Isometric Lie 2-group actions on Riemannian groupoids

TL;DR

This paper investigates isometric actions of Lie 2-groups on Riemannian groupoids, establishing foundational theorems, examples, and infinitesimal descriptions that extend classical geometric concepts to higher groupoid structures.

Contribution

It introduces the concept of isometric Lie 2-group actions on Riemannian groupoids, proving existence theorems, constructing bi-invariant metrics, and developing a 2-equivariant Morse theory framework.

Findings

Existence of 2-equivariant Slice and Tubular Neighborhood Theorems

Construction of bi-invariant groupoid metrics on compact Lie 2-groups

Finite-dimensional algebra of geometric Killing vector fields on separated Riemannian stacks

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.