Glued spaces and lower curvature bounds

TL;DR

This paper surveys theorems and proposes conjectures on gluing constructions under lower curvature bounds across various geometric contexts, including smooth, non-smooth, and synthetic spaces.

Contribution

It introduces new conjectures and synthesizes existing results on gluing techniques under lower curvature bounds in diverse geometric settings.

Findings

Compilation of known theorems on gluing under curvature bounds

Proposed conjectures for future research in synthetic and non-smooth spaces

Insights into curvature bounds in Riemannian, Alexandrov, and RCD spaces

Abstract

In this short note we survey theorems and provide conjectures on gluing constructions under lower curvature bounds in smooth and non-smooth context. Focusing on synthetic lower Ricci curvature bounds we consider Riemannian manifolds, weighted Riemannian manifolds, Alexandrov spaces, collapsed and non-collapsed $RCD$ spaces, and sub-Riemannian spaces.