The orbit space of the action of gauge transformation group on connections
For a principal bundle P with compact base manifold M and structure group a Lie group G which is the direct product of a compact group and a vector space, the action of the group \({\mathcal G}\) of all smooth gauge transformations on the spaces \({\mathcal C}\) of all smooth principal connections is investigated. It is shown that each orbit is a closed splitting tame submanifold and admits smooth tame slices, so that the orbit space \({\mathcal C}/{\mathcal G}\) is stratified in tame submanifolds corresponding to conjugacy classes of holonomy groups of connections. The method follows \textit{W. Kondracki} and \textit{J. Rogulski}, Prepr., Inst. Math. Pol. Acad. Sci., Warsz. 281, 1-110 (1983; Zbl 0522.58025) who showed this result for suitable Sobolev completions (for compact G), but replaces Sobolev space techniques mainly by the Nash-Moser inverse function theorem given by \textit{R. S. Hamilton} in Bull. Am. Math. Soc., New Ser. 7, 65-222 (1982; Zbl 0499.58003).
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