Master's Thesis Defense: A Generalization of Molino Theory to Riemannian Groupoids

Abstract: This thesis extends Molino theory from Riemannian foliations to regular Riemannian groupoids.

Riemannian groupoids describe Riemannian foliations together with their symmetries. In this thesis, we extend classical Molino theory, which concerns the structure of Riemannian foliations on compact manifolds, to the setting of regular Riemannian groupoids with compact unit spaces.

We observe that the orbit foliation associated with a regular Riemannian groupoid defines a Riemannian foliation on the unit space. The main result of the thesis shows that the normal representation of a Riemannian groupoid extends naturally to an action on the Molino structures of the orbit foliation, thereby yielding the fundamental structural description of a regular Riemannian groupoid.

In addition to the main result, we clarify the essential role played by basic Lie algebroids in Molino theory and show that the Molino structure associated with a regular Riemannian groupoid is invariant under Morita equivalence.

As applications, we derive Molino-type structure theorems for compact group actions and orbifolds.

Faculty Advisor: Xiang Tang