Topological properties of the multiplication in a nilpotent Lie group (Q1894609)

Let \(G\) be a connected Lie group and \(H\), \(K\) two closed connected subgroups of \(G\) such that \(H \cap K = \{1\}\). It is obvious that the multiplication \(K \times H \to G\) is an injective immersion. Under the hypothesis that \(G\) is a simply connected nilpotent Lie group, the author shows that the above multiplication is a proper mapping, i.e. the preimages of compact sets are compact. Moreover, in this case, the set KH is a closed smooth submanifold of \(G\) and the multiplication map is a diffeomorphism of \(K \times H\) onto \(KH\). Then the author considers the action of \(K \times H\) on \(G\) given by \(g(k,h) = k^{-1} gh\) and proves that, under certain assumptions, this operation is proper if and only if it is free. Smooth operations which are free and proper have the important property that the orbit space is a smooth manifold. Finally, the author studies some conditions under which the orbit space is diffeomorphic to a Euclidean space.