tangentbundle

In differential geometry, the tangent bundle TM of a smooth n-manifold M is the disjoint union of all tangent spaces T_pM at points p in M, together with a natural smooth structure and a projection map π: TM → M. An element of TM is a pair (p, v) with p in M and v in T_pM. The fiber over p, π^{-1}(p), is the tangent space T_pM.

The tangent bundle captures linearized behavior of M at each point and provides a global way to study vector fields and differential equations on M. The dimension of TM is 2n, since each fiber has dimension n and M has dimension n. If M is Euclidean space R^n, then TM is diffeomorphic to R^n × R^n.

TM is constructed by equipping the disjoint union of tangent spaces with a smooth structure that makes π a smooth fiber bundle with typical fiber R^n. Local trivializations arise from coordinate charts: if (U, φ) is a chart on M with φ: U → R^n, then π^{-1}(U) is diffeomorphic to U × R^n via a map that records, at each p, the coordinates of the tangent vector relative to the chart. Transition maps between overlapping charts are smooth, giving TM a smooth manifold structure.

Sections of TM are maps s: M → TM with π ∘ s = id_M, and they correspond to vector fields on M. For example, a vector field assigns to every p ∈ M a tangent vector in T_pM.

Examples include TM ≅ R^n × R^n for M = R^n, and the tangent bundle T(S^2) consisting of pairs (p, v) with p on the 2-sphere and v tangent to the sphere at p. The tangent bundle serves as the natural domain for differential operators and geometric structures.