For a transitive Lie algebroid A on a connected manifold M and its a representation on a vector bundle F, we study the localization map Y^1: H^1(A,F)-> H^1(L_x,F_x), where L_x is the adjoint algebra at x in M. The main result in this paper is that: Ker Y^1_x=Ker(p^{ 1*})=H^1_{ deR}(M,F_0). Here p^{ 1*} is the lift of H^1(\huaA,F) to its counterpart over the universal covering space of M and H^1_{ deR}(M,F_0) is the F_0=H^0(L,F)-coefficient deRham cohomology. We apply these results to study the associated vector bundles to principal fiber bundles and the structure of transitive Lie bialgebroids.
Categories: math.DG math.SG