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An introduction to weightings along submanifolds

Eckhard Meinrenken

TL;DR

The paper develops a cohesive framework for weightings along submanifolds, unifying coordinate and intrinsic viewpoints and extending to Lie filtrations and Lie groupoids. It introduces the weighted normal bundle $\nu_{\mathcal{W}}(M,N)$ and the weighted deformation space $\delta_{\mathcal{W}}(M,N)$ via graded and Rees algebras, with a zoom action that implements homogeneity. Multiplicative weightings on groupoids are characterized by compatibility conditions (weighted submersions, subgroupoid structure), yielding weighted groupoids $\nu_{\mathcal{W}}(G,H)\rightrightarrows \nu_{\mathcal{W}}(M,N)$ and $\delta_{\mathcal{W}}(G,H)\rightrightarrows \delta_{\mathcal{W}}(M,N)$ and inviting a weighted pseudo-differential calculus. The appendix connects weightings to jet-bundle descriptions, establishing higher-order weightings through submanifolds of $T_{r-1}M$ and a practical criterion due to Gootjes-Dreesbach. Overall, the work provides intrinsic tools for linearization, blow-ups, and hypoelliptic analysis in geometric and groupoid settings.

Abstract

This article is based on a talk given at the Ghent Geometric Analysis Seminar in 2023. We review basic notions from the theory of weightings along submanifolds, with special emphasis on multiplicative weightings for Lie groupoids along subgroupoids.

An introduction to weightings along submanifolds

TL;DR

The paper develops a cohesive framework for weightings along submanifolds, unifying coordinate and intrinsic viewpoints and extending to Lie filtrations and Lie groupoids. It introduces the weighted normal bundle and the weighted deformation space via graded and Rees algebras, with a zoom action that implements homogeneity. Multiplicative weightings on groupoids are characterized by compatibility conditions (weighted submersions, subgroupoid structure), yielding weighted groupoids and and inviting a weighted pseudo-differential calculus. The appendix connects weightings to jet-bundle descriptions, establishing higher-order weightings through submanifolds of and a practical criterion due to Gootjes-Dreesbach. Overall, the work provides intrinsic tools for linearization, blow-ups, and hypoelliptic analysis in geometric and groupoid settings.

Abstract

This article is based on a talk given at the Ghent Geometric Analysis Seminar in 2023. We review basic notions from the theory of weightings along submanifolds, with special emphasis on multiplicative weightings for Lie groupoids along subgroupoids.
Paper Structure (6 sections, 5 theorems, 40 equations)

This paper contains 6 sections, 5 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Definition and examples
  3. Weighted normal bundle
  4. Lie filtrations
  5. Multiplicative weightings
  6. Appendix: Weightings in terms of jet bundle

Key Result

Theorem 3.2

loi:wei. The weighted normal bundle $\nu_\mathcal{W}(M,N)$ is a manifold, in such a way that the functions $f_{[i]}$ for $f\in C^\infty(M)_{(i)}$ are smooth. Similarly, $\delta_\mathcal{W}(M,N)$ is a manifold, in such a way that the functions $\widetilde{f}_{[i]}$ for $f\in C^\infty(M)_{(i)}$ are sm The group $\mathbb{R}^\times$ acts smoothly on the deformation space by the zoom action$t\mapsto \k

Theorems & Definitions (13)

  • Definition 2.1: Coordinate definition loi:weimel:cor
  • Definition 2.2: Intrinsic definition
  • Remark 2.4
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Remark 4.1
  • Theorem 5.1
  • Theorem 5.2: Hudson-M
  • Remark 5.3
  • ...and 3 more