Euler-like vector fields, deformation spaces and manifolds with filtered structure

Let M be a smooth submanifold of a smooth manifold V. Bursztyn, Lima and Meinrenken defined a concept of Euler-like vector field on V associated to the embedding of M into V, and proved that there is a bijection between germs of tubular neighborhoods of M and germs of Euler-like vector fields. We shall present a new view of this result by characterizing Euler-like vector fields algebraically and examining their relation to the deformation to the normal cone from algebraic geometry. Then we shall extend our algebraic point of view to smooth manifolds that are equipped with Lie filtrations, and define deformations to the normal cone and Euler-like vector fields in that context. Our algebraic construction of the deformation to the normal cone gives a new approach to Connes' tangent groupoid and its generalizations to filtered manifolds. In addition, Euler-like vector fields give rise to preferred coordinate systems on filtered manifolds.