Chapter VIII. Analysis on Manifolds

Abstract

This chapter explains how the theory of pseudodifferential operators extends from open subsets of Euclidean space to smooth manifolds, and it gives examples to illustrate the usefulness of generalizing the theory in this way. Section 1 gives a brief introduction to differential calculus on smooth manifolds. The section defines smooth manifolds, smooth functions on them, tangent spaces to smooth manifolds, and differentials of smooth mappings between smooth manifolds, and it proves a version of the Inverse Function Theorem for manifolds. Section 2 extends the theory of smooth vector fields and integral curves from open subsets of Euclidean space to smooth manifolds. Section 3 develops a special kind of quotient space, called an “identification space,” suitable for constructing general smooth manifolds, vector bundles and fiber bundles, and covering spaces out of local data. In particular, smooth manifolds may be defined as identification spaces without knowledge of the global nature of the underlying topological space; the only problem is in addressing the Hausdorff property. Section 4 introduces vector bundles, including the tangent and cotangent bundles to a manifold. A vector bundle determines transition functions, and in turn the transition functions determine the vector bundle via the construction of the previous section. The manifold structures on the tangent and cotangent bundles are constructed in this way. Sections 5–8 concern pseudodifferential operators, including aspects of the theory useful in solving problems in other areas of mathematics. The emphasis is on operators on scalar-valued functions. Section 5 introduces spaces of smooth functions and their topologies, and it defines spaces of distributions; the theory has to compensate for the lack of a canonical underlying measure on the manifold, hence for the lack of a canonical way to view a smooth function as a distribution. Section 5 goes on to study linear partial differential equations on the manifold; although the symbol of the differential operator is not meaningful, the principal symbol is intrinsically defined as a function on the cotangent bundle. The introduction of pseudodifferential operators on smooth manifolds requires new results for the theory in Euclidean space beyond what is in Chapter VII. Section 6 addresses this matter. A notion of transpose is needed, and it is necessary to understand the effect of diffeomorphisms on Euclidean pseudodifferential operators. To handle these questions, it is useful to enlarge the definition of pseudodifferential operator for Euclidean space and to redo the Euclidean theory from the new point of view. Once that program has been carried out, Section 7 patches together pseudodifferential operators in Euclidean space to obtain pseudodifferential operators on smooth separable manifolds. The notions of pseudolocal, properly supported, composition, and elliptic extend, and the theorems are what one might expect from the Euclidean theory. Again the principal symbol is well defined as a function on the cotangent bundle. Section 8 contains remarks about extending the theory to handle operators carrying sections of one vector bundle to sections of another vector bundle, about some other continuations of the theory, and about applications outside real analysis. The section concludes with some bibliographical material.