Tame submanifold: Difference between revisions

Latest revision as of 19:59, 11 May 2008

This article defines a property of a submanifold inside a manifold

Definition

Let $M$ be a manifold of dimension $m$ and $N$ a submanifold of dimension $n$ . Then $N$ is termed tame in $M$ if for every point $x \in N$ , there exists a neighbourhood $U$ of $x$ in $M$ such that the pair $(U, U \cap N)$ is homeomorphic to the pair $(R^{m}, R^{n})$ where $R^{n}$ is viewed as a linear subspace of $R^{m}$ .

Another way of saying this is that the local codimension at each point, equals the codimension of the submanifold as a whole.

Facts

An example of a submanifold which is not tame is the Alexander horned sphere in $R^{3}$ .

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 Let <math>M</math> be a [[manifold]] of dimension <math>m</math> and <math>N</math> a [[submanifold]] of dimension <math>n</math>. Then <math>N</math> is termed ''tame'' in <math>M</math> if for every point <math>x \in N</math>, there exists a neighbourhood <math>U</math> of <math>x</math> in <math>M</math> such that the pair <math>(U, U \cap N)</math> is homeomorphic to the pair <math>(\R^m,\R^n)</math> where <math>\R^n</math> is viewed as a linear subspace of <math>\R^m</math>.
+Another way of saying this is that the local codimension at each point, equals the codimension of the submanifold as a whole.
 ==Facts==
 An example of a submanifold which is ''not'' tame is the [[Alexander horned sphere]] in <math>\R^3</math>.