Part III: Applications of Differential Geometry to Physics

In addition to the exterior derivative d, there is another type of derivative one may define on a manifold possessing no further structure, it is called the Lie Derivative. To define it we need to a smooth map ψ : M → N from a smooth manifold M to another smooth manifold N , which need not have the same dimensions, m and n respectively. Of course an interesting special case will correspond to taking M = N but for clarity and also with other applications in mind, we keep M and N distinct. By smooth map, we mean one given by smooth functions in every smooth coordinate chart. Thus if x are local coordinates for M and y for N , then ψ takes a point p to p = ψ(p) and if p has coordinates x, α = 1, 2, . . .m and p has coordinates y ,β = 1, 2, . . . n, then y = y(x). Associated with ψ are two maps