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Isometric Lie 2-Group Actions on Riemannian Groupoids

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Abstract

We study isometric actions of Lie 2-groups on Riemannian groupoids by exhibiting some of their immediate properties and implications. Firstly, we prove an existence result which allows both to obtain 2-equivariant versions of the Slice Theorem and the Equivariant Tubular Neighborhood Theorem and to construct bi-invariant groupoid metrics on compact Lie 2-groups. We provide natural examples, transfer some classical constructions and explain how this notion of isometric 2-action yields a way to develop a 2-equivariant Morse theory on Lie groupoids. Secondly, we give an infinitesimal description of an isometric Lie 2-group action. We define an algebra of transversal infinitesimal isometries associated to any Riemannian n-metric on a Lie groupoid which in turn gives rise to a notion of geometric Killing vector field on a quotient Riemannian stack. If our Riemannian stack is separated then we prove that the algebra formed by such geometric Killing vector fields is always finite dimensional.

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Acknowledgements

We would like to thank Mateus de Melo, Cristian Ortiz and Luca Vitagliano for several enlightened discussions, comments, and suggestions which allowed us to improve this work. Valencia is grateful to Matias del Hoyo for inviting and supporting him to visit IMPA in Rio de Janeiro in December, 2022. Their mathematical discussions during the academic visit encouraged the authors to start enhancing the presentation and the results stated in the first version of the present work. We are also thankful to the anonymous referees who provided many suggestions and corrections that improved the quality of our results. Herrera-Carmona was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brasil (CAPES) – Finance Code 001. Valencia was supported by Grant 2020/07704-7 Sao Paulo Research Foundation-FAPESP.

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  1. Instituto de Matemática e Estadística, Universidade de São Paulo, Rua do Matão 1010, Cidade Universitária, São Paulo, 05508-090, Brazil

    Juan Sebastián Herrera-Carmona & Fabricio Valencia

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  1. Juan Sebastián Herrera-Carmona
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  2. Fabricio Valencia
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Correspondence to Fabricio Valencia.

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Herrera-Carmona, J.S., Valencia, F. Isometric Lie 2-Group Actions on Riemannian Groupoids. J Geom Anal 33, 323 (2023). https://doi.org/10.1007/s12220-023-01392-w

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