Consider the following pair of principal bundle descriptions of $\mathbb{CP}^2$:
$$
\mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1).
$$
If I have a principal $U(2)$-bundle connection for $\mathbb{CP}^2$, will that correspond to a principal $U(1)$-bundle connection? (Where by correspond I suppose I mean give the same covariant derivative.) 


I am also interested in how this generalises to the $n$-case
$$
\mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1).
$$



Principal Bundle Connection Correspondence for two descriptions of the $\mathbb{CP}^2$
The answer is no. You can see that for example from the connection one forms which are Lie algebra valued. In the first case they are u(2) valued and in the second case they are u(1) valued. However, in the case of CP2 (which also generalizes to CPn), the U(1) and SU(2) factors of the isotropy group U(2) commute, this means that given a U(2) connection on U(2)-->SU(3)-->CP2, you can project it to the U(1) factor to obtain a U(1) connection on U(1)-->S^5-->CP2, such that both connections will have the same horizontal subspaces. 


