On the Equivalence Problem of Curves in a Riemannian Manifold

Abstract

The solution of the equivalence problem of curves with values in Riemannian manifold is discussed. Also domain of validity of Frents theorem is shown to be the spaces of constant curvature. Also we investigate that there are two important classes of curves, namely , Frents curves and curves in normal position which are coincide in dimensions  4 only. A sharp bound for asymptotic stabillity of differential invariants is illustrated, also the complete system of invariants is characterized, and different examples are analyzed in detail.