Tubular neighborhood theorem: Difference between revisions

Revision as of 00:11, 17 January 2008

Statement

Let $M$ be a submanifold (differential sense) of $R^{n}$ , of dimension $m$ . Then, there exists $ϵ > 0$ such that for any point at distance at most $ϵ$ from $M$ , there is a unique expression of the point as a sum $p + v$ where $p \in M$ and $v$ is a normal at $p$ , with $| v | < ϵ$ .

If we define $U$ as the open subset of $R^{n}$ comprising those points of $R^{n}$ at distance less than $ϵ$ from $M$ , then $U$ can be viewed as a concrete realization, in the ambient space $R^{n}$ , of the normal bundle to $M$ in $R^{n}$ . In the situations where the normal bundle to $M$ is trivial, we see that this gives a natural diffeomorphism $U ≅ M \times R^{n - m}$ .

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 ==Statement==
-Let <math>M</math> be a [[submanifold (differential sense)]] of <math>\R^n</math>, of dimension <math>m</math>. Then, there exists <math>\epsilon>0</math> such that for any point at distance at most <math>\epsilon</math> from <math>M</math>, there is a unique expression of the point as a sum <math>p + v</math> where <math>p \in M</math> and <math>v</math> is a normal at <math>p</math>, with <math>\norm{v} < \epsilon</math>.
+Let <math>M</math> be a [[submanifold (differential sense)]] of <math>\R^n</math>, of dimension <math>m</math>. Then, there exists <math>\epsilon>0</math> such that for any point at distance at most <math>\epsilon</math> from <math>M</math>, there is a unique expression of the point as a sum <math>p + v</math> where <math>p \in M</math> and <math>v</math> is a normal at <math>p</math>, with <math> \| v \| < \epsilon</math>.
 If we define <math>U</math> as the open subset of <math>\R^n</math> comprising those points of <math>\R^n</math> at distance less than <math>\epsilon</math> from <math>M</math>, then <math>U</math> can be viewed as a concrete realization, in the ambient space <math>\R^n</math>, of the normal bundle to <math>M</math> in <math>\R^n</math>. In the situations where the normal bundle to <math>M</math> is trivial, we see that this gives a natural diffeomorphism <math>U \cong M \times \R^{n-m}</math>.