# Tubular neighborhood theorem

This fact is an application of the following pivotal fact/result/idea:inverse function theorem

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This fact is an application of the following pivotal fact/result/idea:existence of smooth partitions of unity

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This fact is an application of the following pivotal fact/result/idea:Lebesgue number lemma

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## 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 .