Tubular neighborhood theorem

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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 M be a submanifold (differential sense) of \R^n, of dimension m. Then, there exists \epsilon>0 such that for any point at distance at most \epsilon 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 \| < \epsilon.

If we define U as the open subset of \R^n comprising those points of \R^n at distance less than \epsilon 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 \cong M \times \R^{n-m}.