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Introduction to the $h$-Principle: Second Edition
 
K. Cieliebak University of Augsburg, Augsburg, Germany
Y. Eliashberg Stanford University, Stanford, CA
N. Mishachev Lipetsk Technical University, Lipetsk, Russia
Hardcover ISBN:  978-1-4704-6105-8
Product Code:  GSM/239
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7617-5
Product Code:  GSM/239.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7618-2
Product Code:  GSM/239.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7617-5
eBook: ISBN:  978-1-4704-7618-2
Product Code:  GSM/239.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
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Introduction to the $h$-Principle: Second Edition
K. Cieliebak University of Augsburg, Augsburg, Germany
Y. Eliashberg Stanford University, Stanford, CA
N. Mishachev Lipetsk Technical University, Lipetsk, Russia
Hardcover ISBN:  978-1-4704-6105-8
Product Code:  GSM/239
List Price: $135.00
MAA Member Price: $121.50
AMS Member Price: $108.00
Softcover ISBN:  978-1-4704-7617-5
Product Code:  GSM/239.S
List Price: $89.00
MAA Member Price: $80.10
AMS Member Price: $71.20
eBook ISBN:  978-1-4704-7618-2
Product Code:  GSM/239.E
List Price: $85.00
MAA Member Price: $76.50
AMS Member Price: $68.00
Softcover ISBN:  978-1-4704-7617-5
eBook ISBN:  978-1-4704-7618-2
Product Code:  GSM/239.S.B
List Price: $174.00 $131.50
MAA Member Price: $156.60 $118.35
AMS Member Price: $139.20 $105.20
  • Book Details
     
     
    Graduate Studies in Mathematics
    Volume: 2392024; 363 pp
    MSC: Primary 58

    This is a Revised Edition of: GSM/48

    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 1950s 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 is made on applications to symplectic and contact geometry.

    The present book is the first broadly accessible exposition of the theory and its applications, making it 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.

    This second edition of the book is significantly revised and expanded to almost twice of the original size. The most significant addition to the original book is the new part devoted to the method of wrinkling and its applications. Several other chapters (e.g., on multivalued holonomic approximation and foliations) are either added or completely rewritten.

    Readership

    Graduate students and researchers interested in recent advances in differential topology.

  • Table of Contents
     
     
    • Chapters
    • Intrigue
    • Holonomic approximation
    • Jets and holonomy
    • Thom transversality theorem
    • Holonomic approximation
    • Applications
    • Multivalued holonomic approximation
    • Differential relations and Gromov’s $h$-principle
    • Differential relations
    • Homotopy principle
    • Open Diff $V$-invariant differential relations
    • Applications to closed manifolds
    • Foliations
    • Singularities and wrinkling
    • Singularities of smooth maps
    • Wrinkles
    • Wrinkles submersions
    • Folded solutions to differential relations
    • The $h$-principle for sharp wrinkled embeddings
    • Igusa functions
    • The homotopy principle in symplectic geometry
    • Symplectic and contact basics
    • Symplectic and contact structures on open manifolds
    • Symplectic and contact structures on closed manifolds
    • Embeddings into symplectic and contact manifolds
    • Microflexibility and holonomic $\mathcal {R}$-approximation
    • First applications to microflexibility
    • Microflexible $\mathfrak {A}$-invariant differential relations
    • Further applications to symplectic geometry
    • Convex integration
    • One-dimensional convex integration
    • Homotopy principle for ample differential relations
    • Directed immersions and embeddings
    • First order linear differential operators
    • Nash-Kuiper theorem
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Accessibility – to request an alternate format of an AMS title
Volume: 2392024; 363 pp
MSC: Primary 58

This is a Revised Edition of: GSM/48

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 1950s 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 is made on applications to symplectic and contact geometry.

The present book is the first broadly accessible exposition of the theory and its applications, making it 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.

This second edition of the book is significantly revised and expanded to almost twice of the original size. The most significant addition to the original book is the new part devoted to the method of wrinkling and its applications. Several other chapters (e.g., on multivalued holonomic approximation and foliations) are either added or completely rewritten.

Readership

Graduate students and researchers interested in recent advances in differential topology.

  • Chapters
  • Intrigue
  • Holonomic approximation
  • Jets and holonomy
  • Thom transversality theorem
  • Holonomic approximation
  • Applications
  • Multivalued holonomic approximation
  • Differential relations and Gromov’s $h$-principle
  • Differential relations
  • Homotopy principle
  • Open Diff $V$-invariant differential relations
  • Applications to closed manifolds
  • Foliations
  • Singularities and wrinkling
  • Singularities of smooth maps
  • Wrinkles
  • Wrinkles submersions
  • Folded solutions to differential relations
  • The $h$-principle for sharp wrinkled embeddings
  • Igusa functions
  • The homotopy principle in symplectic geometry
  • Symplectic and contact basics
  • Symplectic and contact structures on open manifolds
  • Symplectic and contact structures on closed manifolds
  • Embeddings into symplectic and contact manifolds
  • Microflexibility and holonomic $\mathcal {R}$-approximation
  • First applications to microflexibility
  • Microflexible $\mathfrak {A}$-invariant differential relations
  • Further applications to symplectic geometry
  • Convex integration
  • One-dimensional convex integration
  • Homotopy principle for ample differential relations
  • Directed immersions and embeddings
  • First order linear differential operators
  • Nash-Kuiper theorem
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Accessibility – to request an alternate format of an AMS title
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