A Generalization of Molino's Theory to Riemannian Groupoids
Abstract
Riemannian groupoids describe Riemannian foliations together with their symmetries. In this article, we extend classical Molino's theory, which concerns the structure of Riemannian foliations on compact manifolds, to the setting of regular Riemannian groupoids with compact connected object manifolds. We observe that the orbit foliation associated with a regular Riemannian groupoid defines a Riemannian foliation on the object manifold. The main result shows that the normal representation of a Riemannian groupoid extends naturally to an action on Molino's 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's theory and identify the normal representation with the degree-one cohomology representation of the adjoint representation up to homotopy. We also establish comparison results under Riemannian Morita equivalence. For regular Riemannian groupoids, we give sufficient conditions for the associated basic Lie algebroids to become isomorphic after pullback to a common refinement.
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BibTeX
@misc{zhang2026generalization,
title = {{A Generalization of Molino's Theory to Riemannian Groupoids}},
author = {Lily Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23341},
eprint = {2609.23341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23341},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Generalization of Molino's Theory to Riemannian Groupoids
AU - Lily Zhang
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23341
UR - https://arxiv.org/abs/2609.23341
AB - Riemannian groupoids describe Riemannian foliations together with their symmetries. In this article, we extend classical Molino's theory, which concerns the structure of Riemannian foliations on compact manifolds, to the setting of regular Riemannian groupoids with compact connected object manifolds. We observe that the orbit foliation associated with a regular Riemannian groupoid defines a Riemannian foliation on the object manifold. The main result shows that the normal representation of a Riemannian groupoid extends naturally to an action on Molino's 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's theory and identify the normal representation with the degree-one cohomology representation of the adjoint representation up to homotopy. We also establish comparison results under Riemannian Morita equivalence. For regular Riemannian groupoids, we give sufficient conditions for the associated basic Lie algebroids to become isomorphic after pullback to a common refinement.
ER -