Geometry of orbit spaces of proper Lie groupoids

Details Geometry of orbit spaces of proper Lie groupoids Geometry of orbit spaces of proper Lie groupoids

2010-12-30

arxiv.org/abs/1101.0180v3

Mathematics

Differential Geometry

31 pages

Scientific paper

In this paper, we study geometric properties of quotient spaces of proper Lie groupoids. First, we construct a natural stratification on such spaces using an extension of the slice theorem for proper Lie groupoids of Weinstein and Zung. Next, we show the existence of an appropriate metric on the groupoid which gives the associated Lie algebroid the structure of a singular riemannian foliation. With this metric, the orbit space inherits a natural length space structure whose properties are studied. Moreover, we show that the orbit space of a proper Lie groupoid can be triangulated. Finally, we prove a de Rham theorem for the complex of basic di?erential forms on a proper Lie groupoid.

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