Partial Groupoid Actions on Smooth Manifolds

Abstract Given a smooth partial action $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> of a Lie groupoid G on a smooth manifold M , we provide necessary and sufficient conditions for $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> to be globalizable with smooth globalization. As an application, we provide results on the differentiable structure of orbit and stabilizer spaces induced by $$\alpha ,$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> which leads to other criteria for its globalization in terms of its orbit maps in the case that $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is free and transitive. Further, under the assumption that $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> is free and proper, we prove that there exists exactly one differentiable structure on the quotient structure of the orbit equivalence space M / G such that the quotient map $$\pi :M\rightarrow M/G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>π</mml:mi> <mml:mo>:</mml:mo> <mml:mi>M</mml:mi> <mml:mo>→</mml:mo> <mml:mi>M</mml:mi> <mml:mo>/</mml:mo> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> is a submersion.