Ivancevic_Applied-Diff-Geom

Introduction 15U i denotes one of the charts in the atlas for M. The extended atlas definesa topological manifold and the differentiablity of the transition maps definea differentiable structure on the tangent bundle manifold.The tangent bundle is where tangent vectors live, and is itself a smoothmanifold. The so–called Lagrangian is a natural energy function on thetangent bundle.Associated with every point x on a smooth manifold M is a tangentspace T x M and its dual, the cotangent space Tx ∗ M. The former consists ofthe possible directional derivatives, and the latter of the differentials, whichcan be thought of as infinitesimal elements of the manifold. These spacesalways have the same dimension n as the manifold does. The collection ofall tangent spaces can in turn be made into a manifold, the tangent bundle,whose dimension is 2n.1.1.4.2 Cotangent Bundle of a Smooth ManifoldRecall that the dual of a vector space is the set of linear functionals (i.e.,real valued linear functions) on the vector space. In particular, if the vectorspace is finite and has an inner product then the linear functionals can berealized by the functions f v (w) = 〈v, w〉.The cotangent bundle T ∗ M is the dual to the tangent bundle T M in thesense that each tangent space has a dual cotangent space as a vector space.The cotangent bundle T ∗ M is a smooth manifold itself, whose dimensionis 2n. The so–called Hamiltonian is is a natural energy function on thecotangent bundle. The total space of a cotangent bundle naturally has thestructure of a symplectic manifold (see below).1.1.4.3 Fibre–, Tensor–, and Jet–BundlesA fibre bundle is a space which locally looks like a product of two spaces butmay possess a different global structure. Tangent and cotangent bundlesare special cases of a fibre bundle. 14A tensor bundle is a direct sum of all tensor products of the tangentbundle and the cotangent bundle. 15 To do calculus on the tensor bundle14 Every fiber bundle consists of a continuous surjective map: π : E −→ B, where smallregions in the total space E look like small regions in the product space B × F. Here Bis called the base space while F is the fiber space. For example, the product space B × F ,equipped with π equal to projection onto the first coordinate, is a fiber bundle. This iscalled the trivial bundle. One goal of the theory of bundles is to quantify, via algebraicinvariants, what it means for a bundle to be non–trivial.15 Recall that a tensor is a certain kind of geometrical entity which generalizes the