Ivancevic_Applied-Diff-Geom

270 Applied Differential Geometry: A Modern IntroductionThis means thatg(h) 0 = span R {A 1 , A 3 = [A 0 , A 1 ]}. (3.120)To satisfy the set of relations (3.118) we need to use expansions over thebasis (3.120) of the external elements A 0 , A 2 ∈ g(h):A 0 = q 01 A 1 + q 13 A 3 , A 2 = q 21 A 1 + q 23 A 3 . (3.121)Substituting expansions (3.121) into (3.118), we get that q 01 = q 23 =λ, q 21 = −λ 2 /2 and q 03 = −2 for some arbitrary real parameter λ ∈ R,that is g(h) = span R {A 1 , A 3 }, where[A 1 , A 3 ] = A 3 /2; A 0 = λA 1 −2A 3 , A 2 = −λ 2 A 1 /2+λA 3 . (3.122)As a result of (3.122) we can state that the holonomy Lie algebra g(h)is a real 2D one, assuming the following (2 × 2)−matrix representation[Blackmore et. al. (1998)]:( )( )1/4 00 1A 1 =, A 3 = ,0 −1/40 0( )( λ/4 −2−λ 2 )/8 λA 0 =, A 2 =0 −λ/40 λ 2 ./8(3.123)Thereby from (3.115), (3.117) and (3.123) we get the reduced curvature1–form Γ ∈ Λ 1 (M) ⊗ g,Γ = (A 0 + uA 1 )dx + ((u x + u 2 /2)A 1 − uA 3 + A 2 )dt, (3.124)generating parallel transport of vectors from the representation space Y ofthe holonomy Lie algebra g(h):dy + Γy = 0, (3.125)upon the integral submanifold ¯M ⊂ M 4 of the ideal I(α), generated bythe set of 2–forms (3.113). The result (3.125) means also that the Burgersdynamical system (3.112) is endowed with the standard Lax type representation,having the spectral parameter λ ∈ R necessary for its integrabilityin quadratures.