Convex Hull Extensions

Abstract

The h-principle. Let p: X → V be a smooth bundle, q-dimensional fibers, over a smooth n-dimensional manifold V, n ≥ 1; s: X (r)→ V is the source map. Let p: R→ X X (r) be a microfibration. We recall the notation introduced in I §3. A section α ∈ Γ(R) (sopoα = idv) is holonomic if there is a C r-section f ∈ Γr(X) such that j r f = poα ∈ Γ(X r). The relation R satisfies the h-principle if for each α ∈ Γ(R) there is a homotopy of sections H: [0,1] ↑ Γ(R), H o = α, such that the section H 1 is holonomic. The h-principle is required to be a relative condition in the following sense. Let K ⊂ V be closed and suppose α is holonomic on K: there is a C r-section g ∈ Γ(X) such that p o α = j rg ∈ ΓK(X(r)). Then in addition we require that for all t ∈ [0,1], Ht t= α ∈ ΓK(R) (constant homotopy over K).KeywordsChapter VIIITangent FieldChapter VersusPath ComponentProof ProcedureThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.