Consider the following pair of principle bundle descriptions of $\mathbb{CP}^2$: $$ \mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1). $$ If I have a principle $U(2)$-bundle connection for $\mathbb{CP}^2$, will that correspond to a principle $U(1)$-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 $$ \mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1). $$

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 $$ \mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1). $$

Consider the following pair of principle bundle descriptions of $\mathbb{CP}^2$: $$ \mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1). $$ If I have a principle $U(2)$-bundle connection for $\mathbb{CP}^2$ will that correspond to a principle $U(1)$-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 $$ \mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1). $$

Consider the following pair of principle bundle descriptions of $\mathbb{CP}^2$: $$ \mathbb{CP}^2 \simeq SU(3)/U(2) \simeq S^5/U(1). $$ If I have a principle $U(2)$-bundle connection for $\mathbb{CP}^2$, will that correspond to a principle $U(1)$-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 $$ \mathbb{CP}^{n} \simeq SU(n+1)/U(n) \simeq S^{2n+1}/U(1). $$