8.7 Almost complex manifolds

This and the next sections deal with spaces which are closely related to complex manifolds. These are somewhat specialized topics and may be omitted on a first reading.

8.7.1 Definitions

There are some differentiable manifolds which carry a similar structure to complex manifolds. To study these manifolds, we somewhat relax the condition (8.16) and require a weaker condition here.

Definition 8.5

Let $M$ be a differentiable manifold. The pair $(M,J)$ , or simply $M$ , is called an almost complex manifold if there exists a tensor field $J$ of type (1, 1) such that at each point $p$ of $M$ , $J_p^2=-{\mathrm{id}}_{T_{p}M}$ . The tensor field $J$ is also called the almost complex structure.

Since $J_p^2=-{\mathrm{id}}_{T_{p}M}$ , $J_p$ has eigenvalues $\pm \mathrm{i}$ . If there are $m+\mathrm{i}$ , then there must be an equal number of $-\mathrm{i}$ , hence $J_p$ is a $2m\times 2m$ matrix and $J_p^2=-I_{2m}$ . Thus, $M$ is an even-dimensional manifold. Note that not all even-dimensional manifolds are almost complex manifolds. For example, $S^4$ is not an almost complex manifold (Steenrod 1951). Note also that we now require a weaker condition $J_p^2=-I_{2m}$ . Of course, the tensor $J_p$ defined by (8.16) satisfies $J_p^2=-I_{2m}$ , hence a complex manifold is an almost complex manifold. There are almost complex manifolds which are not complex manifolds. For example, it is known that $S^6$ admits an almost complex structure, although it is not a complex manifold (Fröhlicher 1955).

Let us complexify a tangent space of an almost complex manifold $(M,J)$ . Given a linear transformation $J_p$ at $T_{p}M$ such that $J_p^2=-I_{2m}$ , we extend $J_p$ to a $\mathbb{C}$ -linear map defined on $T_p{M}^{\mathbb{C}}$ . $J_p$ defined on $T_p{M}^{\mathbb{C}}$ also satisfies $J_p^2=-I_{2m}$ ,

$$J_p^2(X+\mathrm{i}Y)=J_p^{2}X+\mathrm{i}{J}_p^{2}Y=-X+\mathrm{i}(-Y)=-(X+\mathrm{i}Y)$$

where $X,Y\in T_{p}M$ . Let us divide $T_p{M}^{\mathbb{C}}$ into two disjoint vector subspaces, according to the eigenvalue of $J_p$ ,

$$T_p{M}^{\mathbb{C}}=T_p{M}^{+}\oplus T_p{M}^{-}\text{\hspace{1em}(8.121)}$$

where

$$T_p{M}^{\pm }=\{Z\in T_p{M}^{\mathbb{C}}|J_{p}Z=\pm \mathrm{i}Z\}.\text{\hspace{1em}(8.122)}$$

Any vector $V\in T_p{M}^{\mathbb{C}}$ is written as $V=W_1+{\overline{W}}_2$ , where $W_1,W_2\in T_p{M}^{+}$ . Note that $J_{p}V=\mathrm{i}{W}_1-\mathrm{i}{\overline{W}}_2$ . At this stage the reader might have noticed that we can follow the classification scheme of vectors and vector fields developed for the complex manifolds in section 8.2. In fact, the only difference is that on a complex manifold the almost complex structure is explicitly given by (8.18), while on an almost complex manifold, it is required to satisfy the less strict condition $J_p^2=-I_{2m}$ . To classify the complexified tangent spaces and complexified vector spaces, we only need the latter condition. Accordingly, we separate $T_p{M}^{\mathbb{C}}$ into $T_p{M}^{\pm }$ and $\mathfrak{X}(M{)}^{\mathbb{C}}$ into $\mathfrak{X}(M{)}^{\pm }$ , although there does not necessarily exist a basis

of $T_p{M}^{+}$ of the form $\{\partial /\partial z^{\mu }\}$ . For example, we may still define the projection operators

$${\mathcal{P}}^{\pm }\equiv \frac{1}{2}\left({\mathrm{id}}_{T_{p}M}\mp \mathrm{i}{J}_p\right):T_p{M}^{\mathbb{C}}\to T_p{M}^{\pm }.\text{\hspace{1em}(8.123)}$$

We call a vector in $T_p{M}^{+}(T_p{M}^{-})$ a holomorphic (anti-holomorphic) vector and a vector field in $\mathcal{X}(M{)}^{+}(\mathcal{X}(M{)}^{-})$ a holomorphic (anti-holomorphic) vector field.

Definition 8.6

Let $(M,J)$ be an almost complex manifold. If the Lie bracket of any holomorphic vector fields $X,Y\in {\mathcal{X}}^{+}(M)$ is again a holomorphic vector field, $[X,Y]\in {\mathcal{X}}^{+}(M)$ , the almost complex structure $J$ is said to be integrable.

Let $(M,J)$ be an almost complex manifold. Define the Nijenhuis tensor field $N:\mathcal{X}(M)\times \mathcal{X}(M)\to \mathcal{X}(M)$ by

$$N(X,Y)\equiv [X,Y]+J[JX,Y]+J[X,JY]-[JX,JY].\text{\hspace{1em}(8.124)}$$

Given a basis $\{e^{\mu }=\partial /\partial x^{\mu }\}$ and the dual basis $\{\mathrm{d}{x}^{\mu }\}$ , the almost complex structure is expressed as $J=J_{\mu }^{\nu }\mathrm{d}{x}^{\mu }\otimes \partial /\partial x^{\nu }$ . The component expression of $N$ is

$$\begin{array}{rl}& N(X,Y)=\left(X^v{\partial }_v{Y}^{\mu }-Y^v{\partial }_v{X}^{\mu }\right)e_{\mu }\\ & +J_{\lambda }^{\mu }\left\{J_{\kappa }^v{X}^{\kappa }{\partial }_v{Y}^{\lambda }-Y^v{\partial }_v\left(J_{\kappa }^{\lambda }{X}^{\kappa }\right)\right\}{e}_{\mu }\\ & +J_{\lambda }^{\mu }\left\{X^{\nu }{\partial }_{\nu }\left(J_{\kappa }^{\lambda }{Y}^{\kappa }\right)-J_{\kappa }^{\nu }{Y}^{\kappa }{\partial }_{\nu }{X}^{\lambda }\right\}{e}_{\mu }\\ & -\left\{J_{\kappa }^v{X}^{\kappa }{\partial }_v\left(J_{\lambda }^{\mu }{Y}^{\lambda }\right)-J_{\kappa }^v{Y}^{\kappa }{\partial }_v\left(J_{\lambda }^{\mu }{X}^{\lambda }\right)\right\}{e}_{\mu }\\ & =X^{\kappa }{Y}^v[-J_{\lambda }^{\mu }\left({\partial }_v{J}_{\kappa }^{\lambda }\right)+J_{\lambda }^{\mu }\left({\partial }_{\kappa }{J}_v^{\lambda }\right)\\ & -J_{\kappa }^{\lambda }\left({\partial }_{\lambda }{J}_v^{\mu }\right)+J_v^{\lambda }\left({\partial }_{\lambda }{J}_{\kappa }^{\mu }\right)]e_{\mu }.\end{array}\text{\hspace{1em}(8.125)}$$

Thus, $N$ is indeed linear in $X$ and $Y$ and hence a tensor. If $J$ is a complex structure, $J$ is given by (8.18) and the Nijenhuis tensor field trivially vanishes.

Theorem 8.12

An almost complex structure $J$ on a manifold $M$ is integrable if and only if $N(A,B)=0$ for any $A,B\in \mathcal{X}(M)$ .

Proof

Let $Z=X+\mathrm{i}Y$ , $W=U+\mathrm{i}V\in \mathcal{X}(M{)}^{\mathbb{C}}$ . We extend the Nijenhuis tensor field so that its action on vector fields in $\mathcal{X}(M{)}^{\mathbb{C}}$ is given by

$$\begin{array}{rl}N(Z,W)& =[Z,W]+J[JZ,W]+J[Z,JW]-[JZ,JW]\\ & =\{N(X,U)-N(Y,V)\}+\mathrm{i}\{N(X,V)+N(Y,U)\}.\end{array}\text{\hspace{1em}(8.126)}$$

Suppose that $N(A,B)=0$ for any $A,B\in \mathcal{X}(M)$ . From (8.126), it turns out that $N(Z,W)=0$ for $Z,W\in {\mathcal{X}}^{\mathbb{C}}(M)$ . Let $Z,W\in {\mathcal{X}}^{+}(M)\subset \mathcal{X}(M{)}^{\mathbb{C}}$ . Since $JZ=\mathrm{i}Z$ and $JW=\mathrm{i}W$ , we have $N(Z,W)=2\{[Z,W]+\mathrm{i}J[Z,W]\}$ . By assumption, $N(Z,W)=0$ and we find $[Z,W]=-\mathrm{i}J[Z,W]$ or $J[Z,W]=$

$\mathrm{i}[Z,W]$ , that is, $[Z,W]\in {\mathcal{X}}^{+}(M)$ . Thus, the almost complex structure is integrable.

Conversely, suppose that $J$ is integrable. Since ${\mathcal{X}}^{\mathbb{C}}(M)$ is a direct sum of ${\mathcal{X}}^{+}(M)$ and ${\mathcal{X}}^{-}(M)$ , we can separate $Z$ , $W\in {\mathcal{X}}^{\mathbb{C}}(M)$ as $Z=Z^{+}+Z^{-}$ and $W=W^{+}+W^{-}$ . Then

$$N(Z,W)=N\left(Z^{+},W^{+}\right)+N\left(Z^{+},W^{-}\right)+N\left(Z^{-},W^{+}\right)+N\left(Z^{-},W^{-}\right).$$

Since $J{Z}^{\pm }=\pm \mathrm{i}{Z}^{\pm }$ and $J{W}^{\pm }=\pm \mathrm{i}{W}^{\pm }$ , it is easy to see that $N(Z^{+},W^{-})=N(Z^{-},W^{+})=0$ . We also have

$$\begin{array}{rl}N(Z^{+},W^{+})& =[Z^{+},W^{+}]+J[\mathrm{i}{Z}^{+},W^{+}]+J[Z^{+},\mathrm{i}{W}^{+}]-[\mathrm{i}{Z}^{+},\mathrm{i}{W}^{+}]\\ & =2\left[Z^{+},W^{+}\right]-2\left[Z^{+},W^{+}\right]=0\end{array}$$

since $J[Z^{+},W^{+}]=\mathrm{i}[Z^{+},W^{+}]$ . Similarly, $N(Z^{-},W^{-})$ vanishes and we have shown that $N(Z,W)=0$ for any $Z,W\in {\mathcal{X}}^{\mathbb{C}}(M)$ . In particular, it should vanish for $Z,W\in \mathcal{X}(M)$ .

If $M$ is a complex manifold, the complex structure $J$ is a constant tensor field and the Nijenhuis tensor field vanishes. What about the converse? We now state an important (and difficult to prove) theorem.

Theorem 8.13 · Newlander and Nirenberg 1957

Let $(M,J)$ be a $2m$ -dimensional almost complex manifold. If $J$ is integrable, the manifold $M$ is a complex manifold with the almost complex structure $J$ .

In summary we have:

$$\begin{array}{cc}\text{I n t e g r a b l e a l m o s t}& \text{V a n i s h i n g N i j e n h u i s}\\ \text{c o m p l e x s t r u c t u r e}& \text{t e n s o r f i e l d}\end{array}=\text{C o m p l e x m a n i f o l d}.$$

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正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

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