A slice theorem for singular Riemannian foliations, with applications

Abstract

Abstract. We prove a Slice Theorem around closed leaves in a singular Riemannian foliation, and we use it to study the C ∞ -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.

Almost non-negative curvature and rational ellipticity in cohomogeneity two

“…From the slice theorem for SRF due to Mendes and Radeschi [21], it follows as in the proof above that also in the general case of a SRF F which is infinitesimally polar, any riemannian orbifold metric on M/F lifts to a metric on M relative to which F is a SRF.…”

On invariant linearization of Lie groupoids

“…3.3 is a groupoid version of the classic result[H60, Thm.1], asserting that a complete Riemannian submersion is locally trivial, whose converse was later shown in [dH16,Thm.5]. Our result should also be compared with [MR19,Thm.1], where a complete singular Riemannian foliation (M, F, η) is shown to be isomorphic to a linear model over a tube around a leaf -the complete hypothesis is missing in their statement but used along the proof. When the foliation is induced by a complete Riemannian groupoid then the invariant linearization gives a similar result.…”

Lie groupoids and semi-local models of Singular Riemannian foliations

“…The main ingredients of the proof of item (a) were already presented in [2]. Here we put these ingredients together with help of Proposition 4.1 stressing its semi-local description (see also the discussion in Mendes and Radeschi [3] for the case where B is a closed leaf). Item (b) in the Theorem is new and we believe it establishes a natural bridge between the theories of SRFs and Lie groupoids.…”