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Introduction to the $$h$$-Principle
Y. Eliashberg, Stanford University, CA, and N. Mishachev, Lipetsk Technical University, Russia
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2002; 206 pp; hardcover
Volume: 48
ISBN-10: 0-8218-3227-1
ISBN-13: 978-0-8218-3227-1
List Price: US$36 Member Price: US$28.80
Order Code: 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 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.

Graduate students and research mathematicians interested in global analysis and analysis on manifolds.

Reviews

"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

• Intrigue
Holonomic approximation
• Jets and holonomy
• Thom transversality theorem
• Holonomic approximation
• Applications
Differential relations and Gromov's $$h$$-principle
• Differential relations
• Homotopy principle
• Open Diff $$V$$-invariant differential relations
• Applications to closed manifolds
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 of microflexibility
• Microflexible $$\mathfrak{U}$$-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
• Bibliography
• Index