GROUPOID ASSOCIATED TO A SMOOTH MANIFOLD

Abstract

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