Fundamental Vector Fields

Abstract

This chapter addresses fundamental vector fields. The concept of a connection on a principal bundle is essential in the construction of the Cartan model. To define a connection on a principal bundle, one first needs to define the fundamental vector fields. When a Lie group acts smoothly on a manifold, every element of the Lie algebra of the Lie group generates a vector field on the manifold called a fundamental vector field. On a principal bundle, the fundamental vectors are precisely the vertical tangent vectors. In general, there is a relation between zeros of fundamental vector fields and fixed points of the group action. Unless specified otherwise (such as on a principal bundle), a group action is assumed to be a left action.