# Lie groupoids and semi-local models of Singular Riemannian foliations

## TL;DR

This paper develops a local model for Singular Riemannian Foliations near certain submanifolds and constructs a controlling Lie groupoid to understand their transverse geometry.

## Contribution

It introduces a semi-local model for singular foliations and constructs a Lie groupoid that captures their transverse geometric structure.

## Key findings

- Constructed a local model for singular foliations near closed saturated submanifolds.
- Built a Lie groupoid controlling the transverse geometry of the linear approximation.
- Analyzed the closure and Lie algebroid of the constructed Lie groupoid.

## Abstract

We describe a local model for any Singular Riemannian Foliation in a neighbourhood of a closed saturated submanifold of a regular stratum. Moreover we construct a Lie groupoid which controls the transverse geometry of the linear approximation of the Singular Riemannian Foliation around these submanifolds. We also discuss the closure of this Lie groupoid and its Lie algebroid.