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The infinitesimal structure of manifolds with non-continuous Riemannian metrics

Vanessa Ryborz

Abstract

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold $M$ endowed with a Riemannian metric $g$ that is possibly discontinuous, with $g, g^{-1} \in L^\infty_{\mathrm{loc}} $ and $ g \in W^{1,p}_{\mathrm{loc}}$ for $ p < \mathrm{dim} M - 1 $.

The infinitesimal structure of manifolds with non-continuous Riemannian metrics

Abstract

This paper investigates the failure of certain metric measure spaces to be infinitesimally Hilbertian or quasi-Riemannian manifolds, by constructing examples arising from a manifold endowed with a Riemannian metric that is possibly discontinuous, with and for .

Paper Structure

This paper contains 11 sections, 20 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Basic definitions on metric measure spaces
  3. First order calculus on metric measure spaces
  4. Manifolds with continuous Riemannian metrics
  5. The distance on manifolds with non-continuous Riemannian metrics
  6. Construction of examples
  7. A first example of a non-quasi Riemannian manifold
  8. A first example of non-infinitesimal Hilbertianity
  9. An example of a Sobolev Riemannian metric, neither quasi-Riemannian nor infinitesimally Hilbertian
  10. Analysis of quasi-Riemannianity
  11. Analysis of infinitesimal Hilbertianity

Key Result

Theorem 1

For $d \geq 3$, $p \in [1, d-1)$ there exists a $d$-dimensional manifold $M$ and a Riemannian metric $g$ on $M$ such that $g, g^{-1} \in L^\infty$, $g \in W^{1,p}_{\rm loc}(M)$ and such that

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6: burtscher2012length Prop. 4.1 and Thm 3.15
  • Proposition 2.7: burtscher2012length, Proposition 4.10
  • Proposition 2.8: mondino2025equivalence Prop. 4.24
  • ...and 34 more