Toward a Classification of Conformal Hypersurface Invariants

TL;DR

This paper develops a minimal set of conformal invariants, including curvatures and fundamental forms, to characterize hypersurfaces in conformal manifolds, aiding the understanding of boundary data in cosmology and string theory.

Contribution

It introduces a finite, minimal family of hypersurface tensors that generate all conformal invariants up to a fixed derivative order, capturing extrinsic embedding data.

Findings

Constructed a finite set of conformal hypersurface invariants.

Demonstrated these invariants encode extrinsic embedding data.

Provided tools for analyzing boundary data in theoretical physics.

Abstract

Hypersurfaces embedded in conformal manifolds appear frequently as boundary data in boundary-value problems in cosmology and string theory. Viewed as the non-null conformal infinity of a spacetime, we consider hypersurfaces embedded in a Riemannian (or Lorentzian) conformal manifold. We construct a finite and minimal family of hypersurface tensors -- the curvatures intrinsic to the hypersurface and the so-called ``conformal fundamental forms'' -- that can be used to construct natural conformal invariants of the hypersurface embedding up to a fixed order in hypersurface-orthogonal derivatives of the bulk metric. We thus show that these conformal fundamental forms capture the extrinsic embedding data of a conformal infinity in a spacetime.