## Introduction to the H-principleIn 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. |

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### Contents

Holonomic Approximation | 7 |

Thom Transversality Theorem | 15 |

Holonomic Approximation | 21 |

Applications | 37 |

Differential Relations and Gromovs iPrinciple | 53 |

Homotopy Principle | 59 |

Open Diff VInvariant Differential Relations | 65 |

The Homotopy Principle in Symplectic Geometry | 73 |

Embeddings into Symplectic and Contact Manifolds | 111 |

Microflexibility and Holonomic KApproximation | 129 |

First Applications of Microflexibility | 135 |

Further Applications to Symplectic Geometry | 143 |

OneDimensional Convex Integration | 153 |

Homotopy Principle for Ample Differential Relations | 167 |

Directed Immersions and Embeddings | 173 |

First Order Linear Differential Operators | 179 |

Symplectic and Contact Structures on Open Manifolds | 99 |

Symplectic and Contact Structures on Closed Manifolds | 105 |

NashKuiper Theorem | 189 |

199 | |

### Common terms and phrases

A C V A-directed ample arbitrarily small C°-close C°-dense called canonical closed manifolds codimension cohomology class coincides complex manifold construct contact form contact manifold contact structure contact vector field convex integration coordinate cooriented corresponding cube defined denote Diff V-invariant diffeomorphism diffeotopy diffeotopy hT differential equations extends family of holonomic family of sections fiber fibration formal solution function genuine solution Gromov h-principle holds hence Holonomic Approximation Theorem holonomic sections homomorphism homotopy equivalence homotopy Gt Homotopy principle i-principle immersion f implies Inductive Lemma isocontact embedding isocontact immersions isometric isosymplectic immersions isotopy ft isotropic immersions jet space Lagrangian immersions Legendrian linear map f metric microflexible monomorphism n-planes non-degenerate open manifolds p-form parametric version polyhedron positive codimension principal subspace Proof prove r-jet regular homotopy respect Riemannian metrics section F subbundle submanifold symplectic form symplectic manifold symplectic structure tangent tangential homotopy transversal vector bundle