Manifolds with a unique embedding

Volume 117 / 2009

Zbigniew Jelonek Colloquium Mathematicum 117 (2009), 299-317 MSC: 14R10, 14P15, 14P10, 32H02, 32Q28, 57R40. DOI: 10.4064/cm117-2-13

Abstract

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.

Authors

  • Zbigniew JelonekInstytut Matematyczny
    Polska Akademia Nauk
    Św. Tomasza 30
    31-027 Kraków, Poland
    e-mail

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