Ivancevic_Applied-Diff-Geom

150 Applied Differential Geometry: A Modern Introductionorously proven, and in particular there is at present no generally acceptedconstruction of a complete set. A very interesting and intimately relatedproblem is the famous Poincaré conjecture, stating that if a 3D manifoldhas a certain set of topological invariants called its ‘homotopy groups’ equalto those of the 3–sphere S 3 , it is actually homeomorphic to the three-sphere.In four or more dimensions, a complete set of topological invariants wouldconsist of an uncountably infinite number of invariants! A general classificationof topologies is therefore very hard to get, but even without such ageneral classification, each new invariant that can be constructed gives usa lot of interesting new information. For this reason, the construction oftopological invariants of manifolds is one of the most important issues intopology.3.5 (Co)Tangent Bundles of Smooth Manifolds3.5.1 Tangent Bundle and Lagrangian Dynamics3.5.1.1 Intuition Behind a Tangent BundleIn mechanics, to each nD configuration manifold M there is associated its2nD velocity phase–space manifold, denoted by T M and called the tangentbundle of M (see Figure 3.3). The original smooth manifold M is calledthe base of T M. There is an onto map π : T M −→ M, called the projection.Above each point x ∈ M there is a tangent space T x M = π −1 (x) to M at x,which is called a fibre. The fibre T x M ⊂ T M is the subset of T M, such thatthe total tangent bundle, T M =⊔T x M, is a disjoint union of tangentm∈Mspaces T x M to M for all points x ∈ M. From dynamical perspective,the most important quantity in the tangent bundle concept is the smoothmap v : M −→ T M, which is an inverse to the projection π, i.e, π ◦ v =Id M , π(v(x)) = x. It is called the velocity vector–field. Its graph (x, v(x))represents the cross–section of the tangent bundle T M. This explains thedynamical term velocity phase–space, given to the tangent bundle T M ofthe manifold M.3.5.1.2 Definition of a Tangent BundleRecall that if [a, b] is a closed interval, a C 0 −map γ : [a, b] → M is saidto be differentiable at the endpoint a if there is a chart (U, φ) at γ(a) such