Introduction to the H-principle (Google eBook)

Front Cover
American Mathematical Soc., 2002 - Mathematics - 206 pages
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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.
  

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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
Bibliography
199

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