Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 493dimension, fdim Y of Y ), and Y admits a bundle atlas Ψ Y (4.8) wheretrivialization maps ψ ξ (x) and transition functions ρ ξζ (x) are linear isomorphismsof vector spaces. The corresponding bundle coordinates (y i ) obeya linear coordinate transformation lawy ′ i = ρij (x)y j .We have the decomposition y = y i e i (π(y)), where{e i (x)} = ψ −1ξ (x){v i}, x = π(y) ∈ U ξ ,are fibre bases (or frames) for fibres Y x of Y and {v i } is a fixed basis forthe typical fibre V of Y .There are several standard constructions of new vector bundles from oldones:• Given two vector bundles Y and Y ′ over the same base X, their Whitneysum Y ⊕ Y ′ is a vector bundle over X whose fibres are the direct sumsof those of the vector bundles Y and Y ′ .• Given two vector bundles Y and Y ′ over the same base X, their tensorproduct Y ⊗ Y ′ is a vector bundle over X whose fibres are the tensorproducts of those of the vector bundles Y and Y ′ . In a similar way theexterior product Y ∧Y of vector bundles is defined, so that the exteriorbundle of Y is defined as∧Y = X × R ⊕ Y ⊕ ∧ 2 Y ⊕ · · · ⊕ ∧ m Y, (m = fdim Y ).• Let Y → X be a vector bundle. By Y ∗ → X is denoted the dual vectorbundle whose fibres are the duals of those of Y . The interior product(or contraction) of Y and Y ∗ is defined as a bundle map⌋ : Y ⊗ Y ∗ → X × R.Given a linear bundle map Φ : Y ′ → Y of vector bundles over X, itskernel Ker Φ is defined as the inverse image Φ −1 (̂0(X)) of the canonicalzero section ̂0(X) of Y . If Φ is of constant rank, its kernel Ker Φ and itsimage Im Φ are subbundles of the vector bundles Y ′ and Y , respectively.For example, monomorphisms and epimorphisms of vector bundles fulfilthis condition. If Y ′ is a subbundle of the vector bundle Y → X, the factorbundle Y/Y ′ over X is defined as a vector bundle whose fibres are thequotients Y x /Y ′ x, x ∈ X.