Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 21 (2025), 002, 41 pages      arXiv:2405.07692      https://doi.org/10.3842/SIGMA.2025.002

Holography of Higher Codimension Submanifolds: Riemannian and Conformal

Samuel Blitz and Josef Šilhan
Department of Mathematics and Statistics, Masaryk University, Building 08, Kotlářská 2, Czech Republic

Received June 01, 2024, in final form December 16, 2024; Published online January 04, 2025

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.

Key words: Riemannian geometry; conformal geometry; submanifold embeddings; holography.

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