Riemannian metrics on Lie groupoids

Understand

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.