## Horizontal distribution of principal G bundle

If we consider $S^{2n-1}$->$\mathbb{CP}^{n-1}$ as a $S^1$ principal bundle, and given a connection 1-form as $C=\frac{1}{2\pi}\Sigma_i(x_i dx_i-y_i dy_i)$ (where $(x_1,y_1,...,x_{2n},y_{2n})$ are coordinates on $S^{2n-1}$). I think the horizontal distribution is the set of points $(x_1,y_1,...,x_{2n},y_{2n})$ and any real multiply of that point, which is just $\mathbb{RP}^{2n-2}$, is that right?

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 I'm not getting your question - isn't a horizontal distribution in $G\to E\to B$ supposed to be a collection of subspaces $V_p\subset T_p E$ that map isomorphically to $T_{\pi(p)}B$ and satisfies some further properties. In your case, you should have a collection of $2n-2$ planes. Moreover, the $1$-form that you're speaking of should originally begin life as a $1$-form in $\mathbb{CP}^{n-1}$. – Somnath Basu Apr 25 2011 at 6:05