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

My apology in advance if my question is obvious or elementary

We identify elements of $$S^3$$ with their quaternion representation $$x_1+x_2 i +x_3 j +x_4 k$$. 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 $$S_1, S_2$$ coordinate.

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 total space. Now is this structure on $$S^2$$ integrable?

As a similar question, is there an example of a principal bundle $$P\to X,$$ such that $$P$$ is a real manifold and $$X$$ is a complex manifold and a connection admit an invariant almost complex structure which project to a non integrable structure?