Chapter 1 presents theorems on differentiable functions often used in differential topology, such as the implicit function theorem, Sard's theorem and Whitney's approximation theorem.
The next chapter is an introduction to real and complex manifolds. It contains an exposition of the theorem of Frobenius, the lemmata of Poincaré and Grothendieck with applications of Grothendieck's lemma to complex analysis, the imbedding theorem of Whitney and Thom's transversality theorem.
Chapter 3 includes characterizations of linear differentiable operators, due to Peetre and Hormander. The inequalities of Garding and of Friedrichs on elliptic operators are proved and are used to prove the regularity of weak solutions of elliptic equations. The chapter ends with the approximation theorem of Malgrange-Lax and its application to the proof of the Runge theorem on open Riemann surfaces due to Behnke and Stein.
Analysis on Real and Complex Manifolds, 2nd Edition
1. Differentiable Functions in
. Taylor's Formula. Partitions of Unity. Inverse Functions, Implicit Functions and the Rank Theorem. Sard's Theorem and Functional Dependence. Borel's Theorem on Taylor Series. Whitney's Approximation Theorem. An Approximation Theorem for Holomorphic Functions. Ordinary Differential Equations. 2. Manifolds.
Basic Definitions. The Tangent and Cotangent Bundles. Grassmann Manifolds. Vector Fields and Differential Forms. Submanifolds. Exterior Differentiation. Orientation. Manifolds with Boundary. Integration. One Parameter Groups. The Frobenius Theorem. Almost Complex Manifolds. The Lemmata of Poincaré and Grothendieck. Applications: Hartog's Continuation Theorem and the Oka-Weil Theorem. Immersions and Imbeddings: Whitney's Theorems. Thom's Transversality Theorem. 3. Linear Elliptic Differential Operators.
Vector Bundles. Fourier Transforms. Linear Differential Operators. The Sobolev Spaces. The Lemmata of Rellich and Sobolev. The Inequalities of Garding and Friedrichs. Elliptic Operators with C8
Coefficients: The Regularity Theorem. Elliptic Operators with Analytic Coefficients. The Finiteness Theorem. The Approximation Theorem and Its Application to Open Riemann Surfaces