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Riemannian metrics on differentiable stacks

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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.

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Acknowledgements

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. Departamento de Geometria - IME, Universidade Federal Fluminense, Rua Professor Marcos Waldemar de Freitas Reis, Niterói, 24210-201, Brazil

    Matias del Hoyo

  2. Department of Mathematics, University of Illinois at Urbana-Champaign, 1409 W. Green Street, Urbana, IL, 61801, USA

    Rui Loja Fernandes

Authors
  1. Matias del Hoyo
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  2. Rui Loja Fernandes
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Correspondence to Matias del Hoyo.

Additional information

MdH was partially supported by ERC Starting Grant No. 279729 and National Council for Scientific and Technological Development - CNPq grant 303034/2017-3. RLF was partially supported by NSF grants DMS 1308472, DMS 1405671, DMS 1710884 and FCT/Portugal. Both authors acknowledge the support of the Ciências Sem Fronteiras grant 401817/2013-0.

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del Hoyo, M., Fernandes, R.L. Riemannian metrics on differentiable stacks. Math. Z. 292, 103–132 (2019). https://doi.org/10.1007/s00209-018-2154-6

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