The Local Frobenius Theorem

Abstract

Let M be an n-dimensional smooth manifold. The tangent bundle TM of M is the disjoint union of all tangent spaces of points of M. It can be given the structure of a manifold of dimension $2\dim (M)$ as follows. If U is a coordinate neighborhood and x ₁, …, x _n are local coordinates on U, then $T(U) =\{ T_{x}M\,\vert \,x \in U\}$ can be taken to be a coordinate neighborhood of TM.