Holography of Higher Codimension Submanifolds: Riemannian and Conformal

TL;DR

This paper extends holographic methods to higher-codimension submanifolds, revealing new invariants and behaviors in Riemannian and conformal geometry, and exploring formal solutions to extension problems.

Contribution

It introduces a generalized holographic approach for higher-codimension submanifolds, uncovering new invariants and behaviors not present in lower codimension cases.

Findings

Discovery of new invariants obstructing unit defining maps.

Identification of a novel conformal invariant vanishing in codimension 1.

Analysis of formal solutions to extension problems off submanifolds.

Abstract

We provide a natural generalization to submanifolds of the holographic method used to extract higher-order local invariants of both Riemannian and conformal embeddings, some of which depend on a choice of parallelization of the normal bundle. Qualitatively new behavior is observed in the higher-codimension case, giving rise to new invariants that obstruct the order-by-order construction of unit defining maps. In the conformal setting, a novel invariant (that vanishes in codimension 1) is realized as the leading transverse-order term appearing in a holographically-constructed Willmore invariant. Using these same tools, we also investigate the formal solutions to extension problems off of an embedded submanifold.