Conformal Circles and Local Diffeomorphisms

TL;DR

This paper investigates the properties of conformal circles and their mappings, demonstrating that local diffeomorphisms preserving these circles are necessarily conformal, with implications in holography and geometric analysis.

Contribution

It extends the understanding of conformal circle-preserving diffeomorphisms to pseudo-Riemannian manifolds and provides a holographic interpretation within Poincaré-Einstein spaces.

Findings

Local diffeomorphisms preserving conformal circles are conformal.

Extension of Yano and Tomonaga's results to broader settings.

Holographic interpretation for conformal circle mappings.

Abstract

We study unparametrized conformal circles, or called conformal geodesics, study diffeomorphisms mapping conformal circles to conformal circles in pseudo-Riemannian conformal manifolds. We show that such local diffeomorphisms are conformal local diffeomorphisms. Our result extends the result of Yano and Tomonaga. We also present a holographic interpretation for our result on Poincar\'e-Einstein manifolds. The proofs take suitable variations of conformal circles.