Ivancevic_Applied-Diff-Geom

490 Applied Differential Geometry: A Modern Introductioni.e., Φ is a fibrewise map over f which sends a fibre Y x , (for all x ∈ X),to a fibre Y ′ f(x), (for all f(x) ∈ X ′ ). A bundle diffeomorphism is calledan automorphism if it is an isomorphism to itself. In field theory, anyautomorphism of a fibre bundle is treated as a gauge transformation.Given a bundle Y → X, every map f : X ′ → X induces a bundleY ′ = f ∗ Y over X ′ which is called the pull–back of the bundle Y by f, suchthat the following diagram commutesY ✛ f ∗Y ′ππ ′❄X ✛❄Xf′In particular, the product Y × Y ′ over X of bundles π : Y → X and π ′ :Y ′ → X is the pull–backY × Y ′ = π ∗ Y ′ = π ′ ∗ Y.Classical fields are described by sections of fibre bundles. A (global)section of a fibre bundle Y → X is defined as a π−inverse manifold injections : X → Y, s(x) ↦→ Y x , such that π ◦ s = Id X . That is, a section s sendsany point x ∈ X into the fibre Y x ⊂ Y over this point. A section sis an imbedding, i.e., s(X) ⊂ Y is both a submanifold and a topologicalsubspace of Y . It is also a closed map, which sends closed subsets of X ontoclosed subsets of Y . Similarly, a section of a fibre bundle Y → X over asubmanifold of X is defined. Given a bundle atlas Ψ Y and associated bundlecoordinates (x α , y i ), a section s of a fibre bundle Y → X is represented bycollections of local functions {s i = y i ◦ψ ξ ◦s} on trivialization sets U ξ ⊂ X.A fibre bundle Y → X whose typical fibre is diffeomorphic to an Euclideanspace R m has a global section. More generally, its section over aclosed imbedded submanifold (e.g., a point) of X is extended to a globalsection [Steenrod (1951)].In contrast, by a local section is usually meant a section over an opensubset of the base X. A fibre bundle admits a local section around eachpoint of its base, but need not have a global section.For any n ≥ 1 the normal bundle NS n of the n−sphere S n is the fibrebundle (S n , p ′ , E ′ , R 1 ), where E ′ = {(x, y) ∈ R n+1 × R n+1 : ‖x‖ = 1, y =λx, λ ∈ R 1 } and p ′ : E ′ → S n is defined by p ′ (x, y) = x [Switzer (1975)].