Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 533is the right Lie algebra g r of the right–invariant vector–fields on the groupG. The group G acts on this standard fibre by the adjoint representation.4.7 Distributions and Foliations on ManifoldsLet M be an nD smooth manifold. A smooth distribution T of codimensionk on M is defined as a subbundle of rank n − k of the tangent bundle T M.A smooth distribution T is called the involutive distribution if [u, u ′ ] is asection of T whenever u and u ′ are sections of T .Let T be a k−codimensional distribution on M. Its annihilator T ∗ isa kD subbundle of T ∗ M called the Pfaffian system. It means that, on aneighborhood U of every point x ∈ M, there exist k linearly independentsections s 1 , . . . , s k of T ∗ such thatT x | U = ∩ j Ker s j .Let C(T ) be the ideal of ∧(M) generated by sections of T ∗ .A smooth distribution T is involutive iff the ideal C(T ) is differential,that is, dC(T ) ⊂ C(T ).Given an involutive k−codimensional distribution T on M, the quotientT M/T is a kD vector bundle called the transversal bundle of T . There isthe exact sequence0 → T ↩→ T M → T M/T → 0. (4.39)Given a bundle Y → X, its vertical tangent bundle V Y exemplifies aninvolutive distribution on Y .A submanifold N of M is called the integral manifold of a distribution Ton M if the tangent spaces to N coincide with the fibres of this distributionat each point of N.Let T be a smooth involutive distribution on M. For any point x ∈ M,there exists a maximal integral manifold of T passing through x [Kamberand Tondeur (1975)]. In view of this fact, involutive distributions are alsocalled completely integrable distributions.Every point x ∈ M has an open neighborhood U which is a domain ofa coordinate chart (x 1 , . . . , x n ) such that the restrictions of T and T ∗ to U∂∂x n−k∂are generated by the n − k vector–fields∂x, . . . , and the k Pfaffian1forms dx n−k+1 , . . . , dx n respectively.In particular, it follows that integral manifolds of an involutive distributionconstitute a foliation. Recall that a k−codimensional foliation on