Riemannian metrics on Lie groupoids

TL;DR

This paper introduces a new notion of Riemannian metric on Lie groupoids, explores its properties, and uses it to prove a linearization theorem, enhancing understanding of the geometric structure of these groupoids.

Contribution

It defines a compatible Riemannian metric on Lie groupoids and establishes a linearization theorem, strengthening previous results and simplifying proofs.

Findings

Many proper Lie groupoids admit such metrics

The exponential map facilitates a Linearization Theorem

Provides a stronger version of Weinstein-Zung Linearization Theorem

Abstract

We introduce a notion of metric on a Lie groupoid, compatible with multiplication, and we study its properties. We show that many families of Lie groupoids admit such metrics, including the important class of proper Lie groupoids. The exponential map of these metrics allow us to establish a Linearization Theorem for Riemannian groupoids, obtaining both a simpler proof and a stronger version of the Weinstein-Zung Linearization Theorem for proper Lie groupoids. This new notion of metric has a simplicial nature which will be explored in future papers of this series.