Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 269where M 4 is a 4–D submanifold in J 1 (R 2 ; R)) with coordinates (x, t, u (0) =u, u (1) = u x ). The set of 2–forms (3.113) generates the closed ideal I(α),sincedα 1 = dx ∧ α 2 − u (0) dx ∧ α 1 , dα 2 = 0, (3.114)the integral submanifold ¯M = {x, t ∈ R} ⊂ M 4 being defined by thecondition I(α) = 0. We now look for a reduced ‘curvature’ 1–form Γ ∈Λ 1 (M 4 ) ⊗ g, belonging to some (not yet determined) Lie algebra g. This1–form can be represented using (3.113), as follows:Γ = b (x) (u (0) , u (1) )dx + b (t) (u (0) , u (1) )dt, (3.115)where elements b (x) , b (t) ∈ g satisfy [Blackmore et. al. (1998)]∂b (x)∂b= g∂u (0) 2 , (x)∂b= 0, (t)= g∂u (1) ∂u (0) 1 + g 2 u (0) ,∂b (t)= g∂u (1) 2 , [b (x) , b (t) ] = −u (1) g 1 .The set (3.116) has the following unique solution(3.116)b (x) = A 0 +A 1 u (0) ,b (t) = u (1) A 1 + u(0)22 A 1+[A 1 , A 0 ]u (0) +A 2 , (3.117)where A j ∈ g, j = 0, 2, are some constant elements on M of a Lie algebrag under search, satisfying the next Lie structure equations:[A 0 , A 2 ] = 0,[A 0 , [A 1 , A 0 ]] + [A 1 , A 2 ] = 0,[A 1 , [A 1 , A 0 ]] + 1 2 [A 0, A 1 ] = 0.(3.118)From (3.116) one can see that the curvature2–form Ω ∈ span R {A 1 , [A 0 , A 1 ] : A j ∈ g, j = 0, 1}. Therefore, reducingvia the Ambrose–Singer Theorem the associated principal fibered framespace P (M; G = GL(n)) to the principal fibre bundle P (M; G(h)), whereG(h) ⊂ G is the corresponding holonomy Lie group of the connection Γ onP , we need to satisfy the following conditions for the set g(h) ⊂ g to be aLie subalgebra in g : ∇ m x ∇ n t Ω ∈ g(h) for all m, n ∈ Z + .Let us try now to close the above procedure requiring that [Blackmoreet. al. (1998)]g(h) = g(h) 0 = span R {∇ m x ∇ n xΩ ∈ g : m + n = 0} (3.119)