Ivancevic_Applied-Diff-Geom

534 Applied Differential Geometry: A Modern Introductionan nD manifold M is a partition of M into connected leaves F ι with thefollowing property: every point of M has an open neighborhood U which isa domain of a coordinate chart (x α ) such that, for every leaf F ι , the componentsF ι ∩ U are described by the equations x n−k+1 = const,..., x n = const[Kamber and Tondeur (1975)]. Note that leaves of a foliation fail to beimbedded submanifolds in general.For example, every projection π : M → X defines a foliation whoseleaves are the fibres π −1 (x), for all x ∈ X. Also, every nowhere vanishingvector–field u on a manifold M defines a 1D involutive distribution onM. Its integral manifolds are the integral curves of u. Around each pointx ∈ M, there exist local coordinates (x 1 , . . . , x n ) of a neighborhood of xsuch that u is given by u =∂∂x i .4.8 Application: Nonholonomic MechanicsLet T M = ∪ x∈M T x M, be the tangent bundle of a smooth nD mechanicalmanifold M. Recall (from the subsection 4.7 above) that sub–bundle V =∪ x∈M V x , where V x is a vector subspace of T x M, smoothly dependent onpoints x ∈ M, is called the distribution. If the manifold M is connected,dim V x is called the dimension of the distribution. A vector–field X onM belongs to the distribution V if X(x) ⊂ V x . A curve γ is admissiblerelatively to V , if the vector–field ˙γ belongs to V . A differential systemis a linear space of vector–fields having a structure of C ∞ (M) – module.Vector–fields which belong to the distribution V form a differential systemN(V ). A kD distribution V is integrable if the manifold M is foliated to kDsub–manifolds, having V x as the tangent space at the point x. Accordingto the Frobenius Theorem, V is integrable iff the corresponding differentialsystem N(V ) is involutive, i.e., if it is a Lie sub–algebra of the Lie algebraof vector–fields on M. The flag of a differential system N is a sequence ofdifferential systems: N 0 = N, N 1 = [N, N], . . . , N l = [N l−1 , N], . . . .The differential systems N i are not always differential systems of somedistributions V i , but if for every i, there exists V i , such that N i = N(V i ),then there exists a flag of the distribution V : V = V 0 ⊂ V 1 . . . . Suchdistributions, which have flags, will be called regular. Note that the sequenceN(V i ) is going to stabilize, and there exists a number r such thatN(V r−1 ) ⊂ N(V r ) = N(V r+1 ). If there exists a number r such thatV r = T M, the distribution V is called completely nonholonomic, and minimalsuch r is the degree of non–holonomicity of the distribution V .