## Convex Integration Theory: Solutions to the H-Principle in Geometry and Topology§1. Historical Remarks Convex Integration theory, first introduced by M. Gromov [17], is one of three general methods in immersion-theoretic topology for solving a broad range of problems in geometry and topology. The other methods are: (i) Removal of Singularities, introduced by M. Gromov and Y. Eliashberg [8]; (ii) the covering homotopy method which, following M. Gromov's thesis [16], is also referred to as the method of sheaves. The covering homotopy method is due originally to S. Smale [36] who proved a crucial covering homotopy result in order to solve the classification problem for immersions of spheres in Euclidean space. These general methods are not linearly related in the sense that succes sive methods subsumed the previous methods. Each method has its own distinct foundation, based on an independent geometrical or analytical insight. Conse quently, each method has a range of applications to problems in topology that are best suited to its particular insight. For example, a distinguishing feature of Convex Integration theory is that it applies to solve closed relations in jet spaces, including certain general classes of underdetermined non-linear systems of par tial differential equations. As a case of interest, the Nash-Kuiper Cl-isometrie immersion theorem ean be reformulated and proved using Convex Integration theory (cf. Gromov [18]). No such results on closed relations in jet spaees can be proved by means of the other two methods. |

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affine bundle ample relations base manifold bundle pr C-structure Chapter charts codimension codimension 1 tangent compact Complement constant homotopy construction continuous function continuous lift continuous map convex hull extensions Convex Integration theory coordinates Corollary covering homotopy deformation retract denote differential inclusion fiber following properties obtain formal solution Furthermore G R9 G T(TZ G T(X geometrical hence holonomic section homotopy F homotopy of formal i-principle in-path induces integral representation jet spaces Lemma Let f linearly independent local coordinates microfibration neighbourhood notation open set parametrized principal subspace product bundle projection map proof of Theorem properties are satisfied Proposition prove r-jets relation RC relative Relaxation Theorem rth order Serre fibration short maps small homotopy smooth manifold strictly short strictly surrounds sufficiently small Suppose surrounding paths systems of PDEs tangent hyperplane field topology Tr(X weak homotopy equivalence