Calculus on Manifolds

Abstract

The equations of mathematical physics are typically ordinary or partial differential equations for vector or tensor fields over Riemannian manifolds whose group of isometries is a Lie group. It is taken as axiomatic that the equations be independent of the observer, in a sense we shall make precise below; and the consequence of this axiom is that the equations are invariant with respect to the group action. The action of a Lie group on tensor fields over a manifold is thus of primary importance. The action of a Lie group on a manifold M induces in a natural way automorphisms of the algebra of C∞ functions over M and on the algebra of tensor fields over M. The one parameter subgroups of the group induce one parameter subgroups of automorphisms of the tensor fields. The infinitesimal generators of these groups of automorphisms are the Lie derivatives of the action.