Moduli spaces for finite-order jets of Riemannian metrics

TL;DR

This paper constructs and analyzes the moduli space of finite-order jets of Riemannian metrics, providing a stratification framework especially detailed for two-dimensional metrics.

Contribution

It introduces a differentiable moduli space for r-jets of Riemannian metrics and studies its stratification, advancing classification via differential invariants.

Findings

Moduli space is a differentiable space with finite stratification.

Complete stratification analysis for 2D metrics.

Provides a framework for classifying jet metrics using invariants.

Abstract

We construct the moduli space of r-jets at a point of Riemannian metrics on a smooth manifold. The construction is closely related to the problem of classification of jet metrics via differential invariants. The moduli space is proved to be a differentiable space which admits a finite canonical stratification into smooth manifolds. A complete study on the stratification of moduli spaces is carried out for metrics in dimension n=2.