This naturally has a manifold structure, given as follows. For , is obviously isomorphic to , and is thus obviously a manifold. By the definition of a differentiable manifold, for any , there is a neighborhood of and a diffeomorphism . Since this map is a diffeomorphism, its derivative is an isomorphism at all points. Thus is bijective, which endows with a natural structure of a differentiable manifold. Since the transition maps for are differentiable, they are for as well, and is a differentiable manifold. In fact, the projection forgetting the tangent vector and remembering the point, is a vector bundle. A vector field on is simply a section of this bundle.
The tangent bundle is functorial in the obvious sense: If is differentiable, we get a map , defined by on the base, and its derivative on the fibers.
|Date of creation||2013-03-22 13:58:59|
|Last modified on||2013-03-22 13:58:59|
|Last modified by||bwebste (988)|