II. GM-manifolds and unique structure (Q2263033)

Consider an \(r\)-parameter Lie group \(G\) which acts on an \(m\)-dimensional manifold \(M\). Define \(GM=\{\psi_x:g\mapsto g.x|x\in M\}\). From this one obtains the original action \(\Psi:G\times M\to M\) of \(G\) on \(M\). The induced action is given as \(\overline \Psi:G\times GM\to GM\). It is defined as \((g,\psi_x)\mapsto g.\psi_x=\psi_{g.x}\). The author studies the \(G\)-sets and transformation groups, obtaining some topological and differential structures on the set of orbits of \(G\) and on the cross-section space \(M/G\). Then it is shown that \(GM\) is structured by an action of \(G\) induced by the action of \(G\) on \(M\). The existence and uniqueness of a manifold topology and smooth structure on \(GM\) is obtained such that the induced action is smooth.