Group symmetries of the geometry of two sets (Q1812805)

In the paper two local actions of \(r\)-dimensional Lie groups \(G^ r(\lambda)\) and \(G^ r(\delta)\) on an \(m\)-dimensional manifold \(\mathfrak M\) and an \(n\)-dimensional manifold \(\mathfrak N\) are considered. Let their generating maps be given by the functions (1) \(x'=\lambda(x,a)\), \(\xi'=\delta(\xi,a)\), where \(a=(a^ 1,\dots,a^ r)\) are the parameters of the group \(G^ r\), \(x=(x^ 1,\dots,x^ m)\) and \(\xi=(\xi^ 1,\dots,\xi^ n)\) are the coordinates of the manifolds \(\mathfrak M\) and \(\mathfrak N\). If \(r=mn\), \(n\geq m=1\), the problem of finding all functions \(f(x,\xi)=f(x^ 1,\dots,x^ m,\xi^ 1,\dots,\xi^ n\)) up to similarity (change of coordinates) which are nondegenerate and invariant under the actions (1) is solved completely.