On the orbits of an orthogonal group action

Abstract

Let $G$ be the Lie group $SO\left(n,ℝ\right)\times SO\left(n,ℝ\right)$ and let $V$ be the vector space of $n\times n$ real matrices. An action of $G$ on $V$ is given by

$$\left(g,h\right).v:=g^{-1}vh,\left(g,h\right)\in G,v\in V.$$

We consider the orbits of this group action and demonstrate a cross-section to the orbits. We then determine the stabilizer for a typical element in this cross-section and completely describe the fundamental group of an orbit of maximal dimension.