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Let us consider a principal $G$-bundle $P\longrightarrow M$ together with an $H$-reduction $s$, where $H$ is a maximally compact Lie subgroup. As an $H$-reduction, $s\in\Gamma(M,P/H)$, hence we can consider variations of $s$. My question:

Given any variation $s_{t}\in \Gamma(M,P/H)$ s.t. $s_{0}=s$, is it possible to write $s_{t}=exp(tl)s$ for some $t\in(-\epsilon,\epsilon)$, $l\in\Gamma(M,P\times_{Ad}\mathfrak m)$, where $\mathfrak m$ is the complement of $\mathfrak h$?

Since for any $s_{t}$ we know that $s_{t}(x)=g_{(t,x)}s(x)$ with $g_{(t,x)}\in m$, we would be able to do the above globally if we didn't care about smoothness of $l$. Which we do, obviously.

I would also be interested in information regarding special cases as well, in particular when $M$ is a Riemann surface (hence the tag).

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