Integration of Equivariant Forms

This chapter illustrates integration of equivariant forms. An equivariant differential form is an element of the Cartan model. For a circle action on a manifold M, it is a polynomial in u with coefficients that are invariant forms on M. Such a form can be integrated by integrating the coefficients. This can be called equivariant integration. The chapter shows that under equivariant integration, Stokes's theorem still holds. Everything done so far in this book concerning a Lie group action on a manifold can be generalized to a manifold with boundary. An important fact concerning manifolds with boundary is that a diffeomorphism of a manifold with boundary takes interior points to interior points and boundary points to boundary points.