A Slice Theorem for singular Riemannian foliations, with applications

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.

18 pages, 1 figure. Manuscript modified to put an emphasis on the slice theorem. Title changed accordingly. Preliminary section on singular Riemannian foliations added. Proof of Theorem A greatly simpllified. To appear in Transactions of the AMS