Isometric Lie 2-group actions on Riemannian groupoids

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.

31 pages. Substantial changes have been made. The presentation has been improved, new results and examples have been added. Final version accepted for publication