2022/09/18 by Juan Sebastián Herrera-Carmona, Herrera-Carmona, Juan Sebastian, Fabricio Valencia +1 · 1 citation
Mathematics · Medicine · #22A22 #58D19 #58H05 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Ophthalmology and Eye Disorders
paper · pdf · doi:10.48550/arxiv.2209.08643
openalex publication_date 2022/09/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.