Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).[note 1]
where denotes the tangent space to at the point . So, an element of can be thought of as a pair, where is a point in and is a tangent vector to at . There is a natural projection
defined by . This projection maps each tangent space to the single point .
The tangent bundle comes equipped with a natural topology (described in a section below). With this topology, the tangent bundle to a manifold is the prototypical example of a vector bundle (a fiber bundle whose fibers are vector spaces). A section of is a vector field on , and the dual bundle to is the cotangent bundle, which is the disjoint union of the cotangent spaces of . By definition, a manifold is parallelizable if and only if the tangent bundle is trivial.By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sumTM ⊕ E is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for n = 1, 3, 7 (by results of Bott-Milnor and Kervaire).
One of the main roles of the tangent bundle is to provide a domain and range for the derivative of a smooth function. Namely, if f : M → N is a smooth function, with M and N smooth manifolds, its derivative is a smooth function Df : TM → TN.