Tangent Bundle

In differential geometry, the tangent bundle of a differentiable manifold M is the disjoint union of the tangent spaces of M. That is,

where T_xM denotes the tangent space to M at the point x. So, an element of TM can be thought of as a pair (x, v), where x is a point in M and v is a tangent vector to M at x. There is a natural projection

defined by π(x, v) = x. This projection maps each tangent space T_xM to the single point x.

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 TM is a vector field on M, and the dual bundle to TM is the cotangent bundle, which is the disjoint union of the cotangent spaces of M. By definition, a manifold M 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 sum TM ⊕ 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).