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Suppose I have a principal $S^{1}$ bundle over a nice compact symmetric space. The symmetric space arises as a homogeneous space, call it $X=G/H$.

On the Lie algebra level we have the decomposition $\mathfrak g = \mathfrak h \oplus \mathfrak p$ which has several nice bracket relations, notably that $\mathfrak p$ bracketed with itself is contained in $\mathfrak h$.

Now consider our principal $S^{1}$ bundle which I will denote by $Y=G/K$. It too has a decomposition: $\mathfrak g = \mathfrak k \oplus \mathfrak m$

$\mathfrak m$ contains $\mathfrak p$ and has one more dimension as a vector space.

My question is: how is this one dimensional difference in the Lie algebra related to the bundle theory? $S^{1}$ is in some sense one dimensional, yes, but is there more to this?

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