Riemannian metrics on differentiable stacks

TL;DR

This paper develops Riemannian metrics on Lie groupoids and differentiable stacks, enabling linearization, rigidity results, and invariance properties that unify and extend classical geometric theorems.

Contribution

It introduces Riemannian metrics on stacks, proves Morita invariance, and applies these to linearization and tubular neighborhood constructions in differentiable stacks.

Findings

Any split fibration between proper groupoids can be equipped with a Riemannian metric.

Riemannian metrics on stacks are Morita invariant.

Derived rigidity theorems and stacky tubular neighborhoods.

Abstract

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.