# Projection of an invariant almost complex structure to a non-integrable one

My apologies in advance if my question is obvious or elementary.

We identify elements of $$S^3$$ with their quaternion representation $$x_1 + x_2i + x_3j + x_4k$$. We consider two independent vector fields $$S_1(a) = ja$$ and $$S_2(a) = ka$$ on $$S^3$$. On the other hand $$P: S^3\to S^2$$ is a $$S^1$$-principal bundle with the obvious action of $$S^1$$ on $$S^3$$. Then the span of $$S_1, S_2$$ is the standard horizontal space associated to the standard connection of the principal bundle $$S^3 \to S^2$$. Then each horizontal space has an almost complex structure $$J$$. This is the standard structure associated to $$S_1, S_2$$ coordinates which is defined by $$J(S_1) = S_2, J(S_2) = -S_1$$.

Is this structure invariant under the action of $$S^1$$? If yes, we can define a unique almost complex structure on $$S^2$$ which is $$P$$ related to the structure on the total space. Now is this structure on $$S^2$$ integrable?

As a similar question, is there an example of a principal bundle $$P\to X$$, with $$P$$ a real manifold, $$X$$ a complex manifold, and a connection which admits an invariant almost complex structure projecting to a non-integrable structure?

• Let me know if I have misunderstood your question in my answer below. – Michael Albanese May 30 at 21:18
• @MichaelAlbanese Thanks for your answer. It is perfect. Just a question:Let P be a proncipal bundle over X and P is a parallelizable manifold. Does TP afmit dim P sections S_i such that each S_i is invariant under G? – Ali Taghavi May 31 at 3:07
• @MichaelAlbanese In particular Does $S^3$ admit 3 independent sections $S_i$ such that each $S_i$ is $G$ onvariant? – Ali Taghavi May 31 at 8:01

Let $$\pi : P \to X$$ be a smooth principal $$G$$-bundle equipped with a connection. The connection determines a horizontal subbundle $$H$$ of $$TP$$; moreover, $$H \cong \pi^*TX$$. If $$X$$ admits an almost complex structure $$J$$, then so does $$H$$ via the above isomorphism; moreover, the almost complex structure $$J'$$ on $$H$$ constructed in this way is invariant under the action of $$G$$. In particular, $$J'$$ projects to $$J$$ as you put it.
So to come up with an example, just choose a smooth principal $$G$$-bundle $$\pi : P \to X$$ where is $$X$$ is a complex manifold. Now let $$J$$ be a non-integrable almost complex structure on $$X$$ (here we need $$\dim_{\mathbb{R}} X > 2$$), then $$J'$$ will project to $$J$$ which is non-integrable.