(Almost) complex manifold problem.

Question

Can some one give me an exemple of an almost compplex manifold that is not a complex manifold and why? I know that an almost complex manifold is of even real dim and is orientable. I also heard that some $S^{2n}$ don't have a complex structure on them but i might be wrong.

Answer 1

The classic example of an almost complex manifold that is not a complex manifold is the six-sphere $S^6$. Consider $S^6$ in $\mathbb{R}^7 = \mathrm{im}\,\mathbb{O}$ as the set of unit norm imaginary octonions. The almost complex structure on $S^6$ is defined by $J_p v = p \times v$, where $p\in S^6$ and $v\in T_p S^6$ and $\times$ stands for the vector product on $\mathbb{R}^7$. This almost complex structure cannot be induced by a complex atlas on $S^6$ because the Nijenhuis tensor $N_J$ doesn't vanish (cf. the Nirenberg Newlander theorem).

A result of Borel-Serre states that the only spheres endowed with an almost complex structure are $S^2$ and $S^6$. (The one on $S^2$ is a complex structure.) Up to now, it is not known whether there exists a complex structure on $S^6$.

As a reference I mention the Wikipedia page on almost complex manifolds.