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Softcover ISBN:  9781470476175 
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Softcover ISBN:  9781470476175 
Product Code:  GSM/239.S 
List Price:  $89.00 
MAA Member Price:  $80.10 
AMS Member Price:  $71.20 
eBook ISBN:  9781470476182 
Product Code:  GSM/239.E 
List Price:  $85.00 
MAA Member Price:  $76.50 
AMS Member Price:  $68.00 
Softcover ISBN:  9781470476175 
eBook ISBN:  9781470476182 
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 DetailsGraduate Studies in MathematicsVolume: 239; 2024; 363 ppMSC: 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 homotopytheoretic 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.
ReadershipGraduate 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

Onedimensional convex integration

Homotopy principle for ample differential relations

Directed immersions and embeddings

First order linear differential operators

NashKuiper theorem


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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 homotopytheoretic 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.
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

Onedimensional convex integration

Homotopy principle for ample differential relations

Directed immersions and embeddings

First order linear differential operators

NashKuiper theorem