Ivancevic_Applied-Diff-Geom

Applied Bundle Geometry 553FoliationsRecall that related to integrable distributions are foliations.The Frobenius Theorem asserts that integrability and involutivity areequivalent, at least locally. Thus, associated with an involutive distributionis a partition Φ of M into disjoint connected immersed submanifolds calledleaves. This partition Φ is called a foliation. More precisely, a foliation Fof a smooth manifold M is a collection of disjoint immersed submanifoldsof M whose disjoint union equals M. Each connected submanifold of Fis called a leaf of the foliation. Given an integrable distribution ∆, thecollection of maximal integral manifolds for ∆ defines a foliation on M,denoted by F D .A foliation F of M defines an equivalence relation on M whereby twopoints in M are equivalent if they lie in the same leaf of F. The set ofequivalence classes is denoted M/F and is called the leaf space of F. Afoliation F is said to be simple if M/F inherits a manifold structure so thatthe projection from M to M/F is a surjective submersion.In control theory, foliation leaves are related to the set of points that acontrol system can reach starting from a given initial condition. A foliationΦ of M defines an equivalence relation on M whereby two points in M areequivalent if they lie in the same leaf of Φ. The set of equivalence classesis denoted M/Φ and is called the leaf space of Φ.Philip Hall BasisasGiven a set of vector–fields {g 1 ...g m }, define the length of a Lie productl(g i ) = 1, l([A, B]) = l(A) + l(B), (for i = 1, ..., m),where A and B may be Lie products. A Philip Hall basis is an ordered setof Lie products H = {B i } satisfying:(1) g i ∈ H, (i = 1, ..., m);(2) If l(B i ) < l(B j ), then B i < B j ; and(3) [B i , B j ] ∈ H iff(a) B i , B j ∈ H and B i < B j , and(b) either B j = g k for some k or B j = [B l , B r ] with B l , B r ∈ Hand B l ≤ B i .Essentially, the ordering aspect of the Philip Hall basis vectors accountsfor skew symmetry and Jacobi identity to determine a basis.