On invariants of curves in centro-affine geometry

Abstract

Let $GL(n,R)$ be the general linear group of $n \times n$ real matrices. Definitions of $GL(n,R)$-equivalence and the centro-affine type of curves are introduced. All possible centro-affine types are founded. For every centro affine type all invariant parametrizations of a curve are described. The problem of $GL(n,R)$-equivalence of curves is reduced to that of paths. A generating system of the differential field of invariant differential rational functions of a path is described. They can be viewed as centro-affine curvatures of a path. Global conditions of $GL(n,R)$-equivalence of curves are given in terms of the centro-affine type and the generating differential invariants. Independence of elements of the generating differential invariants is proved.