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In Nakahara's book "Geometry, Topology and Physics" (Ch. 8, about the almost complex structure) they write:

Note that any 2$m$-dimensional manifold locally admits a tensor field $J$ [type (1,1)] which squares to $-id_{2m}$ [on $T_pM$].

Now, in view of the definition of almost complex manifold (some pages below in the book):

$M$ is called an a.c.m. if there exists a $(1,1)$-type tensor field $J$ (called the almost complex structure) such that $J_p^2 = -id_{T_pM}$ at each point $p \in M$

it seems to me that the first quoted sentence says something like "every 2$m$-dimensional manifold is almost complex". But, this cannot be true, since $S^4$ is the counterexample.

Where am I wrong?

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    $\begingroup$ locally — yes, globally — no $\endgroup$
    – Grigory M
    Commented Sep 12, 2015 at 9:58

1 Answer 1

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To extend a little bit Grigory's comment:

In geometry and topology you often deal with things that can be defined locally and not globally. Some examples that may already be more familiar to you

  1. Non-vanishing vector fields. Let $M$ be any smooth $n$-manifold and let $U$ be a coordinate chart. Since $U\subset \mathbb{R}^n$ is an open subset, $\partial_{x^1}$ is a non-vanishing vector field on $U$. But we cannot always find a global non-vanishing vector field on $M$.

  2. Orientation. Let $M$ be any smooth $n$-manifold and let $U$ be a coordinate chart. Since $U\subset\mathbb{R}^n$ is an open subset, it is orientable. But $M$ may not be orientable.

  3. Analytic functions. Let $M$ be any compact Riemann surface. Locally there exists many complex analytic functions. Globally there are none.

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  • $\begingroup$ but if such a tensor $J$ exist globally the the manifold is complex. $\endgroup$
    – BLS
    Commented Sep 12, 2015 at 14:47
  • $\begingroup$ Or do you want to say that the first quoted sentence refers to the "local" almost complex structure (in this case every even-dimensional manifold is locally almost complex) and that the second - the definition - refers to the "global" almost complex structure? So $S^4$ is locally almost complex but not globally? And then only if $J$ is globally defined (invariant under change of charts), then the manifold is complex ($J$ is integrable). While in general $J_p$ can be different pointwise in an almost complex manifold. $\endgroup$
    – BLS
    Commented Sep 12, 2015 at 15:01
  • $\begingroup$ Yes. (To your second comment.) $\endgroup$ Commented Sep 13, 2015 at 2:12
  • $\begingroup$ Ok, anyway I think it is better to say that locally every even-dimensional manifold (dimension $2n$) is complex (not just almost complex), since the manifold locally looks like $\mathbb{C}^n$. $\endgroup$
    – BLS
    Commented Sep 13, 2015 at 6:27
  • $\begingroup$ You have to be a lot more careful when you say "manifold locally looks like $\mathbb{C}^n$." In the definition of complex manifold the statement is that charts map specifically to open unit disks in $\mathbb{C}^n$. One of the key things is that if you think of complex geometry your maps should be holomorphic, so differently shaped open sets in $\mathbb{R}^{2n}$, even if they are all contractible, may not be the same complex manifold. $\endgroup$ Commented Sep 13, 2015 at 13:34

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