GROUPOID ASSOCIATED TO A SMOOTH MANIFOLD

In this paper, we introduce the structure of a groupoid associated to a vector eld on a smooth manifold. We show that in the case of the 1-dimensional manifolds, our groupoid has a smooth structure such that makes it into a Lie groupoid. Using this approach, we associated to every vector eld an equivalence relation on the Lie algebra of all vector elds on the smooth manifolds.