Differential Invariants and Symmetry: Riemannian metrics and beyond

TL;DR

This paper explores the application of the Lie-Tresse theorem to the pseudogroup of diffeomorphisms acting on (pseudo-)Riemannian metrics, examining the relationship with Killing vector fields and the broader role of symmetry.

Contribution

It extends the understanding of differential invariants and symmetry in Riemannian geometry, linking classical theorems to the existence of symmetries and their invariants.

Findings

Lie-Tresse theorem applies to diffeomorphism actions on metrics

Relationship established between invariants and Killing vector fields

Symmetry impacts on the structure of Riemannian metrics

Abstract

We discuss Lie-Tresse theorem for the pseudogroup of diffeomorphisms acting on the space of (pseudo-)Riemannian metrics, and relate this to existence of Killing vector fields. Then we discuss the impact of symmetry in the general case.