation

${\displaystyle \dot{u}}\left(s\right)=J\left(u\right)X_t\left(u\right).$ E89

These solutions determine a coboundary operator

$\delta :C^{*}\left(M,h,{\Lambda }_{\varphi }\right)\to C^{*}\left(M,h,{\Lambda }_{\varphi }\right).$ E90

This coboundary operator is defined on the same cochain complex as the Floer coboundary $\delta$ , and the cochain complex has the same grading for both complex $C^{*}\left(M,h,{\Lambda }_{\varphi }\right)$ , which can be identified with the graded ${\Lambda }_{\varphi }$ module of all functions

$\xi :\text{Crit}\left(h\right)\times H_2\left(M\right)\to R$ E91

that satisfy the finiteness condition

$\#\{\left(a,A\right)\text{|}\xi \left(a,A\right)\ne 0,\varphi (A)\ge c\}<\infty$ E92

for all $c\in R$ . The ${\Lambda }_{\varphi }$ -module structure is given by

$\left(\nu *\xi \right)\left(a,A\right)={\displaystyle {\displaystyle \sum }_B^{}}\nu \left(B\right)\xi \left(a,A+B\right),$ E93

the grading is $\mathrm{deg}\left(a,A\right)={\text{ind}}_h\left(a\right)-2c_1(A)$ , and the coboundary operator $\delta$ is defined by

$\left(\delta \xi \right)\left(a,A\right)={\displaystyle {\displaystyle \sum }_b^{}}{n}_h\left(a,b\right)\xi \left(b,A\right),\left(a,A\right)\in \text{Crit}(h)\times H_2(M),$ E94

where $n_h(a,b)$ is the number of connecting orbits from $a$ to $b$ of shift equivalence classes of solutions of

$\{\begin{array}{c}{\displaystyle \dot{u}}\left(s\right)+\nabla u(s)=0,\\ \\ {\displaystyle \underset{s\to -\infty }{\lim }}u\left(s\right)=a,{\displaystyle \underset{s\to +\infty }{\lim }}u\left(s\right)=b,\end{array}$ E95

counted with appropriate signs.

Here we assume that the gradient flow of $h$ is Morse-Smale and so the number of connecting orbits is finite when ${\text{ind}}_h\left(a\right)-{\text{ind}}_h\left(b\right)=1$ . Then the coboundary operator $\delta$ is a ${\Lambda }_{\varphi }$ -module homomorphism of degree one and satisfies $\delta \circ \delta =0$ . Its cohomology is canonically isomorphic to the quantum type cohomology of $M$ with coefficients in ${\Lambda }_{\varphi }$ .

For each element ${\displaystyle \tilde{a}}\in {\displaystyle \widetilde{P(H)}}$ we denote $\mathcal{M}({\displaystyle \tilde{a}},H,\phi )$ by the space of perturbed $\phi$ -cohomomorphic maps $u:\mathbb{C}\to M$ such that $u(r{e}^{2\pi it})$ converges to a periodic solution $a(t)$ of the Hamiltonian system $H_t$ as $r\to \infty$ . The space $\mathcal{M}({\displaystyle \tilde{a}},H,\phi )$ has dimension $n-\mu ({\displaystyle \tilde{a}})$ . Now fix a Morse function $h:M\to R$ such that the downward gradient flow $u:\mathbb{R}\to M$ satisfying (95) is Morse-Smale. For a critical point $b\in \text{Crit}(h)$ the unstable manifold $W^u(b,h)$ of $b$ has dimension ${\text{ind}}_h(b)$ and codimension $2n-{\text{ind}}_h(b)$ in the distribution $D$ .

The submanifold $\mathcal{M}(b,{\displaystyle \tilde{a}})$ of all $u\in M({\displaystyle \tilde{a}},H,\phi )$ with $u(0)\in W^u(b)$ has dimension

$\dim \mathcal{M}\left(b,{\displaystyle \tilde{a}}\right)={\text{ind}}_h\left(b\right)-\mu \left({\displaystyle \tilde{a}}\right)-n.$ E96

If ${\text{ind}}_h\left(b\right)=\mu \left({\displaystyle \tilde{a}}\right)+n$ , then $\mathcal{M}\left(b,{\displaystyle \tilde{a}}\right)$ is $0\text{zero}$ -dimensional and hence the numbers $n(b,{\displaystyle \tilde{a}})$ of its elements can be used to construct the chain map defined by

$\text{Φ}:F{C}^{*}\left(M,H\right)\to C^{*}\left(M,h,{\Lambda }_{\varphi }\right)$ E97

$\left(\text{Φ}\xi \right)\left(b,A\right)\Lambda {\displaystyle {\displaystyle \sum }_{{\text{ind}}_h\left(b\right)=\mu \left({\displaystyle \tilde{a}}\right)+n}^{}}n(b,{\displaystyle \tilde{a}})\xi (A\#{\displaystyle \tilde{a}})$ E98

which is a ${\Lambda }_{\varphi }$ -module homomorphism and raises the degree by $n$ . The chain map $\text{Φ}$ induces a homomorphism on cohomology

$\text{Φ}:F{H}^{*}\left(M,{\Lambda }_{\varphi }\right)\to H^{*}\left(M,h,{\Lambda }_{\varphi }\right)=\frac{\text{Ker}\delta }{\text{Im}\delta }\simeq Q{H}^{*}\left(M,{\Lambda }_{\varphi }\right).$ E99

Similarly, we can construct a chain map,

$\text{Ψ}:C^{*}\left(M,h,{\Lambda }_{\varphi }\right)\to F{C}^{*}\left(M,H\right)$ E100

$\left(\text{Ψ}\xi \right)\left({\displaystyle \tilde{a}}\right):={\displaystyle {\displaystyle \sum }_{\mu \left({\displaystyle \tilde{a}}\right)+n={\text{ind}}_h\left(b\right)-2c_1\left(A\right)}^{}}n(\left(-A\right)\#{\displaystyle \tilde{a}},b)\xi (b,A).$ E101

Then $\text{Φ}\circ \Psi$ and $\text{Ψ}\circ \Phi$ are chain homotopic to the identity. Thus we have an isomorphism $\text{Φ}$ .

We have studied the Gromov-Witten invariants on symplectic manifolds $(M,\omega ,J)$ using the theory of $J$ -holomorphic curves, and the Gromov-Witten type invariants on almost contact metric manifolds $(N,g,\phi ,\eta ,\xi ,\varphi )$ with a closed fundamental 2-form $\varphi$ using the theory of $\phi$ -coholomorphic curves. We also have some relations between them. We can apply the theories to many cases.

Examples 4.2.

1. The product of a symplectic manifold and a unit circle.

2. The circle bundles over symplectic manifolds.

3. The almost cosymplectic fibrations over symplectic manifolds.

4. The preimage of a regular value of a Morse function on a Kähler manifold.

5. The product of two cosymplectic manifolds is Kähler.

6. The symplectic fibrations over almost cosymplectic manifolds.

7. The number of a contactomorphism is greater than or equal to the sum of the Betti numbers of an almost contact metric manifold with a closed fundamental 2-form.

Examples 4.3. Let $N$ be a quintic hypersurface in $\mathbb{C}{P}^4$ which is called a Calabi-Yau threefold. Then $N$ is symply connected, $c_1\left(TN\right)=0$ and its Betti numbers $b_0=b_6=1$ , $b_1=b_5=0$ , $b_2=b_4=1$ and $b_3=204$ .

Let $A$ be the standard generator in $H_2(N)$ and $h\in H^2(N)$ such that $h\left(A\right)=1$ . The moduli space ${\mathcal{M}}_{0,3}(N,A)$ has the dimension zero. The Gromov-Witten invariant ${\text{Φ}}_{0,3}^{N,A}(a_1,a_2,a_3)$ is nonzero only when $\mathrm{deg}\left(a_i\right)=2$ , $i=1,2,3$ . In fact, ${\text{Φ}}_{0,3}^{N,A}\left(h,h,h\right)=5$ [4, 5]. The quantum cohomology of $N$ is $Q{H}^{*}\left(N\right)=H^{*}(N)\otimes \Lambda$ where $\Lambda$ is the universal Novikov ring [5].

Let $(N,g_1,{\omega }_1,J_1)$ be the standard Kähler structure on $N$ and $\left(S^1,g_2,{\phi }_2=0,{\eta }_2=d\theta ,{\xi }_2=\frac{d}{d\theta },{\varphi }_2=0\right)$ the standard contact structure on $S^1$ . Then the product $M=N\times S^1$ has a canonical cosymplectic structure $(M,g,\phi ,\eta ,\xi ,\varphi )$ as in Section 3. The quantum type cohomology of $M$ is

$Q{H}^{*}\left(M\right)=Q{H}^{*}\left(N\right)\otimes Q{H}^{*}\left(S^1\right)$ E102

Let ${\psi }_1:N\to N$ be a Hamiltonian symplectomorphism with nondegenerate critical points. Then $\#Fix({\psi }_1)\ge {\displaystyle {\displaystyle \sum }_{i=0}^6}{b}_i\left(N\right)=208$ .

Let ${\psi }_2:M\to M$ be a Hamiltonian contactomorphism with nondegenerate critical points. Then $\#Fix({\psi }_2)\ge {\displaystyle {\displaystyle \sum }_{i=0}^7}{b}_i\left(M\right)=416$ .