# Principal Bundle Connection Correspondence for two descriptions of the $\mathbb{CP}^2$

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

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"Principal bundle" :) –  Q.Q.J. Feb 17 '10 at 21:39

I am I right in assuming then that any $U(1)$-connection induces a $U(2)$ connection via the embedding $\mathfrak{u(1)} \to\mathfrak{u(2)}$? –  Aston Smythe Jan 26 '10 at 23:39