Formes normales pour les champs conformes pseudo-riemanniens

TL;DR

This paper develops normal forms for conformal vector fields near singularities on pseudo-Riemannian manifolds, showing linearizability or conformal flatness in real-analytic Lorentzian cases and analogous results for smooth metrics with bounded flow derivatives.

Contribution

It introduces new normal form results for conformal vector fields on pseudo-Riemannian manifolds, including conditions for linearization and conformal flatness.

Findings

Real-analytic Lorentzian case: linearizable or conformally flat

Smooth metrics: similar results with bounded flow derivative

Normal forms are locally conjugate to model space forms

Abstract

We establish normal forms for conformal vector fields on pseudo-Riemannian manifolds in the neighborhood of a singularity. For real-analytic Lorentzian manifolds, we show that the vector field is analytically linearizable or the manifold is conformally flat. In either case, the vector field is locally conjugate to a normal form on a model space. For smooth metrics of general signature, we obtain the analogous result under the additional assumption that the differential of the flow at the fixed point is bounded.