On local congruence of immersions in homogeneous or nonhomogeneous spaces (Q352515)

The aim of the paper is to study local congruence of immersions in the following sense:NEWLINENEWLINE { Definition:} Suppose that \(\psi_1,\psi_2:X\to M\) are immersions of a manifold \(X\) into a \(G\)-space \(M\). We say that \(\psi_1\) and \(\psi_2\) are congruent at \(x_0\in X\) if there exist an open neighborhood \(U\subset X\) of \(x_0\) and a transformation \(g\in G\) such that \(\psi_1(x)=g\circ\psi_2(x)\) for all \(x\in U\).NEWLINENEWLINE The main goal of the paper is to provide a theoretical justification of a method that the author devised for solving the local congruence problem of immersed submanifolds of a \(G\)-space that may or may not be homogeneous. Roughly speaking, this method is a hybrid of the classical Cartan method and the method of invariant coframe fields constructed by equivariant moving frames. One of the main results of the paper finds a necessary and sufficient condition for immersions to be congruent when there exists on the ambient \(G\)-space a \(G\)-invariant coframe field of constant structure. The method is then demonstrated by rediscovering the speed and curvature invariants of Euclidean planar curves, the Schwartzian derivative of holomorphic immersions in the projective line, and equivalents of the first and second fundamental forms of surfaces in \(\mathbb R^3\) subject to rotations.