A Slice Theorem for singular Riemannian foliations, with applications

TL;DR

This paper proves a Slice Theorem for singular Riemannian foliations around closed leaves and explores the algebra of smooth basic functions, showing it is finitely generated by polynomials in the infinitesimal case.

Contribution

It introduces a Slice Theorem for singular Riemannian foliations and extends results on the algebra of basic functions to the inhomogeneous setting.

Findings

Established a Slice Theorem around closed leaves.

Showed the algebra of smooth basic functions is finitely generated by polynomials in the infinitesimal case.

Generalized results by G. Schwarz to inhomogeneous singular Riemannian foliations.

Abstract

We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the $C^\infty$-algebra of smooth basic functions, generalizing to the inhomogeneous setting a number of results by G.~Schwarz. In particular, in the infinitesimal case we show that this algebra is generated by a finite number of polynomials.