The Definition of an Almost Complex Manifold (Nakahara) I am having problems making sense of Michio Nakahara's definition of the Almost Complex Structure/Almost Complex Manifold, such as it appears in Geometry, Topology and Physics (2nd Edition).
On p. 318-319, he writes

The tensor field $J$ is called the almost complex structure of a complex manifold $M$. Note that any $2m$-dimensional manifold locally admits a tensor field $J$ which squares to $-I_{2m}$. However, $J$ may be patched across charts and defined globally only on a complex manifold.

What confuses me is when he later on defines the almost complex manifold on p. 342:

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 $M$, $J_p^2 = - \text{id}_{T_p M}$. The tensor field $J$ is also called the almost complex structure.

What confuses me is that this definition seems to be in contradiction to the first one. As far as I can understand it, $J$ in the second definition is defined globally in the sense that it is defined at every single point on the manifold $M$. Yet if this was the case, then by the first definition, it would not only be an almost complex manifold we were looking at, but would in fact be a full complex manifold.
My question therefore is, when Nakahara on p. 318-319 writes that $J$ can only be defined globally on a complex manifold, in what sense does he mean globally? And what does he mean by "be patched across charts"? In what sense cannot $J$ for an almost complex but still non-complex manifold not be "patched across charts and defined globally"?
Many thanks.
 A: $\newcommand{\Cpx}{\mathbf{C}}$tl; dr: The second extract is the standard definition. As for resolving the first extract, if Nakahara was talking about "a complex manifold $(M, J)$" just before, then the first is essentially correct. The extract does read a little strangely in isolation, and "only" is not quite what was meant, or else Nakahara meant, "...only on an almost complex manifold."

As a matter of terminology, it's more logically consistent (and, I think, suggestive) to speak of a complex structure as a tensor field $J$ satisfying $J^{2} = -I$, and to call an integrable complex structure a holomorphic structure.
The major snag is, the term "complex" has incompatible meaning depending whether it contrasts with "almost-complex" or "holomorphic". The bottom line is, when you hear someone say "almost-complex...", gently persuade them to say "complex" instead, and to say "holomorphic" if that's what they mean.
In the interest of setting a good example, here are Nakahara's "translated" extracts:
On p. 318-319:

The tensor field $J$ is called the complex structure of a holomorphic manifold $M$. Note that any $2m$-dimensional manifold locally admits a tensor field $J$ which squares to $-I_{2m}$. However, $J$ may be patched across charts and defined globally only on a holomorphic manifold.

On p. 342:

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


There are two logical issues threaded through these extracts:


*

*Is the tensor field $J$ (with $J^{2} = -I$ pointwise) locally defined or globally defined?

*Is the tensor field $J$ integrable (in the sense of having vanishing Nijenhuis tensor, i.e., being locally induced by multiplication by $i$ in local holomorphic coordinates)?
If $M$ is even-dimensional, then locally $M$ is modeled on $\Cpx^{m}$ for some positive integer $m$, so locally $M$ admits a holomorphic structure, i.e., an integrable complex structure $J$. (This fact is trivial, and uninteresting because generally one cannot choose the overlap maps to be holomorphic, or even to respect $J$.)
If $M$ is even-dimensional and oriented (i.e., equipped with a distinguished orientation), $M$ may or may not admit a complex structure compatible with the orientation. The two-sphere $S^{2}$ and six-sphere $S^{6}$ are known at admit (global) complex structures (for either orientation), and all other even-dimensional spheres are known not to admit a complex structure (for either orientation).
If $M$ is complex, $M$ may or may not admit a holomorphic structure, i.e., an integrable complex structure, i.e., a holomorphic atlas. The six-sphere is the most famous complex manifold not known to admit a holomorphic structure (recent claims notwithstanding).
