Ivancevic_Applied-Diff-Geom

266 Applied Differential Geometry: A Modern Introductionsatisfying such equations [Blackmore et. al. (1998)]:dα i = a i k (α) ∧ αk ,dβ j = f j k αk + ω j s ∧ β s ,(3.97)where a i k (α) ∈ Λ1 (M), f j k ∈ Λ0 (M × Y ) and ω j s ∈ Λ 1 (M × Y ) for alli, k = 1, ..., m α , j, s = 1, ..., n. Since the identity d 2 β j ≡ 0 takes place forall j = 1, ..., n, from (3.97) we deduce the following relationship:(dω j k + ωj s ∧ ω s k)∧ β k +(df j s + ω j k f k s + f j l al s(α)As a result of (3.98) we get [Blackmore et. al. (1998)])∧ α s ≡ 0. (3.98)dω j k + ωj s ∧ ω s k∈ I(α, β),df j s + ω j k f k s + f j l al s(α) ∈ I(α, β)(3.99)for all j, k = 1, ..., n, s = 1, ..., m α . The second inclusion in (3.99) gives apossibility to define the 1–forms θ j s = f j l al s(α) satisfying the inclusiondθ j s + ω j k ∧ θk s ∈ I(α, β) ⊕ f j l cl s(α), (3.100)which we obtained having used the identities d 2 α j ≡ 0, j = 1, ..., m α , inthe form c j s(α) ∧ α s ≡ 0,c j s(α) = da j s(α) + a j l (α) ∧ al s(α), (3.101)following from (3.97). Let us suppose further that as s = s 0 the 2–formsc j s 0(α) ≡ 0 for all j = 1, ..., m α . Then as s = s 0 , we can define a set of1–forms θ j = θ j s 0∈ Λ 1 (M × Y ), j = 1, ..., n, satisfying the exact inclusions:dθ j + ω j k ∧ θk = Θ j ∈ I(α, β) (3.102)together with a set of inclusions for 1–forms ω j k ∈ Λ1 (M × Y )dω j k + ωj s ∧ ω s k = Ω j k∈ I(α, β) (3.103)As it follows from the general theory [Sulanke and Wintgen (1972)] of connectionson the fibred frame space P (M; GL(n)) over a base manifold M,we can interpret the equations (3.103) as the equations defining the curvature2–forms Ω j k ∈ Λ2 (P ), as well as interpret the equations (3.102) asthose, defining the torsion 2–forms Θ j ∈ Λ 2 (P ). Since I(α) = 0 = I(α, β)upon the integral submanifold ¯M ⊂ M, the reduced fibred frame spaceP ( ¯M; GL(n)) will have the flat curvature and be torsion free, being as a