We show that if X, Y are smooth, compact k-dimensional submanifolds of ℝⁿ and 2k+2 ≤ n, then each diffeomorphism ϕ: X → Y can be extended to a diffeomorphism Φ: ℝⁿ → ℝⁿ which is tame (to be defined in this paper). Moreover, if X, Y are real analytic manifolds and the mapping ϕ is analytic, then we can choose Φ to be also analytic. We extend this result to some interesting categories of closed (not necessarily compact) subsets of ℝⁿ, namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of ℝⁿ (with morphisms being semi-algebraic continuous mappings). In each case we assume that X, Y are k-dimensional and ϕ is an isomorphism, and under the same dimension restriction 2k+2 ≤ n we assert that there exists an extension Φ :ℝⁿ → ℝⁿ which is an isomorphism and it is tame. The same is true in the category of smooth algebraic subvarieties of ℂⁿ, with morphisms being holomorphic mappings and with morphisms being polynomial mappings.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-13,
author = {Zbigniew Jelonek},
title = {Manifolds with a unique embedding},
journal = {Colloquium Mathematicae},
volume = {116},
year = {2009},
pages = {299-317},
zbl = {1179.14039},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-13}
}
Zbigniew Jelonek. Manifolds with a unique embedding. Colloquium Mathematicae, Tome 116 (2009) pp. 299-317. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm117-2-13/