Conformal theory of curves with tractors

TL;DR

This paper develops a comprehensive tractor calculus framework for analyzing curves in conformal geometry, providing new invariants, parametrizations, and tools applicable across various signatures, including null curves.

Contribution

It introduces a tractor-based approach to define conformal invariants and parametrizations, extending classical methods to indefinite signatures and null curves.

Findings

Defined absolute conformal invariants via tractor Frenet frames

Extended conformal curve analysis to indefinite signatures and null curves

Provided tools for conserved quantities along conformal curves

Abstract

We present the general theory of curves in conformal geometry using tractor calculus. This primarily involves a tractorial determination of distinguished parametrizations and relative and absolute conformal invariants of generic curves. The absolute conformal invariants are defined via a tractor analogue of the classical Frenet frame construction and then expressed in terms of relative ones. This approach applies likewise to conformal structures of any signature; in the case of indefinite signature we focus especially on the null curves. It also provides a conceptual tool for handling distinguished families of curves (conformal circles and conformal null helices) and conserved quantities along them.