Tangent Bundle

The tangent bundle TM of a Smooth Manifold $M$ is the Space of Tangent Vectors to points in the manifold, i.e., it is the set $(x,v)$ where $x\in M$ and $v$ is tangent to $x\in M$. For example, the tangent bundle to the Circle is the Cylinder.

See also Cotangent Bundle, Tangent Vector