Subfibers

Subfibers are a concept in the study of fibered spaces, particularly fiber bundles. Given a fiber bundle π: E → B with typical fiber F, a subfiber is a subset E′ ⊆ E for which the restriction π|_{E′}: E′ → B is again a fiber bundle, whose fiber F′ is a subspace of F. If such a subbundle exists with a fixed subspace F′ ⊆ F and the bundle structure reduces accordingly, E′ is called a subbundle of E over B with fiber F′. Equivalently, there exists an open covering {U_i} of B and diffeomorphisms φ_i: π^{-1}(U_i) → U_i × F such that φ_i(E′ ∩ π^{-1}(U_i)) = U_i × F′ for a fixed F′.

Examples illustrate the idea. In a trivial bundle B × F, the set B × F′ with F′ ⊆ F is a subfiber. In a vector bundle E → B, a subvector bundle consists of subspaces E′_b ⊆ E_b that vary smoothly and form a bundle with fiber F′.

Notes and properties. Subfibers may exist locally without global globalization; their existence depends on the structure group and transition functions preserving F′. A subbundle arises when the structure group reduces to the subgroup that preserves F′, enabling a global E′ → B with fiber F′.

See also: fiber bundle, subbundle, vector bundle, subfibration.