Thanks in advance to everyone! I am a little bit confused regarding connection on principal bundles. I explain my problem. Consider $S^2$ with the standard metric and the Levi Civita connection and the hopf bundle $S^3 \to S^2$. I can consider an horizontal distribution on $S^3$ that induced the same (standard) covariant derivative and parallel transport on $S^2$. What I’m trying to do now is the following. I have a different metric on $S^2$ and a connection that is compatible with that metric, but not torsion free. I would like to find the horizontal distribution on $S^3$ that induce the same connection on the base space. How should I do? I tried to use the Cartan equations with torsion, but I’m not sure the result is correct!
Thank you!