Riemannian metrics on Lie groupoids

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.

29 pages; Final version accepted for publication in Journal für die reine und angewandte Mathematik (Crelle)

Cited by in corpus (10)

• Deformations of Lie groupoids
• Orbispaces as differentiable stratified spaces
• On deformations of compact foliations
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• On Hausdorff integrations of Lie algebroids
• Geodesics on Riemannian stacks
• Principal bundles and connections modelled by Lie group bundles
• Orbifold Euler characteristics of non-orbifold groupoids
• On invariant linearization of Lie groupoids
• Differentiating $L_{\infty }$ groupoids