Nash \(C^ 1\)-embedding theorem for Carnot-Carathéodory metrics
The Nash \(C^1\)-embedding theorem for Riemannian manifolds is generalized to contact Carnot-Caratheodory manifolds. Namely, it is proved the following: Let \(V\) and \(W\) be manifolds with contact structures \(S \subset T(V)\) and \(T \subset T(W)\) and let \(g\) and \(h\) be positive definite quadratic forms defined on \(S\) and \(T\) respectively, where \(T\) is globally represented as \(\text{Ker }\beta\) for some 1-form \(\beta\) on \(W\). Assume, moreover, that \(h\) agrees with the curvature form \(\omega = d\beta|_T\) in the sense that they together define a Hermitian structure on \(T\). If \(\dim W \geq 2\dim V + 3\) and \(V\) is compact then there exists a \(C^1\)-embedding \(f : V \to W\) which is contact and isometric, in the sense that \(Df\) sends \((S,g)\) isometrically onto \((T,h)\).
scientific article; zbMATH DE number 778515