ABSTRACT
In this chapter, we are interested in making use of a selected number of results from differential geometry, which are necessary for the development of quantum methods for condensed molecular systems subjected to holonomic constraints. The objective is to introduce the concepts of covariant and contravariant vector (and tensor) spaces, the Einstein sum convention, the metric tensor, integration, differentiation, and classical dynamics in manifolds. In all the cases treated in this book, a manifold is simply a Euclidean space with n dimensions, or a subspace obtained generally by remapping the original n-dimensional Euclidean space with curvilinear coordinates, and then constraining one or more of these. We denote with the symbol Rn, a Euclidean space with n dimensions when it is mapped with orthogonal Cartesian coordinates. Another possible source of manifolds is the use of a Lie group or semigroup of continuous transformations, introduced in Chapter 4. The Lie group approach is used for the treatment of rigid bodies composed of n point particles. These types of manifolds are most likely of intense interest to the molecular theorist. A typical example of a manifold of this type is the rigid rotor space, where the R6 space is remapped by transforming into center of mass coordinates (which we can remove from the dynamics), using spherical polar coordinates for the pseudo-one body problem that is left, and fixing the distance r to a constant r0. The resulting manifold is the set of all points on the surface of a sphere of radius r0, also known as a two-sphere and denoted with the symbol S2.
