ON THE WEDGE PRODUCT

Abstract

This chapter focuses on the wedge product. Tensor products in cotangent space can be defined by the values of the functionals they yield at tensors in rank two tangent spaces. Exterior differential forms on a space constitute the skew symmetric subspace of tensor cotangent space. A mapping from one space to another induces a linear mapping of tangent space from which the adjoint mapping of linear cotangent space can be obtained. The adjoint mapping of the wedge product is the wedge product of the adjoint mappings. Invariant measures on the right coset spaces, such as the Stiefel and Grassmann manifolds, are obtained by omitting the factors in the wedge product associated with the respective isotropy groups. The invariant measure on the sphere or coset space is obtained from the invariant measure on the group by omitting the third differential form from the wedge product.