The Equivalence Problem of Curves in a Riemannian Manifold

TL;DR

This paper addresses the problem of classifying curves in Riemannian manifolds by developing invariants and conditions for equivalence, extending Frenet's theorem, and analyzing specific curve classes in various dimensions.

Contribution

It introduces new invariants for curves in general Riemannian manifolds and characterizes their equivalence, extending classical results to broader settings.

Findings

Frenet's theorem applies only to constant curvature spaces.

New invariants are necessary for general Riemannian manifolds.

Complete systems of invariants and their generation are characterized.

Abstract

The equivalence problem of curves with values in a Riemannian manifold, is solved. The domain of validity of Frenet's theorem is shown to be the spaces of constant curvature. For a general Riemannian manifold new invariants must thus be added. There are two important generic classes of curves; namely, Frenet curves and a new class, called curves "in normal position". They coincide in dimensions $\leq 4$ only. A sharp bound for asymptotic stability of differential invariants is obtained, the complete systems of invariants are characterized, and a procedure of generation is presented. Different classes of examples (specially in low dimensions) are analyzed in detail.