Riemannian Groupoids

Abstract

This thesis studies the geometry of Lie groupoids. In particular, it introduces a new type of Riemannian metric on groupoid that is compatible with the groupoid structure. This metric is a simplification of an earlier definition from the literature. We use this simplified definition to reprove the linearization theorem for proper groupoids. Finally, associated to these multiplicative metrics, there is a new type of Ehressman connection on the source map of the groupoid. This thesis also contains a preliminary study of these new connections.