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Holography of Higher Codimension Submanifolds: Riemannian and Conformal

Samuel Blitz, Josef Šilhan

TL;DR

This work extends holographic techniques to submanifolds of arbitrary codimension, revealing that higher-codimension embeddings yield invariants that depend on a chosen parallelization of the normal bundle. By developing canonical defining maps and defining densities in both Riemannian and conformal geometries, the authors identify order-by-order obstructions that generalize classical invariants like the Willmore energy. A key outcome is the emergence of a Willmore-type holographic obstruction density in the conformal, higher-codimension setting, along with a conformal extrinsic Laplacian that governs its leading behavior. The paper further investigates extension problems off the submanifold, providing symmetric-extension results and highlighting intrinsic obstructions in the restricted-extension problem, all within a tractor-calculus framework. These results significantly advance extrinsic conformal geometry and furnish tools for systematically probing submanifold embeddings via holographic data.

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.

Holography of Higher Codimension Submanifolds: Riemannian and Conformal

TL;DR

This work extends holographic techniques to submanifolds of arbitrary codimension, revealing that higher-codimension embeddings yield invariants that depend on a chosen parallelization of the normal bundle. By developing canonical defining maps and defining densities in both Riemannian and conformal geometries, the authors identify order-by-order obstructions that generalize classical invariants like the Willmore energy. A key outcome is the emergence of a Willmore-type holographic obstruction density in the conformal, higher-codimension setting, along with a conformal extrinsic Laplacian that governs its leading behavior. The paper further investigates extension problems off the submanifold, providing symmetric-extension results and highlighting intrinsic obstructions in the restricted-extension problem, all within a tractor-calculus framework. These results significantly advance extrinsic conformal geometry and furnish tools for systematically probing submanifold embeddings via holographic data.

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.
Paper Structure (21 sections, 16 theorems, 180 equations)

This paper contains 21 sections, 16 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Notations and conventions
  3. Orthonormal frames for normal bundles of Riemannian submanifold embeddings
  4. Classical submanifold geometry
  5. Rotation minimizing frames
  6. Gauge-fixed frames
  7. Defining maps for parallelized Riemannian submanifold embeddings
  8. First order
  9. Second order
  10. Higher orders
  11. Conformal geometry
  12. Orthonormal frames for normal bundles of conformal submanifold embeddings
  13. Defining densities for parallelized conformal submanifold embeddings
  14. First order
  15. Second order
  16. ...and 6 more sections

Key Result

Proposition 2.1

Let $\Lambda^{d-k} \hookrightarrow (M,g)$ be a smooth submanifold embedding with $1 \leq k \leq d-1$. Then the normal curvature $\mathcal{R}$ vanishes if and only if there exists in a neighborhood around every point $p \in \Lambda$ an orthonormal frame for the normal bundle $N \Lambda$ with vanishin

Theorems & Definitions (24)

  • Proposition 2.1
  • Definition 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Lemma 3.4
  • Corollary 3.5
  • Remark 5.1
  • ...and 14 more