
Book DetailsGraduate Studies in MathematicsVolume: 48; 2002; 206 ppMSC: Primary 58
Now Available in Second Edition: GSM/239 In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the fifties that the solvability of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopytheoretic nature. One says in this case that the corresponding differential relation satisfies the \(h\)principle. Two famous examples of the \(h\)principle, the NashKuiper \(C^1\)isometric embedding theory in Riemannian geometry and the SmaleHirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the \(h\)principle.
The authors cover two main methods for proving the \(h\)principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the \(h\)principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry.
Gromov's famous book “Partial Differential Relations”, which is devoted to the same subject, is an encyclopedia of the \(h\)principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists and analysts will also find much value in this very readable exposition of an important and remarkable topic.
ReadershipGraduate students and research mathematicians interested in global analysis and analysis on manifolds.

Table of Contents

Chapters

Intrigue

Part 1. Holonomic approximation

Chapter 1. Jets and holonomy

Chapter 2. Thom transversality theorem

Chapter 3. Holonomic approximation

Chapter 4. Applications

Part 2. Differential relations and Gromov’s $h$principle

Chapter 5. Differential relations

Chapter 6. Homotopy principle

Chapter 7. Open Diff $V$invariant differential relations

Chapter 8. Applications to closed manifolds

Part 3. The homotopy principle in symplectic geometry

Chapter 9. Symplectic and contact basics

Chapter 10. Symplectic and contact structures on open manifolds

Chapter 11. Symplectic and contact structures on closed manifolds

Chapter 12. Embeddings into symplectic and contact manifolds

Chapter 13. Microflexibility and holonomic $\mathcal {R}$approximation

Chapter 14. First applications of microflexibility

Chapter 15. Microflexible $\mathfrak {U}$invariant differential relations

Chapter 16. Further applications to symplectic geometry

Part 4. Convex integration

Chapter 17. Onedimensional convex integration

Chapter 18. Homotopy principle for ample differential relations

Chapter 19. Directed immersions and embeddings

Chapter 20. First order linear differential operators

Chapter 21. NashKuiper theorem


Reviews

The reveiwed book is the first broadly accessible exposition of the theory written for mathematicians who are interested in an introduction into the \(h\)principle and its applications ... very readable, many motivations, examples and exercises are included ... a very good text for graduate courses on geometric methods for solving partial differential equations and inequalities.
Zentralblatt MATH 
In my opinion, this is an excellent book which makes an important theory accessible to graduate students in differential geometry.
Jahresbericht der DMV

 Book Details
 Table of Contents
 Reviews
In differential geometry and topology one often deals with systems of partial differential equations, as well as partial differential inequalities, that have infinitely many solutions whatever boundary conditions are imposed. It was discovered in the fifties that the solvability of differential relations (i.e. equations and inequalities) of this kind can often be reduced to a problem of a purely homotopytheoretic nature. One says in this case that the corresponding differential relation satisfies the \(h\)principle. Two famous examples of the \(h\)principle, the NashKuiper \(C^1\)isometric embedding theory in Riemannian geometry and the SmaleHirsch immersion theory in differential topology, were later transformed by Gromov into powerful general methods for establishing the \(h\)principle.
The authors cover two main methods for proving the \(h\)principle: holonomic approximation and convex integration. The reader will find that, with a few notable exceptions, most instances of the \(h\)principle can be treated by the methods considered here. A special emphasis in the book is made on applications to symplectic and contact geometry.
Gromov's famous book “Partial Differential Relations”, which is devoted to the same subject, is an encyclopedia of the \(h\)principle, written for experts, while the present book is the first broadly accessible exposition of the theory and its applications. The book would be an excellent text for a graduate course on geometric methods for solving partial differential equations and inequalities. Geometers, topologists and analysts will also find much value in this very readable exposition of an important and remarkable topic.
Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Chapters

Intrigue

Part 1. Holonomic approximation

Chapter 1. Jets and holonomy

Chapter 2. Thom transversality theorem

Chapter 3. Holonomic approximation

Chapter 4. Applications

Part 2. Differential relations and Gromov’s $h$principle

Chapter 5. Differential relations

Chapter 6. Homotopy principle

Chapter 7. Open Diff $V$invariant differential relations

Chapter 8. Applications to closed manifolds

Part 3. The homotopy principle in symplectic geometry

Chapter 9. Symplectic and contact basics

Chapter 10. Symplectic and contact structures on open manifolds

Chapter 11. Symplectic and contact structures on closed manifolds

Chapter 12. Embeddings into symplectic and contact manifolds

Chapter 13. Microflexibility and holonomic $\mathcal {R}$approximation

Chapter 14. First applications of microflexibility

Chapter 15. Microflexible $\mathfrak {U}$invariant differential relations

Chapter 16. Further applications to symplectic geometry

Part 4. Convex integration

Chapter 17. Onedimensional convex integration

Chapter 18. Homotopy principle for ample differential relations

Chapter 19. Directed immersions and embeddings

Chapter 20. First order linear differential operators

Chapter 21. NashKuiper theorem

The reveiwed book is the first broadly accessible exposition of the theory written for mathematicians who are interested in an introduction into the \(h\)principle and its applications ... very readable, many motivations, examples and exercises are included ... a very good text for graduate courses on geometric methods for solving partial differential equations and inequalities.
Zentralblatt MATH 
In my opinion, this is an excellent book which makes an important theory accessible to graduate students in differential geometry.
Jahresbericht der DMV