3. Almost contact metric manifolds

Let be a real ð2n þ 1Þ-dimensional smooth manifold. An almost cocomplex structure on M is defined by a smooth ð1, 1Þ type tensor ϕ, a smooth vector field ξ, and a smooth 1-form η on M such that for each point x∈M,

$$
\rho\_x^2 = -I + \eta\_x \otimes \xi\_x,\\
\eta\_x(\xi\_x) = 1,\tag{40}
$$

where I : TxM ! TxM is the identity map of the tangent space TxM.

A Riemannian manifold M with a metric tensor g and with an almost co-complex structure ðϕ, ξ, ηÞ such that

$$\mathcal{g}(X,Y) = \mathcal{g}(\varphi X, \varphi Y) + \eta(X)\eta(Y), X, Y \in \Gamma(TM),\tag{41}$$

is called an almost contact metric manifold.

The fundamental 2-form φ of an almost contact metric manifold ðM, g,ϕ, ξ, ηÞ is defined by

$$
\phi(X,Y) = \mathcal{g}(X, \mathcal{q}Y) \tag{42}
$$

for all <sup>X</sup>, <sup>Y</sup>∈ΓðTMÞ. The <sup>ð</sup>2<sup>n</sup> <sup>þ</sup> <sup>1</sup>Þ-form <sup>φ</sup><sup>n</sup>∧<sup>η</sup> does not vanish on <sup>M</sup>, and so <sup>M</sup> is orientable. The Nijehuis tensor [8, 11] of the (1,1) type tensor ϕ is the (1,2) type tensor field N<sup>ϕ</sup> defined by

$$N\_{\varphi}(X,Y) = [\![\varphi X,\!\varphi Y]\!] - [X,Y] - \varphi[\![\varphi X,Y]\!] - \varphi[X,\!\varphi Y] \tag{43}$$

for all X, Y∈ΓðTMÞ, where ½X, Y� is the Lie bracket of X and Y. An almost cocomplex structure ðϕ, ξ, ηÞ on M is said to be integrable if the tensor field N<sup>ϕ</sup> ¼ 0, and is normal if N<sup>ϕ</sup> þ 2dη⊗ξ ¼ 0.

Definition. An almost contact metric manifold ðM, g,ϕ, η, ξ, φÞ is said to be


An example of compact Sasakian manifolds is an odd-dimensional unit sphere S<sup>2</sup>nþ<sup>1</sup> , and the one of the co-Kähler (almost cosymplectic) manifolds is a product MS<sup>1</sup> where M is a compact Kähler (symplectic) manifold, respectively.

Let <sup>ð</sup>M<sup>2</sup>n1þ<sup>1</sup> <sup>1</sup> , <sup>g</sup>1,ϕ1, <sup>η</sup>1, <sup>ξ</sup>1<sup>Þ</sup> and <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup> <sup>2</sup> , g2,ϕ2, η2, ξ2Þ be almost contact metric manifolds. For the product M :¼ M1M2, Riemannian metric on M is defined by

$$\mathcal{g}\left( (X\_1, Y\_1), (X\_2, Y\_2) \right) = \mathcal{g}\_1(X\_1, X\_2) + \mathcal{g}\_2(Y\_1, Y\_2). \tag{44}$$

An almost complex structure on M is defined by

$$J(X,Y) = \left(\wp\_1(X) + \eta\_2(Y)\xi\_1, \wp\_2(Y) - \eta\_1(X)\xi\_2\right). \tag{45}$$

Then J <sup>2</sup> <sup>¼</sup> <sup>−</sup><sup>I</sup> and the fundamental 2-form <sup>φ</sup> on <sup>M</sup> is <sup>φ</sup> <sup>¼</sup> <sup>φ</sup><sup>1</sup> <sup>þ</sup> <sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>η</sup>1∧η2. If <sup>φ</sup>1, <sup>φ</sup><sup>2</sup> and <sup>η</sup><sup>1</sup> and η<sup>2</sup> are closed, then φ is closed. Thus we have

Theorem 3.1. Let <sup>ð</sup>M<sup>2</sup>n1þ<sup>1</sup> <sup>1</sup> , g1,ϕ1, η1, ξ1Þ be almost contact metric manifolds, j ¼ 1, 2, and ðM, g,φ, JÞ be the product constructed as above.


3. If Mi, i ¼ 1, 2, are cosymplectic, then M is Kähler.

Let <sup>ð</sup>M<sup>2</sup>n<sup>1</sup> <sup>1</sup> , <sup>g</sup>1, <sup>J</sup>1<sup>Þ</sup> be a symplectic manifold, and <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup> <sup>2</sup> , g2,ϕ2, η2, ξ2Þ be an almost contact metric manifold. Then ξ<sup>1</sup> ¼ η<sup>1</sup> ¼ 0, and ω<sup>1</sup> ¼ φ<sup>1</sup> on M1.

Theorem 3.2. Let ðM, g,ϕ, η, ξÞ be the product constructed as above.


## 3.1. Quantum type cohomology

The fundamental 2-form φ of an almost contact metric manifold ðM, g,ϕ, ξ, ηÞ is defined by

for all <sup>X</sup>, <sup>Y</sup>∈ΓðTMÞ. The <sup>ð</sup>2<sup>n</sup> <sup>þ</sup> <sup>1</sup>Þ-form <sup>φ</sup><sup>n</sup>∧<sup>η</sup> does not vanish on <sup>M</sup>, and so <sup>M</sup> is orientable. The Nijehuis tensor [8, 11] of the (1,1) type tensor ϕ is the (1,2) type tensor field N<sup>ϕ</sup> defined by

for all X, Y∈ΓðTMÞ, where ½X, Y� is the Lie bracket of X and Y. An almost cocomplex structure ðϕ, ξ, ηÞ on M is said to be integrable if the tensor field N<sup>ϕ</sup> ¼ 0, and is normal if

Definition. An almost contact metric manifold ðM, g,ϕ, η, ξ, φÞ is said to be

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

An example of compact Sasakian manifolds is an odd-dimensional unit sphere S<sup>2</sup>nþ<sup>1</sup>

one of the co-Kähler (almost cosymplectic) manifolds is a product MS<sup>1</sup> where M is a compact

ϕ1ðXÞ þ η2ðYÞξ1,ϕ2ðYÞ−η1ðXÞξ<sup>2</sup>

<sup>2</sup> <sup>¼</sup> <sup>−</sup><sup>I</sup> and the fundamental 2-form <sup>φ</sup> on <sup>M</sup> is <sup>φ</sup> <sup>¼</sup> <sup>φ</sup><sup>1</sup> <sup>þ</sup> <sup>φ</sup><sup>2</sup> <sup>þ</sup> <sup>η</sup>1∧η2. If <sup>φ</sup>1, <sup>φ</sup><sup>2</sup> and <sup>η</sup><sup>1</sup> and

<sup>1</sup> , g1,ϕ1, η1, ξ1Þ be almost contact metric manifolds, j ¼ 1, 2, and ðM, g,φ, JÞ

1. almost cosymplectic (or almost co-Kähler) if dφ ¼ 0 and dη ¼ 0,

2. contact (or almost Sasakian) if φ ¼ dη,

Kähler (symplectic) manifold, respectively.

<sup>1</sup> , <sup>g</sup>1,ϕ1, <sup>η</sup>1, <sup>ξ</sup>1<sup>Þ</sup> and <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup>

g 

An almost complex structure on M is defined by

η<sup>2</sup> are closed, then φ is closed. Thus we have

Theorem 3.1. Let <sup>ð</sup>M<sup>2</sup>n1þ<sup>1</sup>

1. If φ<sup>i</sup> and η<sup>i</sup>

be the product constructed as above.

2. J is an almost complex structure on M.

3. an almost C-manifold if dφ ¼ 0, dη≠0, and dη≠φ,

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

product M :¼ M1M2, Riemannian metric on M is defined by

JðX, YÞ ¼

, i = 1,2, are closed, then φ is closed.

ðX1, Y1Þ,ðX2, Y2Þ

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

N<sup>ϕ</sup> þ 2dη⊗ξ ¼ 0.

98 Manifolds - Current Research Areas

Let <sup>ð</sup>M<sup>2</sup>n1þ<sup>1</sup>

Then J

φðX, YÞ ¼ gðX,ϕYÞ (42)

, and the

NϕðX, YÞ¼½ϕX,ϕY�−½X, Y�−ϕ½ϕX, Y�−ϕ½X,ϕY� (43)

<sup>2</sup> , g2,ϕ2, η2, ξ2Þ be almost contact metric manifolds. For the

¼ g1ðX1, X2Þ þ g2ðY1, Y2Þ: (44)

: (45)

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 <sup>ð</sup>M<sup>2</sup>nþ<sup>1</sup> , g,ϕ, η, ξÞ be an almost contact metric manifold. Then the distribution H ¼ {X∈TMjηðXÞ ¼ 0} is an n-dimensional complex vector bundle on M.

Now fix the vector bundle H! M. As the symplectic manifolds, a (1,1) type tensor field <sup>ϕ</sup> : <sup>H</sup> ! <sup>H</sup> with <sup>ϕ</sup><sup>2</sup> <sup>¼</sup> <sup>−</sup><sup>I</sup> is said to be tamed by <sup>φ</sup> if <sup>φ</sup>ðX,ϕX<sup>Þ</sup> <sup>&</sup>gt; 0 for <sup>X</sup>∈H\{0} is said to be compatible if φðϕX,ϕYÞ ¼ φðX, YÞ.

Assume that the almost contact metric manifold M has a closed fundamental 2-form φ, i.e., dφ ¼ 0. An almost contact metric manifold M with the φ is called semipositive if for every A∈π2ðMÞ, φðAÞ > 0, c1ðHÞðAÞ≥3−n, then c1ðHÞðAÞ > 0 [13]. A smooth map u : ðΣ, jÞ!ðM,ϕÞ from a Riemann surface ðΣ, jÞ into ðM,ϕÞ is said to be ϕ-coholomorphic if du∘j ¼ ϕ∘du.

Let A∈H2ðM; ZÞ be a two-dimensional integral homology class in M. Let M0,3ðM; A,ϕÞ be the moduli space of stable rational ϕ-coholomorphic maps with three marked points, which represent class A.

Lemma 3.1.1. For a generic almost complex structure <sup>ϕ</sup> on the distribution, <sup>C</sup><sup>n</sup> ! <sup>H</sup> ! <sup>M</sup>, the moduli space M0,3ðM; A,ϕÞ is a compact stratified manifold with virtual dimension 2c1ðHÞ½A� þ 2n.

Consider the evaluation map given by

$$\text{rev}: \mathcal{M}\_{0,3}(M; A, \varphi) \to M^3,\tag{46}$$

$$\text{ev}(\Sigma; z\_1, z\_2, z\_3, u) = \left(\mu(z\_1), \mu(z\_2), \mu(z\_3)\right). \tag{47}$$

We have a Gromov-Witten type invariant given by

$$\Phi\_{0,3}^{M,A,\psi}: H^\*(M^3) \to Q \tag{48}$$

$$\mathfrak{O}\_{0,3}^{M,A,\psi}(\alpha) = \int\_{\mathcal{M}\_{0,3}(M;A,\varphi)} \mathrm{ev}^\*(\alpha) = \mathrm{ev}\_\*[\mathcal{M}\_{0,3}(M;A,\varphi)] \cdot PD(\alpha) \tag{49}$$

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

We define a quantum type product � on H� <sup>ð</sup>MÞ, for <sup>α</sup>∈H<sup>k</sup> ðMÞ and β∈H ðMÞ,

$$\alpha \* \beta = \sum\_{A \in H\_2(M)} (\alpha \* \beta)\_A q^{c\_1(\mathfrak{H})[A]/N},\tag{50}$$

where N is called the minimal Chern number defined by

$$<\mathcal{c}\_1(\mathfrak{H}), H\_2(M)> = N\mathbb{Z} \tag{51}$$

The <sup>ð</sup><sup>α</sup> � <sup>β</sup>ÞA∈H<sup>k</sup>þl−2c1ðHÞ½A� ðMÞ is defined for each C∈Hkþl−2c1ðHÞ½<sup>A</sup>�ðMÞ,

$$\int\_{\mathbb{C}} (\alpha \ast \beta)\_A = \Phi\_{0,3}^{M,A,\psi}(\alpha \otimes \beta \otimes \gamma), \gamma = PD(\mathbb{C}).\tag{52}$$

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

$$QH^\*(M) := H^\*(M) \otimes \mathbb{Q}[\eta] \tag{53}$$

where Q½q� is the ring of Laurent polynomials in q of degree 2N with coefficients in the rational numbers Q. By linearly extending the product � on QH� ðMÞ, we have

Theorem 3.1.2. The quantum type cohomology QH� ðMÞ of the manifold M is an associative ring under the product �.

Let <sup>ð</sup>M<sup>2</sup>n<sup>1</sup> <sup>1</sup> , <sup>g</sup>1, <sup>J</sup>1, <sup>ω</sup>1<sup>Þ</sup> be a symplectic manifold and <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup> <sup>2</sup> , g2,ϕ2, η2, ξ2, , φ2Þ be an either almost cosymplectic or contact or C-manifold.

Let the product <sup>ð</sup>M<sup>2</sup>nþ<sup>1</sup> , g,ϕ, η, ξ, φÞ be construct as Theorem 3.2 where n ¼ n<sup>1</sup> þ n2. Now we will only consider the free parts of the cohomologies. By the Künneth formula, H� ðMÞ≃H� ðM1Þ⊗H� <sup>ð</sup>M2<sup>Þ</sup> in particular, <sup>H</sup>2ðMÞ≃H2ðM1<sup>⊕</sup> H1ðM1Þ⊗H1ðM2Þ ⊕H2ðM2ÞÞ.

Assume that a two-dimensional classA ¼ A<sup>1</sup> þ A2∈H2ðM1Þ⊕H2ð2Þ⊂H2ðMÞ.

Lemma 3.1.3. Let ðM, g,ϕ, η, ξ, φÞ be the product M ¼ M1M<sup>2</sup> constructed as above. For a generic almost cocomplex structure ϕ on M

(1) the moduli space M0, <sup>3</sup>ðM; A,ϕÞ is homeomorphic to the product

$$
\mathcal{M}\_{0,3}(M\_1, A\_1, I\_1) \mathcal{M}\_{0,3}(M\_2, A\_2, \varphi),
\tag{54}
$$

$$\dim \mathcal{M}\_{0,3}(M, A, \varphi) = 2[\mathfrak{c}\_1(TM\_1)(A\_1) + \mathfrak{c}\_1(\mathfrak{H}\_2)(A\_2)] + 2(n\_1 + n\_2). \tag{55}$$

Theorem 3.1.4. For the product ðM, g,ϕ, η, ξ, φÞ¼ðM1, g1, J1, ω1ÞðM2, g2,ϕ2, η2, ξ2, φ2Þ, if A ¼ A<sup>1</sup> þ A2∈H2ðM1Þ⊕H2ðM2Þ⊂H2ðMÞ, then the Gromov-Witten type invariants satisfy the following equality

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 101

$$
\mathfrak{d}\_{0,3}^{M,A,\psi} = \mathfrak{d}\_{0,3}^{M\_1,A\_1,I\_1} \cdot \mathfrak{d}\_{0,3}^{M\_2,A\_2,\psi\_2} \,. \tag{56}
$$

The complex ðn<sup>1</sup> þ n2Þ-dimensional vector bundle

$$TM\_1 \oplus \mathfrak{H}\_2 \longrightarrow M = M\_1 M\_2 \tag{57}$$

has the first Chern class c1ðTM1⊕H2Þ ¼ c1ðTM1Þ þ c1ðH2Þ. The minimal Chern numbers N<sup>1</sup> and N<sup>2</sup> are given by N1Z ¼< c1ðTM1Þ, H2ðM1Þ > and N2Z ¼< c1ðH2Þ, H2ðM2Þ > : (58)

For cohomology classes

which is the number of these intersection points counted with signs according to their orienta-

ðMÞ is defined for each C∈Hkþl−2c1ðHÞ½<sup>A</sup>�ðMÞ,

ðMÞ :¼ H�

where Q½q� is the ring of Laurent polynomials in q of degree 2N with coefficients in the rational

will only consider the free parts of the cohomologies. By the Künneth formula,

Lemma 3.1.3. Let ðM, g,ϕ, η, ξ, φÞ be the product M ¼ M1M<sup>2</sup> constructed as above. For a generic

Theorem 3.1.4. For the product ðM, g,ϕ, η, ξ, φÞ¼ðM1, g1, J1, ω1ÞðM2, g2,ϕ2, η2, ξ2, φ2Þ, if A ¼ A<sup>1</sup> þ A2∈H2ðM1Þ⊕H2ðM2Þ⊂H2ðMÞ, then the Gromov-Witten type invariants satisfy the follow-

<sup>ð</sup>M2<sup>Þ</sup> in particular, <sup>H</sup>2ðMÞ≃H2ðM1<sup>⊕</sup>

Assume that a two-dimensional classA ¼ A<sup>1</sup> þ A2∈H2ðM1Þ⊕H2ð2Þ⊂H2ðMÞ.

<sup>α</sup> � <sup>β</sup> <sup>¼</sup> ∑

<sup>ð</sup><sup>α</sup> � <sup>β</sup>Þ<sup>A</sup> <sup>¼</sup> <sup>Φ</sup><sup>M</sup>,A,<sup>ϕ</sup>

QH�

A∈H2ðMÞ

<sup>ð</sup>MÞ, for <sup>α</sup>∈H<sup>k</sup>

ðMÞ and β∈H ðMÞ,

< c1ðHÞ, H2ðMÞ >¼ NZ (51)

<sup>0</sup>, <sup>3</sup> ðα⊗β⊗γÞ, γ ¼ PDðCÞ: (52)

ðMÞ, we have

, g,ϕ, η, ξ, φÞ be construct as Theorem 3.2 where n ¼ n<sup>1</sup> þ n2. Now we

H1ðM1Þ⊗H1ðM2Þ

M0,3ðM1, A1, J1ÞM0, <sup>3</sup>ðM2, A2,ϕÞ, (54)

dimM0,3ðM, A,ϕÞ ¼ 2½c1ðTM1ÞðA1Þ þ c1ðH2ÞðA2Þ� þ 2ðn<sup>1</sup> þ n2Þ: (55)

ðMÞ⊗Q½q� (53)

ðMÞ of the manifold M is an associative ring

<sup>2</sup> , g2,ϕ2, η2, ξ2, , φ2Þ be an either almost

⊕H2ðM2ÞÞ.

<sup>ð</sup><sup>α</sup> � <sup>β</sup>ÞAqc1ðHÞ½A�=<sup>N</sup>, (50)

tions.

We define a quantum type product � on H�

The <sup>ð</sup><sup>α</sup> � <sup>β</sup>ÞA∈H<sup>k</sup>þl−2c1ðHÞ½A�

100 Manifolds - Current Research Areas

under the product �.

Let the product <sup>ð</sup>M<sup>2</sup>nþ<sup>1</sup>

ðM1Þ⊗H�

almost cocomplex structure ϕ on M

Let <sup>ð</sup>M<sup>2</sup>n<sup>1</sup>

ðMÞ≃H�

ing equality

H�

where N is called the minimal Chern number defined by

∫ C

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

numbers Q. By linearly extending the product � on QH�

<sup>1</sup> , <sup>g</sup>1, <sup>J</sup>1, <sup>ω</sup>1<sup>Þ</sup> be a symplectic manifold and <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup>

(1) the moduli space M0, <sup>3</sup>ðM; A,ϕÞ is homeomorphic to the product

Theorem 3.1.2. The quantum type cohomology QH�

cosymplectic or contact or C-manifold.

$$\alpha = \alpha\_1 \otimes \alpha\_2 \in H^{k\_1}(M\_1) \otimes H^{k\_2}(M\_2) \subset H^k(M), \tag{59}$$

$$\beta = \beta\_1 \otimes \beta\_2 \in H^{l\_1}(M\_1) \otimes H^{l\_2}(M\_2) \subset H^{l}(M), \tag{60}$$

k<sup>1</sup> þ k<sup>2</sup> ¼ k, the quantum type product α � β is defined by

$$\alpha \ast \beta = \sum\_{\substack{A\_1 \otimes H\_2(M\_1) \\ A\_2 \otimes H\_2(M\_2)}} (\alpha\_1 \ast \beta\_1)\_{A\_1} \eta^{c\_1(A\_1)/N\_1} \otimes (\alpha\_2 \ast \beta\_2)\_{A\_2} \eta^{c\_1(A\_2)/N\_2} \tag{61}$$

where qi is a degree 2Ni auxiliary variable, i ¼ 1, 2, and the cohomology class ðα<sup>i</sup> � β<sup>i</sup> ÞAi ∈Hkiþli−2c1ðAiÞðMiÞ is defined by the Gromov-Witten type invariants as follows:

$$\int\_{\mathbb{C}\_{i}} (\alpha\_{i} \ast \beta\_{i})\_{A\_{i}} = \Phi\_{0,3}^{M\_{i}, A\_{i}, \psi\_{i}} (\alpha\_{i} \otimes \beta\_{i} \otimes \gamma\_{i}) \tag{62}$$

where Ci∈Hkiþli−2c1ðAiÞðMiÞ, γ<sup>i</sup> ¼ PDðCiÞ and ϕ<sup>1</sup> :¼ J1,i ¼ 1, 2, respectively. The quantum type cohomology of M is defined by the tensor product

$$\mathbb{Q}\mathcal{Q}H^\*(M) = H^\*(M)\otimes\mathbb{Q}[q\_1, q\_2],\tag{63}$$

where Q½q1, q2� is the ring of Laurent polynomials of variables q<sup>1</sup> and q<sup>2</sup> with coefficients in Q. Extend the product � linearly on the quantum cohomology QH� ðMÞ; similarly, we define the quantum cohomology rings

$$\begin{cases} QH^\*(M\_1) = H^\*(M\_1) \otimes Q[q\_1], \\\\ QH^\*(M\_2) = H^\*(M\_2) \otimes Q[q\_2]. \end{cases} \tag{64}$$

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

$$
\mathbb{Q}H^\*(M) = \mathbb{Q}H^\*(M\_1)\otimes\mathbb{Q}H^\*(M\_2).\tag{65}
$$

Let <sup>ð</sup>M, <sup>g</sup>,ϕ, <sup>φ</sup><sup>Þ</sup> be the product of a compact symplectic manifold <sup>ð</sup>M<sup>2</sup>n<sup>1</sup> <sup>1</sup> , g1, J1, ω1Þ and an either almost cosymplectic or contact or C-manifold <sup>ð</sup>M<sup>2</sup>n2þ<sup>1</sup> <sup>2</sup> , g2,ϕ2, η2, ξ2, φ2Þ. We choose integral bases, e0,e1, …,ek<sup>1</sup> of H� ðM1Þ and f <sup>0</sup>, f <sup>1</sup>, …, f <sup>k</sup><sup>2</sup> of H� <sup>ð</sup>M2<sup>Þ</sup> such that <sup>e</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup> ðM1Þ, <sup>f</sup> <sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup> ðM2Þ and each basis element has a pure degree. We introduce a linear polynomial of k<sup>1</sup> þ 1 variables t0, t1, …, tk<sup>1</sup> , with coefficients in H� ðM1Þ

$$a\_t := t\_0 e\_0 + t\_1 e\_1 + \dots + t\_{k\_1} e\_{k\_1},\tag{66}$$

and a linear polynomial of k<sup>2</sup> þ 1 variables s0,s1, ⋯,sk<sup>2</sup> with coefficients in H� ðM2Þ

$$a\_s := s\_0 f\_0 + s\_1 f\_1 + \dots + s\_{k\_2} f\_{k\_2} \,. \tag{67}$$

By choosing the coefficients in Q, the cohomology of M is

$$H^\*(M) \# H^\*(M\_1) \otimes H^\*(M\_2). \tag{68}$$

Then, H� ðMÞ has an integral basis {ei⊗f <sup>i</sup> ji ¼ 0, …, k1, j ¼ 0, …, k2}. The rational Gromov-Witten type potential of the product ðM, ωÞ is a formal power series in the variables {ti,sjji ¼ 0, …, k1, j ¼ 0, …, k2} with coefficients in the Novikov ring Λω as follows:

Ψ<sup>M</sup> <sup>0</sup> <sup>ð</sup>t,sÞ ¼ ∑ A ∑ m 1 m! Φ<sup>M</sup>,A,<sup>ϕ</sup> <sup>0</sup>,<sup>m</sup> ðat⊗as, …, at⊗asÞe −∫ A φ <sup>¼</sup> ∑ A1 ∑ m<sup>1</sup> 1 m1! Φ<sup>M</sup>1,A1,J<sup>1</sup> <sup>0</sup>,m<sup>1</sup> ðat, …, atÞe − ∫ A1 ω1 � ∑ A2 ∑ m<sup>2</sup> 1 m2! Φ<sup>M</sup>2,A2,J<sup>2</sup> <sup>0</sup>,m<sup>2</sup> ðas, …, asÞe − ∫ A2 φ2 <sup>¼</sup> <sup>Ψ</sup><sup>M</sup><sup>1</sup> <sup>0</sup> <sup>ð</sup>tÞ � <sup>Ψ</sup><sup>M</sup><sup>2</sup> <sup>0</sup> ðsÞ: (69)

Theorem 3.1.6. The rational Gromov-Witten type potential of ðM,ϕÞ is the product of the rational Gromov-Witten potentials of M<sup>1</sup> and M2, that is,

$$
\Psi\_0^M(t,s) = \Psi\_0^{M\_1}(t) \cdot \Psi\_0^{M\_2}(s). \tag{70}
$$

#### 3.2. Floer type cohomology

In this subsection, we assume that our manifold <sup>ð</sup>M<sup>2</sup>nþ<sup>1</sup> , g,ϕ, η, ξ,φÞ is either a almost cosymplectic, contact, or C-manifold.

Let Ht ¼ Htþ<sup>1</sup> : M ! R be a smooth 1-periodic family of Hamiltonian functions. Denoted by Xt : M ! TM the Hamiltonian vector field of Ht.

The vector fields Xt generate a family of Hamiltonian contactomorphisms ψ<sup>t</sup> : M ! M satisfying <sup>d</sup> dtψ<sup>t</sup> ¼ Xt∘ψ<sup>t</sup> and ψ<sup>0</sup> ¼ id.

Let a : R=Z ! M be a contractible loop, then there is a smooth map u : D ! M, defined on the unit disk <sup>D</sup> <sup>¼</sup> {z∈Cjjzj≤1}, which satisfies <sup>u</sup>ðe<sup>2</sup>πitÞ ¼ <sup>a</sup>ðtÞ. Two such maps <sup>u</sup>1, <sup>u</sup><sup>2</sup> : <sup>D</sup> ! <sup>M</sup> are called equivalent if their boundary sum <sup>u</sup>1#ð−u2<sup>Þ</sup> : <sup>S</sup><sup>2</sup> ! <sup>M</sup> is homologus to zero in <sup>H</sup>2ðMÞ.

Let ~a :¼ ða, ½u�Þ be an equivalence class and denoted by LMg the space of equivalence classes. The space LMg is the universal covering space of the space LM of contractible loops in M whose group of deck transformation is H2ðMÞ.

The symplectic type action functional aH : LMg ! R is defined by

$$a\_H(a, [u]) = -\underset{D}{\int} u^\* \phi - \underset{0}{\int} H\_t(a(t))dt,\tag{71}$$

then satisfies aHðA#~aÞ ¼ aHð~aÞ−φðAÞ:

bases, e0,e1, …,ek<sup>1</sup> of H�

102 Manifolds - Current Research Areas

of k<sup>1</sup> þ 1 variables t0, t1, …, tk<sup>1</sup> , with coefficients in H�

By choosing the coefficients in Q, the cohomology of M is

ðMÞ has an integral basis {ei⊗f <sup>i</sup>

Ψ<sup>M</sup>

Φ<sup>M</sup>1,A1,J<sup>1</sup>

1 m1!

Gromov-Witten potentials of M<sup>1</sup> and M2, that is,

<sup>¼</sup> ∑ A1 ∑ m<sup>1</sup>

3.2. Floer type cohomology

ing <sup>d</sup>

cosymplectic, contact, or C-manifold.

dtψ<sup>t</sup> ¼ Xt∘ψ<sup>t</sup> and ψ<sup>0</sup> ¼ id.

Xt : M ! TM the Hamiltonian vector field of Ht.

<sup>0</sup> <sup>ð</sup>t,sÞ ¼ ∑

A ∑ m 1 m!

<sup>0</sup>,m<sup>1</sup> ðat, …, atÞe

Ψ<sup>M</sup>

In this subsection, we assume that our manifold <sup>ð</sup>M<sup>2</sup>nþ<sup>1</sup>

<sup>f</sup> <sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup>

Then, H�

ðM1Þ and f <sup>0</sup>, f <sup>1</sup>, …, f <sup>k</sup><sup>2</sup> of H�

as :¼ s0f <sup>0</sup> þ s1f <sup>1</sup> þ ⋯ þ sk<sup>2</sup> f <sup>k</sup><sup>2</sup>

ðM1Þ⊗H�

type potential of the product ðM, ωÞ is a formal power series in the variables

Φ<sup>M</sup>,A,<sup>ϕ</sup>

<sup>0</sup> <sup>ð</sup>tÞ � <sup>Ψ</sup><sup>M</sup><sup>2</sup>

Theorem 3.1.6. The rational Gromov-Witten type potential of ðM,ϕÞ is the product of the rational

Let Ht ¼ Htþ<sup>1</sup> : M ! R be a smooth 1-periodic family of Hamiltonian functions. Denoted by

The vector fields Xt generate a family of Hamiltonian contactomorphisms ψ<sup>t</sup> : M ! M satisfy-

Let a : R=Z ! M be a contractible loop, then there is a smooth map u : D ! M, defined on the unit disk <sup>D</sup> <sup>¼</sup> {z∈Cjjzj≤1}, which satisfies <sup>u</sup>ðe<sup>2</sup>πitÞ ¼ <sup>a</sup>ðtÞ. Two such maps <sup>u</sup>1, <sup>u</sup><sup>2</sup> : <sup>D</sup> ! <sup>M</sup> are called equivalent if their boundary sum <sup>u</sup>1#ð−u2<sup>Þ</sup> : <sup>S</sup><sup>2</sup> ! <sup>M</sup> is homologus to zero in <sup>H</sup>2ðMÞ.

− ∫ A1 ω1 � ∑ A2 ∑ m<sup>2</sup>

<sup>¼</sup> <sup>Ψ</sup><sup>M</sup><sup>1</sup>

<sup>0</sup> <sup>ð</sup>t,sÞ ¼ <sup>Ψ</sup><sup>M</sup><sup>1</sup>

<sup>0</sup>,<sup>m</sup> ðat⊗as, …, at⊗asÞe

1 m2!

<sup>0</sup> ðsÞ:

<sup>0</sup> <sup>ð</sup>tÞ � <sup>Ψ</sup><sup>M</sup><sup>2</sup>

Φ<sup>M</sup>2,A2,J<sup>2</sup>

and a linear polynomial of k<sup>2</sup> þ 1 variables s0,s1, ⋯,sk<sup>2</sup> with coefficients in H�

H�

ðMÞ≅H�

{ti,sjji ¼ 0, …, k1, j ¼ 0, …, k2} with coefficients in the Novikov ring Λω as follows:

ðM2Þ and each basis element has a pure degree. We introduce a linear polynomial

ðM1Þ

at :¼ t0e<sup>0</sup> þ t1e<sup>1</sup> þ ⋯ þ tk<sup>1</sup> ek<sup>1</sup> , (66)

<sup>ð</sup>M2<sup>Þ</sup> such that <sup>e</sup><sup>0</sup> <sup>¼</sup> <sup>1</sup>∈H<sup>0</sup>

ðM2Þ

: (67)

− ∫ A2 φ2

<sup>0</sup> ðsÞ: (70)

, g,ϕ, η, ξ,φÞ is either a almost

ðM2Þ: (68)

ji ¼ 0, …, k1, j ¼ 0, …, k2}. The rational Gromov-Witten

−∫ A φ

<sup>0</sup>,m<sup>2</sup> ðas, …, asÞe

ðM1Þ,

(69)

Lemma 3.2.1. Let ðM, φÞ the manifold with a closed fundamental 2-form φ and fix a Hamiltonian function H∈C<sup>∞</sup>ðR=ZM<sup>Þ</sup> . Let <sup>ð</sup>a, <sup>½</sup>u�Þ∈LM and V∈TaLM <sup>¼</sup> <sup>C</sup><sup>∞</sup>ðR=Z, <sup>a</sup>�TMÞ. Then

$$(da\_{\mathcal{H}})\_{(a,[u])}(V) = \int\_0 \phi \left(\dot{a} - \mathcal{X}\_{\mathcal{H}\_l}(a), V\right) dt. \tag{72}$$

We denote by <sup>~</sup> <sup>P</sup>ðHÞ⊂LM<sup>g</sup> the set of critical points of aH and by <sup>P</sup>ðHÞ⊂LM the corresponding set of periodic solutions.

Consider the downward gradient flow lines of aH with respect to an L<sup>2</sup> -norm on LM. The solutions are

$$
\mu: \mathbb{R}^2 \to M, (\mathbf{s}, t) \mapsto \mu(\mathbf{s}, t) \tag{73}
$$

of the partial differential equation

$$
\partial\_t(\mu) + \wp(\mu) \left( \partial\_t \mu - \mathcal{X}\_t(\mu) \right) = 0 \tag{74}
$$

with periodicity condition

$$
\mu(\mathbf{s}, t+1) = \mu(\mathbf{s}, t) \tag{75}
$$

and limit condition

$$\lim\_{s \to -\infty} \mu(s, t) = a(t), \lim\_{s \to +\infty} \mu(s, t) = b(t), \tag{76}$$

where a, b∈PðHÞ.

Let <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> :<sup>¼</sup> <sup>M</sup>ð~a, <sup>~</sup>b, <sup>H</sup>,ϕ<sup>Þ</sup> be the space of all solutions <sup>u</sup>ðs, <sup>t</sup><sup>Þ</sup> satisfying (74)–(76) with

$$
\tilde{a}\#\!u = \tilde{b}.\tag{77}
$$

The solutions are invariant under the action uðs, tÞ↦uðs þ r, tÞ of the time shift r∈R. Equivalent classes of solutions are called Floer connecting orbits.

For a generic Hamiltonian function <sup>H</sup>, the space <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> is a finite dimensional manifold of dimension

$$\dim \mathcal{M}(\tilde{a}, \tilde{b}) = \mu(\tilde{a}) \text{-} \mu(\tilde{b}), \tag{78}$$

where the function <sup>μ</sup> : <sup>~</sup> <sup>P</sup>ðHÞ ! <sup>Z</sup> is a version of the Maslov index defined by the path of unitary matrices generated by the linealized Hamiltonian flow along aðtÞ on D .

If Ht≡H is a C<sup>2</sup> -small Morse function, then a critical point ða, ½u�Þ of Ht is a constant map uðDÞ ¼ a with index indHðaÞ.

If <sup>μ</sup>ð~aÞ−μð~bÞ ¼ 1, then the space <sup>M</sup>ð~a, <sup>~</sup>b<sup>Þ</sup> is a one-dimensional manifold with <sup>R</sup> action by time shift and the quotient <sup>M</sup>ð~a, <sup>~</sup>bÞ=<sup>R</sup> is a finite set. In fact, <sup>μ</sup>ð~aÞ∈π1ðUðnÞÞ≃Z.

If <sup>μ</sup>ð~aÞ−μð~bÞ ¼ 1, <sup>~</sup>a, <sup>~</sup>b<sup>∈</sup> <sup>~</sup> <sup>P</sup>ðHÞ, then we denote

$$\eta(\tilde{a}, \tilde{b}) := \# \left( \frac{\mathcal{M}(\tilde{a}, \tilde{b})}{\mathbb{R}} \right), \tag{79}$$

where the connection orbits are to be counted with signs determined by a system of coherent orientation <sup>s</sup> of the moduli space <sup>M</sup>ð~a, <sup>~</sup>bÞ. These numbers give us a Floer type cochain complex.

Let FC� ðM, HÞ be the set of functions

$$
\xi: P(\tilde{H}) \to R
\tag{80}
$$

that satisfy the finiteness condition

$$\#\{\check{\mathfrak{x}}\in\mathcal{P}(\check{H})\,|\,\xi(\check{\mathfrak{x}})\,\mathfrak{\*}0,a\_{H}(\check{\mathfrak{x}})\leq c\}\leqslant\ll\tag{81}$$

for all c∈R.

Now we define a coboundary operator

$$\delta^k: \mathcal{FC}^k(M, H) \to \mathcal{FC}^{k+1}(M, H), \tag{82}$$

$$(\delta^k \xi)(\tilde{a}) = \sum\_{\mu(\tilde{a}) = \mu(\tilde{b}) + 1} \eta(\tilde{a}, \tilde{b}) \xi(\tilde{b}) \tag{83}$$

where ξ∈FCk <sup>ð</sup>M, <sup>H</sup>Þ, <sup>μ</sup>ð~aÞ ¼ <sup>k</sup> <sup>þ</sup> 1 and <sup>μ</sup>ð~bÞ ¼ <sup>k</sup>.

Lemma 3.2.2. Let ðM,ϕÞ be a semipositive almost contact metric manifold with a closed functional 2 forms. The coboundary operators satisfy δ<sup>k</sup>þ<sup>1</sup> <sup>∘</sup>δ<sup>k</sup> <sup>¼</sup> 0, for all k.

Definition - Theorem 3.2.3. (1) For a generic pair ðH,ϕÞ on M, the cochain complex ðFC� , δÞ defines cohomology groups

Symplectic Manifolds: Gromov-Witten Invariants on Symplectic and Almost Contact Metric Manifolds http://dx.doi.org/10.5772/65663 105

$$FH^\*(M, \phi, H, \varphi) := \frac{\text{Ker}\,\delta}{\text{Im}\,\delta} \tag{84}$$

which are called the Floer type cohomology groups of the ðM,φ, H,ϕÞ.

(2) The Floer type cohomology group FH� ðM, φ, H,ϕÞ is a module over Novikov ring Λφ and is independent of the generic choices of H and ϕ.
