Introduction to the $H$-Principle
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...
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PrefaceIntrigue1Pt. 1Holonomic ApproximationCh. 1Jets and Holonomy7Ch. 2Thom Transversality Theorem15Ch. 3Holonomic Approximation21Ch. 4Applications37Pt. 2Differential Relations and Gromov's h-PrincipleCh. 5Differential Relations53Ch. 6Homotopy Principle59Ch. 7Open Diff V-Invariant Differential Relations65Ch. 8Applications to Closed Manifolds69Pt. 3The Homotopy Principle in Symplectic GeometryCh. 9Symplectic and Contact Basics75Ch. 10Symplectic and Contact Structures on Open Manifolds99Ch. 11Symplectic and Contact Structures on Closed Manifolds105Ch. 12Embeddings into Symplectic and Contact Manifolds111Ch. 13Microflexibility and Holonomic R-Approximation129Ch. 14First Applications of Microflexibility135Ch. 15Microflexible [actual symbol not reproducible]-Invariant Differential Relations139Ch. 16Further Applications to Symplectic Geometry143Pt. 4Convex IntegrationCh. 17One-Dimensional Convex Integration153Ch. 18Homotopy Principle for Ample Differential Relations167Ch. 19Directed Immersions and Embeddings173Ch. 20First Order Linear Differential Operators179Ch. 21Nash-Kuiper Theorem189Bibliography199Index203
\ From The CriticsEliashberg and Mishachev (credentials not listed) discuss two methods for proving the h-principle: holonomic approximation and convex integration. Applications to symplectic and contact geometry are emphasized. A brief text, the book is suited for a graduate-level course on geometric methods for solving partial differential equations and inequalities. Numerous diagrams illustrate the principles and concepts described in the text. Annotation c. Book News, Inc., Portland, OR (booknews.com)\ \
