# Introduction to the \(h\)-Principle

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*Y. Eliashberg; N. Mishachev*

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.

#### Reviews & Endorsements

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

#### Table of Contents

# Table of Contents

## Introduction to the $h$-Principle

- Cover Cover11 free
- Other titles in this series i2 free
- Title page v6 free
- Contents ix10 free
- Preface xv16 free
- Intrigue 120 free
- Part I. Holonomic approximation 524
- Jets and holonomy 726
- Thom transversality theorem 1534
- Holonomic approximation 2140
- Applications 3756
- Part II. Differential relations and Gromov’s ℎ-principle 5170
- Differential relations 5372
- Homotopy principle 5978
- Open Diff 𝑉-invariant differential relations 6584
- Applications to closed manifolds 6988
- Part III. The homotopy principle in symplectic geometry 7392
- Symplectic and contact basics 7594
- Symplectic and contact structures on open manifolds 99118
- Symplectic and contact structures on closed manifolds 105124
- Embeddings into symplectic and contact manifolds 111130
- Microflexibility and holonomic ℛ-approximation 129148
- First applications of microflexibility 135154
- Microflexible 𝔘-invariant differential relations 139158
- Further applications to symplectic geometry 143162
- Part IV. Convex integration 151170
- One-dimensional convex integration 153172
- Homotopy principle for ample differential relations 167186
- Directed immersions and embeddings 173192
- First order linear differential operators 179198
- Nash-Kuiper theorem 189208
- Bibliography 199218
- Index 203222 free
- Back Cover Back Cover1226