Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 103–132 | Cite as

Riemannian metrics on differentiable stacks

  • Matias del HoyoEmail author
  • Rui Loja Fernandes


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.



We are grateful to IMPA, UU and UIUC for hosting us at several stages of this project. We thank H. Bursztyn, E. Lerman, I. Marcut and I. Moerdijk for fruitful discussions, and to M. Crainic, J.N. Mestre and I. Struchiner for sharing with us a preliminary version of their preprint [6]. We also thank the referee for his comments and suggestions, that helped improve this manuscript.


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Authors and Affiliations

  1. 1.Departamento de Geometria - IMEUniversidade Federal FluminenseNiteróiBrazil
  2. 2.Department of MathematicsUniversity of Illinois at Urbana-ChampaignUrbanaUSA

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