structure. Applying this product structure to the tangent space at each point and globalizing yields the canonical vector field. Informally, although the manifold is curved, each tangent space at a point , , is flat, so the tangent bundle manifold is locally a product of a curved and a flat Thus the tangent bundle of the tangent bundle is locally (using for "choice of coordinates" and for "natural identification"):

:and the map is the projection onto the first coordinates:
:Splitting the first map via the zero section and