*A complete proof of the tubular neighborhood theorem for submanifolds of euclidean space is presented. I was unable to find it in the litterature.
*

Let me introduce some notations. A submanifold of dimension of is a subset that is locally -diffeomorphic to open balls of . Similarily, a manifold with boundary is locally diffeomorphic to open balls of the half space .

If is a differentiable map between manifolds, we denote by the differential of at . Each tangent space has an orthogonal complement in ; the *normal bundle *of is . In the following, we assume is compact.

Given , we let and , where is the euclidean distance. The set is an -neighborhood of the cross-section in , and is a *tubular neighborhood* of in . We will prove the following theorem.

**Tubular neighborhood theorem. **Let be a compact submanifold of , without boundary. For sufficiently small, and are manifolds, diffeomorphic under the map .

**Corollary 1.** If is sufficiently small, then for each there exists an unique closest point to on .

Note, however, that this corollary may not hold when is only a manifold. We will require to be at least . The proof will make clear why this is necessary, but I also present a counter-example.

**Counterexample ( manifolds). **Let be the graph of in . It is indeed a compact manifold since is . However, the points have, for all , two closest points on . To see this, first note that if had an unique closest point to , then that point would be , by the parity of . Now, consider the function , its graph being the lower half of a circle centered at and crossing . We find , meaning that the graph of is under near . This is a contradiction, as me may thus shrink the circle to find two intersection points on closer to than .

## Proof of the theorem.

In the following, is a compact submanifold of of dimension .

**Lemma 1.** The normal bundle is a submanifold of and .

**Proof.** Let and consider a neighborhood of in . It may be chosen so that , for some with surjective. Restricting some more, we can find a diffeomorphism . Using and and , we construct a map having as a regular value and such that . It will follow from the preimage theorem that is a submanifold of dimension . Furthermore, is an open neighborhood of for all and we will have found that is a manifold.

The map is defined as , , where and is a basis of . Because the vectors form a basis of , the zero set is precisely and we find that . To differentiate , we use the fact that is . In its matrix form,

where both and are surjective whenever . Thus is indeed surjective for all . The assertion follows from . QED.

**Lemma 2.** For all , is a submanifold with boundary.

**Proof.** Let . For any , we have is surjective. By the preimage theorem, we find that is a submanifold of with boundary . QED.

**Lemma 3.** The map is a local diffeomorphism onto its image .

Proof. The differential of is simply , an isomorphism since is the direct sum of and . By the inverse function theorem, it follows that is a local diffeomorphism onto its image.

Now, it is clear that . If , then by compacity of we can find a closest point . It is straightforward to verify that , and thus , . QED.

**Lemma 4 (See Spivak, 1970). **If is taken sufficiently small, then is a diffeomorphism.

**Proof. **It suffices to show that is bijective, whenever is sufficiently small. The local diffeomorphism will then be a global diffeomorphism. Note that is injective on . Let be the set of points showcasing the non-injectivity of . This set is disjoint from the compact set . Therefore, if we can show that is closed, we will find and taking will suffice.

Let be a sequence of points in converging to some . By continuity of , we must have . We cannot have , as this would contradict the fact that is a local diffeomorphism. Therefore and . QED.

**References:**

Milnor, J.W. (1965) Topology from a differentiable point of view.

Spivak, M. (1970) A comprehensive introduction to differential geometry.

Pingback: Constructive approximation of compact hypersurfaces | Math. Stat. Notes