The Nash-Kuiper C -isometric embedding theorem in exercises

The Nash-Kuiper C 1 -isometric embedding theoremin exercisesDecomposition into primitive metricsWe use the term Riemannian semi-metric for a field of non-negative quadraticforms on the tangent bundle of a manifold.A Riemannian semi-metric g on R n is called primitive if g = α(x)(dl) 2 ,where l = l(x) is a linear function on R n and α is a non-negative functionwith compact support.A Riemannian semi-metric g on a manifold V is called primitive if thereexists a local parametrization u : R n → U ⊂ V such that supp g ⊂ U andu ∗ g is a primitive semi-metric on R n .1. Prove that any Riemannian semi-metric g on a compact manifold V canbe decomposed into a finite sum of primitive semi-metrics.Smoothness of a limitLet g be a Riemannian metric on a manifold V and h the Euclidean metricon R q . Given two maps f, ˜f : V → R q we denote by d g (f, ˜f) the functiond g (f, ˜f)TV \ V → R , v → d g (f, ˜f)(v) = || f ∗v − ˜f ∗ v|| h|| v|| g,where h is the standard metric on R q .2. Let f i : V → R q Cbe a sequence of (smooth) maps. Prove that if f 0i →f andd g (f i , f i+1 ) < c i , with ∑ c i < ∞ , then f is a C 1 C-smooth map and f i →f 1.iThe function r(˜g, g)Given a positive metric g and non-negative ˜g we denote by r(˜g, g) the functionTV \ V → R , v → r(˜g, g)(v) = || v|| eg|| v|| g.3. Show that that1

(r1) r(˜g, g 1 ) ≤ r(˜g, g 2 ) if g 1 ≥ g 2and(r2) r(˜g, g) ≤ r(˜g + g 1 , g + g 1 ) if r(˜g, g) ≤ 1 .One-dimensional approximationAn embedding (V, g) → (R q , h) is called short (resp. strictly short) if theform g − f ∗ h ≤ h is non-negative, i.e. semi-metric (resp. positive definite).4. Let g be any Riemannian metric on the interval I. Prove that if q > 1then for any ε > 0 any short embedding f : (I, g) → (R q , h) can be C 0 -approximated by ε-isometric embeddings. Moreover, for any ρ > 0 the approximatingmap ˜f can be chosen in such a way thatwhere ∆ = g − f ∗ h.Adding a primitive metricd g (f, ˜f) < r(∆, g) + ρProve the following generalization of Exercise 4:5. Suppose that dim V = n < q. Let f : (V, g) → (R q , h) be a short embeddingsuch that ∆ = g − f ∗ h is a primitive semi-metric on V . Then for any ε theembedding f can be C 0 -approximated by ε-isometric embeddings. Moreover,for any ρ > 0 the approximating map ˜f can be chosen to satisfy the inequalityApproximation Theoremd g (f, ˜f) < r(∆, g) + ρ .An embedding f : (V, g) → (R q , h) is called ε-isometric if(1 − ε)g < f ∗ h < (1 + ε)g .Prove the following Approximation theorem:6. Let n < q. For any ε > 0, any short embedding f : (V n , g) → (R q , h)can be C 0 -approximated by ε-isometric embeddings. Moreover, given a fixeddecomposition of a Riemannian semi-metric∆ = g − f ∗ h2

into a sum of N primitive metrics then for any constant ρ > 0 one can choosean approximating embedding ˜f which satisfies the inequalityd g (f, ˜f) < N r(∆, g) + ρ .Nash-Kuiper’s C 1 -isometric embedding theoremDeduce from Exercises 6 and 2 the Nash-Kuiper C 1 -isometric embeddingtheorem:7. If n < q then any strictly short immersionf : (V n , g) → (R q , h) ,where h is the standard metric on R q , can be C 0 -approximated by isometricC 1 -smooth immersions. Moreover, if the initial immersion f is an embeddingthen f can be approximated by isometric C 1 -embeddings.Bound for the regularity in the Nash-Kuiper theorem8. Show that the h-principle type results fail for C 2 -isometric immersions.3