Subbundle

Subbundle is a concept in the theory of vector bundles over a smooth manifold. Let E → M be a smooth vector bundle. A subbundle F ⊂ E is a smooth vector subbundle of E over M, such that for every x ∈ M the fiber F_x is a linear subspace of E_x and the inclusion F → E is a bundle map over M. Equivalently, F is a subbundle if there exists an open cover {U_i} of M with local trivialisations φ_i: E|_{U_i} → U_i × V and subspaces V_i ⊆ V of fixed dimension, such that φ_i(F|_{U_i}) = U_i × V_i for all i and the transition maps preserve these subspaces V_i. In particular, the rank of F is constant.

Examples include the tangent bundle TM together with distributions of constant rank (a choice of subbundle of TM); the vertical subbundle of the tangent bundle of a fiber bundle; and holomorphic subbundles in complex geometry (where E and F carry holomorphic structures).

Properties: F has a well-defined rank k, and its fibers vary smoothly with x. Sections of F are vector fields on M that at each point take values in F_x. If F is a subbundle of E, there is a natural quotient bundle E/F with fibers E_x/F_x. If a complement G ⊂ E exists as a subbundle, then E ≅ F ⊕ G; in smooth categories this can be arranged locally or globally with extra structure, such as a connection or a Riemannian metric giving an orthogonal complement.

Terminology: the term subbundle is distinguished from a subsheaf in the holomorphic category; a subbundle is a subbundle in the sense of vector bundles, with a compatible smooth or holomorphic structure. In geometric constructions, subbundles encode constraints or foliations and interact with bundle theory, connections, and characteristic classes.