  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.
