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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?
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?