Manifolds and Exterior Differential Forms

Abstract

Our discussion of differential forms is spread over the next four chapters. The fundamentals of the algebraic theory and theory of integration of differential forms form the substance of Chapter 6. This material is a necessary expansion of the introductory material in James (1954). Chapter 7 is concerned with explicit derivation of differential forms for invariant measures on various manifolds. Although somewhat premature, in case the manifold is a locally compact metric matrix group, the basic idea has already been discussed in Chapter 3, Section 3.6. We discuss several manifolds, the Stiefel and Grassman manifolds, which are not groups, in Chapter 7. These examples require some long detailed calculations and this material has been segregated into a separate chapter. The fact that differential forms are so useful in the derivation of density functions is due to the fact that the manifold of n × h matrices naturally decomposes into a topological product of manifolds on which groups of transformations act. The theory of Sections 3.3 and 3.4 suggests that the invariant measures should factor. Since Lebesgue measure is absolutely continuous relative to the invariant measures, Lebesgue measure also factors. Chapter 8 discusses the well-known methods of decomposing a n × h matrix. Chapter 9 factors Lebesgue measure over several of these products of manifolds thereby obtaining density functions of several multivariate statistics.