Conformal Differential Invariants

TL;DR

This paper computes the algebraic structure of differential invariants for conformal metrics, providing tools to distinguish generic conformal structures and solving the local recognition problem.

Contribution

It explicitly determines the Hilbert polynomial, Poincare function, and the field of rational invariants for conformal structures under diffeomorphisms, advancing the understanding of their classification.

Findings

Computed the Hilbert polynomial for conformal invariants

Derived the Poincare function counting invariants

Described the field of rational invariants separating generic orbits

Abstract

We compute the Hilbert polynomial and the Poincare function counting the number of fixed jet-order differential invariants of conformal metric structures modulo local diffeomorphisms, and we describe the field of rational differential invariants separating generic orbits of the diffeomorphism pseudogroup action. This resolves the local recognition problem for conformal structures.