Convex integration theory. Solutions to the \(h\)-principle in geometry and topology. Reprint of the 1998 original. (English) Zbl 1223.58007

Spring, David

Modern Birkhäuser Classics. Basel: Birkhäuser (ISBN 978-3-0348-0059-4/pbk; 978-3-0348-0060-0/ebook). viii, 213 p. (2010).

Let \(p:X\to V\) be a smooth bundle and \(X^{(r)}\) the associated space of \(r\)-jets. For some subset \(\mathcal R\) in the space \(X^{(r)}\), a continuous mapping \(\rho:\mathcal R\to X^{(r)}\) satisfies the \(h\)-principle if each section of \(\rho\) is homotopic through sections to a holonomic section \(\alpha\), i.e., one for which there is a \(C^r\)-section \(f\) of \(p\) such that \(\rho\alpha\) is the \(r\)-jet of \(f\).
The bulk of the book is devoted to an exposition of the basic results in convex integration theory along with their proofs, including several passes at the \(h\)-principle in various contexts, beginning with the 1-jet case which provides the reader with a good introduction to the theory as well as an important application. The last two chapters give applications to the theory of PDEs and the connection between convex integration theory and Filippov’s relaxation theorem of optimal control theory. Other applications are in symplectic topology or deal with divergence free vector fields in 3-manifolds, isometric immersions, totally real embeddings, and classical immersion-theoretic topics such as immersions, submersions and \(k\)-mersions.
The book is a reprint of the 1998 original (see Zbl 0997.57500).

Reviewer: David B. Gauld (Auckland)

MSC:

58C35	Integration on manifolds; measures on manifolds
58A20	Jets in global analysis
58E25	Applications of variational problems to control theory
53C42	Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
58-02	Research exposition (monographs, survey articles) pertaining to global analysis

Keywords:

convex integration; \(h\)-principle; \(r\)-jet; smooth bundle; open relations in jet spaces; ample relations in jet spaces; closed relations in jet spaces; geometry of jet spaces; systems of PDEs; Filippov’s relaxation theorem; immersions; submersions

Citations:

Zbl 0997.57500