Equivalence of paths with respect to symplectic group action (Q1397519)

Let \(V\) be an irreducible affine algebraic variety, \(G\) an algebraic group regularly acting on \(V\), and \(\{x_i\}\) and \(\{y_i\}\) be two finite sets of points of \(V\). One of the important problems of invariant theory is the determination of conditions under which the systems \(\{x_i\}\) and \(\{y_i\}\) are \(G\)-equivalent. A similar problem can be considered in differential geometry for curves. Let \(V\) be a smooth variety, \(G\) a Lie group smoothly acting on \(V\), and \(x\) and \(y\) be two smooth curves in \(V\). It is required to find conditions ensuring the \(G\)-equivalence of curves \(x\) and \(y\). In the paper, these problems are solved in the case where \(G\) is a symplectic group using methods of differential algebra and invariant theory.