Riemannian metrics on differentiable stacks

Abstract

We study Riemannian metrics on Lie groupoids in the relative setting. We show that any split fibration between proper groupoids can be made Riemannian, and we use these metrics to linearize proper groupoid fibrations. As an application, we derive rigidity theorems for Lie groupoids, which unify, simplify and improve similar results for classic geometries. Then we establish the Morita invariance for our metrics, introduce a notion for metrics on stacks, and use them to construct stacky tubular neighborhoods and to prove a stacky Ehresmann theorem.

26 pages, final version. Main Theorem 4.2.3 on linearization of groupoid fibrations was strengthened. Application to deformation of foliations was removed from here and elaborated in the independent note arXiv:1807.10748

References in corpus (3)

• Topological and Smooth Stacks
• Regular Poisson manifolds of compact types (PMCT 2)
• On deformations of compact foliations

Cited by in corpus (13)

• Orbispaces as differentiable stratified spaces
• Morita equivalences of vector bundles
• Configuration Lie groupoids and orbifold braid groups
• Cartan Connections on Lie Groupoids and their Integrability
• On deformations of compact foliations
• Integrability of quotients in Poisson and Dirac geometry
• Integration of quadratic Lie algebroids to Riemannian Cartan-Lie groupoids
• Geodesics on Riemannian stacks
• On invariant linearization of Lie groupoids
• Shifted coisotropic structures for differentiable stacks
• Weak representations, representations up to homotopy, and VB-groupoids
• Multiplicative Gray stability
• The integration problem for principal connections