Josef Šilhán

Fefferman–Graham ambient metrics of Patterson–Walker metrics

Given an n-dimensional manifold N with an affine connection D, we show that the associated Patterson-Walker metric g on TN admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which the ambient metric exists to all orders and can be realised in a natural and explicit way. In particular, it follows that Patterson-Walker metrics have vanishing Fefferman-Graham obstruction tensors. As an application of the concrete ambient metric realisation we show in addition that Patterson-Walker metrics have vanishing Q-curvature. We further show that the relationship between the geometric constructions mentioned above is very close: the explicit Fefferman-Graham ambient metric is itself a Patterson-Walker metric.

A Projective-to-Conformal Fefferman-Type Construction

We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into Spin(n + 1, n + 1). The construction associates a split-signature (n, n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.

Second order symmetries of the conformal laplacian and R-separation

Let (M, g) be an arbitrary pseudo-Riemannian manifold of dimension at least 3, let Δ := ∇agab∇b be the Laplace-Beltrami operator and let ΔY be the conformal Laplacian. In some references, Kalnins and Miller provide an intrinsic characterization for R-separation of the Laplace equation ΔΨ = 0 in terms of second order conformal symmetries of Δ. The main goal of this paper is to generalize this result and to explain how the (resp. conformal) symmetries of ΔY + V (where V is an arbitrary potential) can be used to characterize the R-separation of the Schrodinger equation (ΔY + V)Ψ = EΨ (resp. the Schrodinger equation at zero energy (ΔY + V)Ψ = 0). Using a result exposed in our previous paper, we obtain characterizations of the R-separation of the equations ΔYΨ = 0 and ΔYΨ = EΨ uniquely in terms of (conformal) Killing tensors pertaining to (conformal) Killing-Stackel algebras.