On the almost complex structure

Question

We say that $J$ is an almost complex structure on a 2n-dimensional manifold M is a map $J:TM\rightarrow TM$ such that $J^2=−id_{TM}.$ What is the difference between an almost complex structure and a complex structure. ?

Answer 1

Without more context, it's hard to know what exactly you do not yet understand about the definition of an almost complex structure, so I'll address only the question about the distinction between almost complex and complex structures on manifolds.

Firstly, a (linear) complex structure on a real vector space $\Bbb V$ is a map $J \in \operatorname{End}(\Bbb V)$ such that $J^2 = -\operatorname{id}_{\Bbb V}$. This definition mimics the defining property $i^2 = -1$ of the imaginary unit, and indeed, the complex structure on $\Bbb V$ is characterized by $i \cdot X = J X$ for all $X \in \Bbb V$. Both complex structures and almost complex structures entail putting a linear complex structure on the tangent space at each point, but one structure is more restrictive than the other. One can check that $\Bbb V$ admits a complex structure if and only if $\dim \Bbb V$ is even, so only even-dimensional manifolds can have (almost) complex structures.

Now, a complex structure on a manifold $M$ is a maximal atlas $\{(U_\alpha, \phi_\alpha)\}$ of chart maps $\phi_\alpha : U_\alpha \to \Bbb C^n$, $U_\alpha \subseteq M$, whose transition maps $\phi_{U_\beta}^{-1} \circ \phi_{U_\alpha}$ are all holomorphic functions. So, just as a smooth structure on a manifold lets us view patches on the manifold as behaving like open subsets of $\Bbb R^n$ while giving us a notion of smooth functions, a complex structure lets us view patches on the manifold as behaving like open subsets of $\Bbb C^n$, in particular with a notion of holomorphic functions.

Notice that a complex structure determines a unique (linear) complex structure on each tangent space $T_p M$, this time by declaring $J X := i X$ (or, more precisely, $J X := T \phi_\alpha^{-1} \cdot (i T\phi_\alpha \cdot X)$). The hypothesis that the transition maps are holomorphic ensures that $J$ so defined does not depend of the choice of chart $(U_\alpha, \phi_\alpha)$ in which one works.

We can ask whether this construction can be reversed: If we have (linear) complex structures on each tangent space $T_p M$ (and smoothly varying with $p$), is there necessarily a complex structure on $M$ so that our linear complex structures are all determined by that complex structure?

The answer is no, which perhaps shouldn't be surprising: The holomorphicity condition involves the first derivatives of functions, whereas asking for linear complex structures on each tangent space does not entail a differential condition. We call these structures---choices of complex structure on each tangent space---almost complex structures. In this language,

• a complex structure on a manifold determines a unique almost complex structure, but
• not all almost complex structures come from complex structures.

For reasons outside the scope of this answer, an almost complex structure that does come from a complex structure is said to be integrable.

It's then natural to ask which almost complex structures do arise from complex structures, i.e., which are integrable, and there is a computationally straightforward answer: The Newlander-Nirenberg Theorem assers that a complex structure $J$ is integrable iff $$[X, Y] + J([JX, Y] + [X, JY]) - [JX, JY] = 0$$ for all vector fields $X, Y$. Notice that since the Lie bracket of two vector fields depends on their first derivatives, integrability of $J$ is a first-order differential condition. Checking directly shows that the left-hand side of the above condition is actually linear over $C^{\infty}(M)$, i.e., it defines a $(1, 2)$-tensor invariant $N_J$ of the almost complex structure $J$ called the Nijenhuis tensor.

Remark Since $N_J(X, Y)$ is skew in $X, Y$, in dimension $2$ $N_J$ is determined completely by $N_J(X, JX)$ for any nonvanishing (local) vector field $X$. But substituting in the above formula and simplifying shows that $N_J = 0$, an hence an almost complex structure on a (real) surface does always come from a complex structure. That is not the case in dimension $> 2$.