Analysis on Real and Complex Manifolds

by Narasimhan

• ISBN13: 9780444877765
• ISBN10: 0444877762
• Edition: 2nd
• Format: Hardcover
• Copyright: 1985-12-01
• Publisher: Elsevier Science

Summary

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 Poincareacute; 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.

Table of Contents

Chapters

Differentiable Functions in R n

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

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 Poincare and Grothendieck

Applications: Hartog's Continuation Theorem and the Oka-Weil Theorem

Immersions and Imbeddings: Whitney's Theorems

Thom's Transversality Theorem

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 C infin; Coefficients: The Regularity Theorem

Elliptic Operators with Analytic Coefficients

The Finiteness Theorem

The Approximation Theorem and Its Application to Open Riemann Surfaces

References

Subject Index

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