Riemannian metrics on Lie groupoids. (English) Zbl 1385.53076
The authors introduce a notion of Riemannian metric for Lie groupoids which is compatible with the groupoid multiplication. It is shown that such a metric can be attached to Lie groups, étale groupoids, transitive groupoids, locally trivial Lie group bundles, as well as Hausdorff proper Lie groupoids. The main result of the paper is that the exponential map associated with this metric can be used to prove the linearization theorem for Riemannian groupoids as such. Corollaries of this result are Ehresmann’s theorem for submersions, the Reeb stability theorem for foliations, the Bochner linearisation theorem as well as the Tube theorem for proper Lie group actions. The authors also announce forthcoming work of theirs, where this notion of Riemannian metric gives rise to a metric on smooth stacks.
Reviewer: Iakovos Androulidakis (Athína)
MSC:
| 53D17 | Poisson manifolds; Poisson groupoids and algebroids |
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