Infinitesimal Conformal Rigidity on Damek-Ricci Spaces

TL;DR

This paper proves that all conformal vector fields on Damek-Ricci spaces are Killing, demonstrating a strong form of infinitesimal conformal rigidity through a novel local PDE-based approach.

Contribution

It introduces a new local analytic method to establish conformal rigidity on Damek-Ricci spaces, differing from classical Einstein-based proofs.

Findings

All conformal vector fields are Killing on Damek-Ricci spaces

The proof uses explicit PDE analysis on the Lie group model

The approach avoids global arguments and transformation groups

Abstract

We show that every conformal vector field on a Damek-Ricci space is necessarily Killing, establishing a strong form of infinitesimal conformal rigidity. Although this rigidity phenomenon is classically known in the Einstein setting, our proof follows a completely different approach. We formulate the conformal Killing condition as an explicit system of partial differential equations on the solvable Lie group model and analyze it directly. This local and analytic method yields a constructive proof of rigidity without relying on global arguments or transformation groups.