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Approximation theorems for Nash mappings and Nash manifolds. (English) Zbl 0601.58005
A semialgebraic submanifold M of \({\mathbb{R}}^ n\) of class \(C^ r\) \((r=0,...,\infty,\omega)\) is called a \(C^ r\)-Nash manifold, and a \(C^ r\)-Nash map is a \(C^ r\)-map of a semialgebraic graph. Using partitions of unity with Nash functions and stratifications by Nash manifolds, several results of fundamental importance are derived: Every \(C^ r\)- Nash map between \(C^ r\)-Nash manifolds can be approximated by a \(C^{\omega}\)-Nash map in the \(C^ r\)-topology. (A limit \(f_ k\to 0\) in this topology means the uniform convergence \(v_ 1...v_ sf_ k\to 0\) for any \(C^ r\)-Nash vector fields \(v_ 1,...,v_ s\) in the number \(s\leq r.)\) Every \(C^ r\)-Nash manifold M in \({\mathbb{R}}^ n\) \((1\leq r<\infty)\) can be approximated by \(C^{\omega}\)-Nash manifolds in \(C^ r\)-topology, and for any compact \(C^{\omega}\)-Nash submanifold \(N\subset M\), the approximations can be chosen identical on N. As a result, \(C^ r\)-Nash diffeomorphism classes are identical with \(C^{\omega}\)-Nash diffeomorphism classes of \(C^ r\)-Nash manifolds. (For the whole class of abstract Nash manifolds, an analogous result is not true.) Moreover, a \(C^ 0\)-vector bundle over a \(C^ r\)-Nash manifold (0\(\leq r\leq \omega)\) possesses a unique \(C^ r\)-Nash bundle structure.
Reviewer: J.Chrastina

MSC:
58A07 Real-analytic and Nash manifolds
14Pxx Real algebraic and real-analytic geometry
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