Ivancevic_Applied-Diff-Geom

Applied Jet Geometry 811Furthermore, if s : X → Y is a global section, there exists a connectionΓ such that s is an integral section of Γ. This connection is defined asan extension of the local section s(x) ↦→ j 1 s(x) of the affine jet bundleJ 1 (X, Y ) → Y over the closed imbedded submanifold s(X) ⊂ Y .Note that every connection Γ on the bundle Y −→ X defines a system offirst–order differential equations on Y (in the spirit of [Bryant et. al. (1991);Krasil’shchik et. al. (1985); Pommaret (1978)]) which is an imbedded subbundleΓ(Y ) = Ker D Γ of the jet bundle J 1 (X, Y ) −→ Y . It is given by thecoordinate relationsy i α = Γ i (y). (5.33)Integral sections for Γ are local solutions of (5.33), and vice versa.We can introduce the following basic forms involving a connection Γand a soldering form σ:• the curvature of a connection Γ is given by the horizontal verticalvaluedtwo–form:R = 1 2 d ΓΓ = 1 2 Ri αµdx α ∧ dx µ ⊗ ∂ i ,R i αµ = ∂ α Γ i µ − ∂ µ Γ i α + Γ j α∂ j Γ i µ − Γ j µ∂ j Γ i α; (5.34)• the torsion of a connection Γ with respect to σ:Ω = d σ Γ = d Γ σ = 1 2 Ωi αµdx α ∧ dx µ ⊗ ∂ i• the soldering curvature of σ:= (∂ α σ i µ + Γ j α∂ j σ i µ − ∂ j Γ i ασ j µ)dx α ∧ dx µ ⊗ ∂ i ; (5.35)ε = 1 2 d σσ = 1 2 εi αµdx α ∧ dx µ ⊗ ∂ i= 1 2 (σj α∂ j σ i µ − σ j µ∂ j σ i l)dx α ∧dx µ ⊗ ∂ i . (5.36)They satisfy the following relations:Γ ′ = Γ + σ, R ′ = R + ε + Ω, Ω ′ = Ω + 2ε.In particular, the curvature (5.34) of the linear connection (5.21) readsRαµ(y) i = −R i jαµ(x)y j ,R i jαµ = ∂ α Γ i jµ − ∂ µ Γ i jα + Γ k jµΓ i kα − Γ k jαΓ i kµ.