Cotangentbundel

Cotangentbundel, or cotangent bundle, of a smooth manifold M is the vector bundle T* M whose fiber over a point p in M is the cotangent space T*_p M, the dual of the tangent space T_p M. The total space T* M is the disjoint union of all cotangent spaces and comes with a natural projection π: T* M → M. If dim M = n, then T* M is a smooth manifold of dimension 2n.

Locally, choosing coordinates x^1, ..., x^n on M induces coordinates on T* M given by (x^i, p_i), where p_i are coordinates on the fiber corresponding to the basis dx^i. The cotangent bundle therefore has a canonical, coordinate-dependent but geometrically intrinsic, structure.

Two central geometric objects live on T* M. The canonical one-form θ is defined so that θ_{(p, α)}(v) = α(Dπ_{(p, α)} v); in coordinates this becomes θ = Σ_i p_i dx^i. Its exterior derivative ω = dθ is a symplectic form, expressed as ω = Σ_i dx^i ∧ dp_i. The pair (T* M, ω) makes the cotangent bundle a natural symplectic manifold, a fundamental setting in classical and geometric mechanics.

Functorial aspects include the cotangent lift: a smooth map f: M → N induces a map T* f: T* N → T* M. For M = R^n, T* R^n is diffeomorphic to R^n × R^n with coordinates (q^i, p_i).

Applications span Hamiltonian mechanics, symplectic geometry, geometric quantization, and representation theory, where the cotangent bundle often serves as a canonical phase space.