Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 499Let T Y → Y be the tangent bundle of a bundle Y → X. The followingdiagram commutesπ YT Y❄YT ππ✲ T Xπ X❄✲ Xwhere T π : T Y → T X is a fibre bundle. Note that T π is still the bundlemap of the bundle T Y → Y to T X over π and the fibred map of the bundleT Y → X to T X over X. There is also the canonical surjectionπ T : T Y −→ T X −→ Y, given by π T = π X ◦ T π = π ◦ π Y .Now, given the fibre coordinates (x α , y i ) of a fibre bundle Y , the correspondinginduced coordinates of T Y are(x α , y i , ẋ α , ẏ i ), ẏ ′ i =∂y ′ i∂y j ẏj .This expression shows that the tangent bundle T Y → Y of a fibre bundleY has the vector subbundleV Y = Ker T πwhere T π is regarded as the fibred map of T Y → X to T X over X. Thesubbundle V Y consists of tangent vectors to fibres of Y . It is called thevertical tangent bundle of Y and provided with the induced coordinates(x α , y i , ẏ i ) with respect to the fibre bases {∂ i }.The vertical cotangent bundle V ∗ Y → Y of a fibre bundle Y → X isdefined as the dual of the vertical tangent bundle V Y → Y . Note that itis not a subbundle of the cotangent bundle T ∗ Y , but there is the canonicalsurjectionζ : T ∗ Y −→ V ∗ Y, ẋ α dx α + ẏ i dy i ↦→ ẏ i dy i , (4.12)where {dy i } are the bases for the fibres of V ∗ Y which are duals of theholonomic frames {∂ i } for the vertical tangent bundle V Y .With V Y and V ∗ Y , we have the following short exact sequences ofvector bundles over a fibre bundle Y → X:0 → V Y ↩→ T Y → Y × T X → 0, (4.13)0 → Y × T ∗ X ↩→ T ∗ Y → V ∗ Y → 0 (4.14)