# Non-integrable almost-complex structures for homogeneous spaces

Let $$M$$ be a smooth homogeneous $$G$$-space for a Lie group $$G$$, and let $$J$$ be a $$G$$-invariant almost-complex structure for $$M$$. Do there exist succinct sufficient (and necessary) conditions for $$J$$ to be integrable? Besides the six sphere, what other examples of a non-integrable invariant almost-complex structure for a smooth homogeneous space are there? Do there exist non-integrable almost-complex structures for any flag manifolds?

• When you give flag manifolds as examples of homogeneous spaces, do you mean homogeneous for $G$ the compact group? (I'm pretty sure you are.) Nov 19, 2011 at 21:10
• Yes, that's what I mean Nov 19, 2011 at 22:53

On a homogeneous space $G/H$ the computation is a little trickier. You take $\omega=g^{-1}dg$, the left invariant Maurer Cartan form on $G$, and then $\omega+\mathfrak{h}$ is semibasic for the quotient map $G \to G/H$ and splits into a complex linear part and a conjugate linear part on each tangent space of $G/H$. Let $\eta$ be the complex linear part. Pick a splitting of $\mathfrak{g}$ into $\mathfrak{g}/\mathfrak{h}$ and some complement, identified with $\mathfrak{h}$, and let $\Omega$ be the projection of $\omega$ to complement. The equation $d \omega + (1/2)[\omega,\omega]=0$ gives an equation $d \eta = - \Omega \wedge \eta + a \eta \wedge \eta + b \eta \wedge \bar{\eta} + c \bar{\eta} \wedge \bar{\eta}$. The Nijenhuis tensor vanishes just when $c=0$. I am pretty sure that the generic invariant almost complex structure on a flag manifold (invariant under the compact form of the automorphism group) is not integrable, but I would have to check.
Presumably, you are looking for something other than "the Nijenhaus tensor, possibly simplified by the action of $(\rho_{G})_*$"?
As for examples: A nearly Kähler manifold, which is not Kähler, always has a non-integrable almost complex structure. You mention $$S^6$$, there are three more compact simply connected homogoneous nearly Kähler manifolds on the spaces: $$\mathbb{CP}^3,S^3\times S^3$$ and also on the flag manifold $$\mathrm{SU}(3)/{\mathbb{T}^2}$$. You can also consider a compact semi-simple Lie group $$G$$ with maximal torus $$T$$. Then $$\mathfrak{g}\otimes \mathbb{C}$$ splits as a sum of $$\mathfrak{t}^2\otimes \mathbb{C}$$ and a sum of root spaces. A $$G$$-invariant almost complex structure on $$G/T$$ is then same as an appropriate choice of a subset of root spaces. The integrability is then read off from the commutator relationships between the root spaces.