Thesis Defense: A Generalization of Molino Theory to Riemannian Groupoids

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

For a Hausdorff regular Riemannian groupoid with compact connected base manifold, the induced orbit foliation is a Riemannian foliation. Passing to the transverse orthonormal frame bundle and the lifted foliation, this thesis recovers the Molino package: the lifted foliation is transversely parallelizable and admits an $O(q)$-equivariant fibre bundle whose fibres are the closures of its leaves, and the restriction of the lifted foliation to each fibre is a Lie foliation with dense holonomy group.

Moreover, over the Molino base, the construction yields a transitive basic Lie algebroid whose isotropy Lie algebras are identified with the structural Lie algebras of the restricted Lie foliations on the Molino fibres. Equivalently, these fibrewise structural Lie algebras assemble into the smooth isotropy bundle of the basic Lie algebroid.

This thesis also develops corollaries in the compact group equivariant, orbifold, and effective proper étale cases.
 

Faculty Advisor: Xiang Tang