The role of the Hopf Fibration in Kaluza-Klein theory

I have been learning recently about the Hopf Fibration and its relation to physics. My professor has told me that it is one of the simplest methods of dimensional reduction in Kaluza-Klein theory. After looking through various sources on KK theory, I see similarities and can imagine where the Hopf Fibration might fit, for example, the Hopf Fibration maps from $S^3$ to $S^2$, and this corresponds to the 3 spatial dimensions we observe, if we take time to be independent of the fibration, and of course the $U(1)$ fibres are the extra dimension compactified.

My question is, in practical terms, how does the Hopf Fibration actually come in to the theory? The line element in the KK metric has the form begin{align*}
ds^2 = g_{mu nu}dx^mu dx^nu + left( dx^4 – A_mu dx^mu right)^2 end{align*} and this is similar in form to a metric I derived on the total space of the Hopf Fibration (particularly in the term in brackets, which looks just like the connection in the metric of the HF). When dimensionally reducing the theory using the Hopf Fibration, what would you do calculationally? I feel lost here – my first thought would be to just swap out the connection above with the one from the HF but I feel like i’m almost definitely wrong.

Also wouldn’t the HF only be applicable if space had the topological structure of $S^2$ ?

Any help would be greatly appreciated – thanks in advance!