# 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.) Commented Nov 19, 2011 at 21:10
• Yes, that's what I mean Commented 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.
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.
Presumably, you are looking for something other than "the Nijenhaus tensor, possibly simplified by the action of $(\rho_{G})_*$"?