Modified conformal extensions

TL;DR

This paper introduces a geometric construction of split-signature conformal structures with twistor spinors, extending known metrics and characterizing Einstein metrics and symmetries through projective data, with explicit ambient metrics and vanishing Q-curvature.

Contribution

It provides a novel geometric construction and characterization of conformal structures with twistor spinors, including explicit ambient metrics and conditions for Einstein metrics.

Findings

Explicit construction of split-signature conformal structures with twistor spinors.

Complete description of Einstein metrics and symmetries in terms of projective data.

Explicit Fefferman--Graham ambient metric with vanishing Q-curvature.

Abstract

We present a geometric construction and characterization of $2n$-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson--Walker metric construction for $n$-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an explicit geometrically constructed Fefferman--Graham ambient metric and show vanishing of $Q$-curvature.