On the orbits of an orthogonal group action (Q970212)

Let \(G\) be the Lie group \(SO(n,\mathbb{R})\times SO(n,\mathbb{R})\) and \(V\) be the vector space of \(n\times n\) real matrices. A smooth action of \(G\) on \(V\) is defined by \((g,h).v=g^{-1}vh\) for \((g,h)\in G\) and \(v\in V\). For \(v\in V\) the \textit{orbit of} \(v\) is defined by \(G.v=\{(g,h).v\mid (g,h)\in G\}\) and the \textit{stabilizer of} \(v\) is defined by \(G_v=\{(g,h)\in G\mid (g,h).v=v\}\). A subset \(D\) of \(V\) is called a \textit{crossed section} to the orbits if every orbit intersects \(D\) in the sense that for each \(v\in V\) there is an element \((g,h)\in G\) and an element \(d\in D\) such that \((g,h).v=d\). In this paper it is proved that the diagonal matrices with nonnegative entries give a crossed section to this action and the stabilizers of the elements in this crossed section are determined. In later sections a review of some covering spaces, covering groups and fundamental groups is given and the fundamental group of a generic orbit is obtained.