Tubular neighborhood theorem - Revision history

Vipul: 4 revisions

2008-05-18T20:12:48Z

4 revisions

← Older revision	Revision as of 20:12, 18 May 2008
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Vipul: Tubular neighbourhood theorem moved to Tubular neighborhood theorem

2008-04-10T23:07:30Z

Tubular neighbourhood theorem moved to Tubular neighborhood theorem

← Older revision	Revision as of 23:07, 10 April 2008
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Vipul at 00:12, 17 January 2008

2008-01-17T00:12:14Z

← Older revision		Revision as of 00:12, 17 January 2008
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			{{applicationof\|inverse function theorem}}

			{{applicationof\|existence of smooth partitions of unity}}

			{{applicationof\|Lebesgue number lemma}}

	==Statement==		==Statement==

Vipul at 00:11, 17 January 2008

2008-01-17T00:11:22Z

← Older revision		Revision as of 00:11, 17 January 2008
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	==Statement==		==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>.		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>.

Vipul at 00:10, 17 January 2008

2008-01-17T00:10:44Z

New page

==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>.

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>.