tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold  is a manifold , which assembles all the tangent vectors in . As a set, it is given by the disjoint union of the tangent spaces of M. That is,

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 .