Ivancevic_Applied-Diff-Geom

552 Applied Differential Geometry: A Modern Introductionas the rank of ∆ at x,.containing x such that the tangent bundle, T N, isexactly ∆ restricted to N, i.e., T N = ∆| N . Such a submanifold is calledthe maximal integral manifold through x.It is natural to consider distributions generated by the vector–fieldsappearing in the sequence of flows (3.48). In this case, consider the distributiondefined by∆ = span{f; g 1 ...g m },where the span is taken over the set of smooth real–valued functions. Denoteby ¯∆ the involutive closure of the distribution ∆, which is the closureof ∆ under bracketing. Then, ¯∆ is the smallest subalgebra of X k (M) whichcontains {f; g 1 ...g m }. We will often need to ‘add’ distributions. Since distributionsare, pointwise, vector spaces, define the sum of two distributions,Similarly, define the intersection(∆ 1 + ∆ 2 )(x) = ∆ 1 (x) + ∆ 2 (x).(∆ 1 ∩ ∆ 2 )(x) = ∆ 1 (x) ∩ ∆ 2 (x).More generally, we can arrive at a distribution via a family of vector–fields, which is a subset V ⊂ X k (M). Given a family of vector–fields V, wemay define a distribution on M by∆ V (x) = 〈X(x)|X ∈ V〉 R .Since X k (M) is a Lie algebra, we may ask for the smallest Lie subalgebraof X k (M) which contains a family of vector–fields V. It will be denoted asLie(V), and will be represented by the set of vector–fields on M generatedby repeated Lie brackets of elements in V. Let V (0) = V and then iterativelydefine a sequence of families of vector–fields byV (i+1) = V (i) ∪ {[X, Y ]|X ∈ V (0) = V and Y ∈ V (i) }.Now, every element of Lie(V) is a linear combination of repeated Lie bracketsof the formwhere Z i ∈ V for i = 1, ..., k.[Z k , [Z k−1 , [· · ·, [Z 2 , Z 1 ] · ··]]]