Vector Fields and Flows on Differentiable Stacks

Abstract

This paper introduces the notions of vector field and flow on a general differentiable stack. Our main theorem states that the flow of a vector field on a compact proper differentiable stack exists and is unique up to a uniquely determined 2-cell. This extends the usual result on the existence and uniqueness of flows on a manifold as well as the author's existing results for orbifolds. It sets the scene for a discussion of Morse Theory on a general proper stack and also paves the way for the categorification of other key aspects of differential geometry such as the tangent bundle and the Lie algebra of vector fields.

41 pages

References in corpus (6)

• A New Cohomology Theory for Orbifold
• Orbifolds as Groupoids: an Introduction
• Introduction to the language of stacks and gerbes
• Differentiable Stacks and Gerbes
• Morse Inequalities for Orbifold Cohomology
• Pontrjagin-Thom maps and the homology of the moduli stack of stable curves

Cited by in corpus (9)

• Lie 2-algebras of vector fields
• Orbispaces as differentiable stratified spaces
• The stack of Yang-Mills fields on Lorentzian manifolds
• Stacky Hamiltonian actions and symplectic reduction
• Multiplicative vector fields on bundle gerbes
• Lie groupoids and the Frolicher-Nijenhuis bracket
• Proper Orbifold Cohomology
• Multiplicative Connections and Their Lie Theory
• Equivariant Bifurcation from Relative Equilibria via Isomorphic Vector Fields