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Introduction to the $h$-Principle
 
Y. Eliashberg Stanford University, Stanford, CA
N. Mishachev Lipetsk Technical University, Lipetsk, Russia
Introduction to the h-Principle
Now available in new edition: GSM/239
eBook ISBN:  978-1-4704-1796-3
Product Code:  GSM/48.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Introduction to the h-Principle
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Introduction to the $h$-Principle
Y. Eliashberg Stanford University, Stanford, CA
N. Mishachev Lipetsk Technical University, Lipetsk, Russia
Now available in new edition: GSM/239
eBook ISBN:  978-1-4704-1796-3
Product Code:  GSM/48.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 482002; 206 pp
    MSC: Primary 58

    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 homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the \(h\)-principle. Two famous examples of the \(h\)-principle, the Nash-Kuiper \(C^1\)-isometric embedding theory in Riemannian geometry and the Smale-Hirsch 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.

    Readership

    Graduate 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. One-dimensional 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. Nash-Kuiper 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 482002; 206 pp
MSC: Primary 58

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 homotopy-theoretic nature. One says in this case that the corresponding differential relation satisfies the \(h\)-principle. Two famous examples of the \(h\)-principle, the Nash-Kuiper \(C^1\)-isometric embedding theory in Riemannian geometry and the Smale-Hirsch 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.

Readership

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. One-dimensional 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. Nash-Kuiper 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
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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