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

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

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 "Principal bundle" :) – Q.Q.J. Feb 17 2010 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 2010 at 23:39 It is possible. You can induce a connection on an associated bundle, thus if you define an action of U(1) on U(2), then you get an associated U(2) bundle on CP2 and you can compute the induced connection. In our case, the action of U(1) on U(2) is trivial because they commute, thus the induction will be trivial. However, if you want to get any U(2) bundle over CP2 (such that the total space is not necessarily SU(3)), then you may define another action of U(1) on U(2) then induce. – David Bar Moshe Jan 27 2010 at 4:29