Manifolds with a unique embedding

Zbigniew Jelonek¹

¹ Instytut Matematyczny Polska Akademia Nauk Św. Tomasza 30 31-027 Kraków, Poland

Colloquium Mathematicum, Tome 117 (2009) no. 2, pp. 299-317.

Résumé

We show that if $X, Y$ are smooth, compact $k$-dimensional submanifolds of $\mathbb R^n$ and $2k+2\le n$, then each diffeomorphism $\phi: X\to Y$ can be extended to a diffeomorphism $\Phi: \mathbb R^n\to \mathbb R^n$ which is tame (to be defined in this paper). Moreover, if $X, Y$ are real analytic manifolds and the mapping $\phi$ is analytic, then we can choose $\Phi$ to be also analytic.We extend this result to some interesting categories of closed (not necessarily compact) subsets of $\mathbb R^n$, namely, to the category of Nash submanifolds (with Nash, real-analytic and smooth morphisms) and to the category of closed semi-algebraic subsets of $\mathbb R^n$ (with morphisms being semi-algebraic continuous mappings). In each case we assume that $X, Y$ are $k$-dimensional and $\phi$ is an isomorphism, and under the same dimension restriction $2k+2\le n$ we assert that there exists an extension $\Phi :\mathbb R^n\to\mathbb R^n$ which is an isomorphism and it is tame.The same is true in the category of smooth algebraic subvarieties of $\mathbb C^n$, with morphisms being holomorphic mappings and with morphisms being polynomial mappings.

Détail

DOI : 10.4064/cm117-2-13

Keywords: smooth compact k dimensional submanifolds mathbb each diffeomorphism phi extended diffeomorphism phi mathbb mathbb which tame defined paper moreover real analytic manifolds mapping phi analytic choose phi analytic extend result interesting categories closed necessarily compact subsets mathbb namely category nash submanifolds nash real analytic smooth morphisms category closed semi algebraic subsets mathbb morphisms being semi algebraic continuous mappings each assume k dimensional phi isomorphism under dimension restriction assert there exists extension phi mathbb mathbb which isomorphism tame category smooth algebraic subvarieties mathbb morphisms being holomorphic mappings morphisms being polynomial mappings