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The first purpose of this note is to comment on a recent article of Bursztyn,Lima and Meinrenken, in which it is proved that if M is a smooth submanifold ofa manifold V, then there is a bijection between germs of tubular neighborhoodsof M and germs of "Euler-like" vector fields on V. We shall explain how toapproach this bijection through the deformation to the normal cone that isassociated to the embedding of M into V. The second purpose is to studygeneralizations to smooth manifolds equipped with Lie filtrations. Following inthe footsteps of several others, we shall define a deformation to the normalcone that is appropriate to this context, and relate it to Euler-like vectorfields and tubular neighborhood embeddings.

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