# 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 neccessary) 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.) – Allen Knutson Nov 19 '11 at 21:10
• Yes, that's what I mean – Ago Szekeres Nov 19 '11 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 $(\rho_{G})_*$"?