Difference between revisions of "Tubular neighborhood theorem"
From Diffgeom
m (4 revisions) 

(No difference)

Latest revision as of 20:12, 18 May 2008
This fact is an application of the following pivotal fact/result/idea: inverse function theorem
View other applications of inverse function theorem OR Read a survey article on applying inverse function theorem
This fact is an application of the following pivotal fact/result/idea: existence of smooth partitions of unity
View other applications of existence of smooth partitions of unity OR Read a survey article on applying existence of smooth partitions of unity
This fact is an application of the following pivotal fact/result/idea: Lebesgue number lemma
View other applications of Lebesgue number lemma OR Read a survey article on applying Lebesgue number lemma
Statement
Let be a submanifold (differential sense) of , of dimension . Then, there exists such that for any point at distance at most from , there is a unique expression of the point as a sum where and is a normal at , with .
If we define as the open subset of comprising those points of at distance less than from , then can be viewed as a concrete realization, in the ambient space , of the normal bundle to in . In the situations where the normal bundle to is trivial, we see that this gives a natural diffeomorphism .