Let be a compact manifold of codimension . We show that can be well approximated by a part of an algebraic manifold.
For all , there exists and a polynomial function defined on such that is diffeomorphic to and
This was proved by Seifert in a 1936 german language paper. It was later generalized to the Nash-Tognoli theorem which implies that non-singular real algebraic sets have precisely the same topological invariants as compact manifolds .
Here, however, our motivations are more elementary and practical. Our proof of theorem 1 points towards constructive approximation processes and shows the separability of the space of all compact hypersurfaces, under an appropriate topology. This is relevant in statistics and computer graphics, for smooth hypersurface regression and reconstruction. Growing sequences of finite dimensional search spaces of smooth manifolds are required in these applications.
In preparation for our proof, in section 2, we also discuss the Jordan-Brouwer theorem, the orientability of and that for some function having regular value . We show that these three facts are essentially equivalent, in the sense that any one can be quite easily obtained from another.
I was unable to find an elementary proof, in english or french, of theorem 1 in the litterature. The work presented here is the result of my own efforts.
1.1 Notations and conventions
A function between arbitrary subsets and of is said to be smooth if it can locally be extended to a map from an open domain. A diffeomorphism is an invertible smooth map such that is also smooth. An hypersurface of is a subset such that each point has a neighborhood in that is diffeomorphic to .
In the following, is a compact and connected hypersurface of .
1.2 The global inverse function theorem
We need the following global variant of the inverse function theorem.
Lemma 1. (Global inverse function theorem)
Let be a smooth map between manifolds of the same dimensions and let . If is injective on and if its differential is injective for all , then there exists a neighborhood of such that is open in and is a diffeomorphism.
By the usual inverse function theorem, the statements holds when is a singleton. Thus for every point , there exists open sets and such that is a diffeomorphism. Now, all that is left to show is that there exists a neighborhood of on which is injective. Then with , we will have open in and a diffeomorphism.
To construct , we first let and we place ourselves in . Consider the set of points showcasing the non-injectivity of . Note that is closed, since is locally injective on . Thus is open, both in and , and contains some neighborhood of of the form . Taking yields the result.
1.3 Remarks about the Jordan-Brouwer theorem
Recall the Jordan-Brouwer theorem: is the disjoint union of two non-empty connected components, one of which is bounded. I show that it implies quite easily the following facts: there exists a function having regular value such that ; and is orientable. Reciprocally, points and each imply the Jordan-Brouwer theorem.
1.3.1 (1) and (2) from Jordan-Brouwer
Now, let be the euclidean distance and let . Recall that (see this previous post) admits an -neighborhood , for all sufficiently small, such that
- every has a unique closest point ;
- the map is smooth.
Assume that the Jordan-Brouwer theorem holds. Let be the bounded connected component of and let be the other one. At each point , we let be the normal unit vector to that points towards , the exterior region. It can be verified that is smooth, expressing it using local parametrizations of . It provides a practical parametrization of the -neighborhood and orients , showing point (2).
For sufficiently small is diffeomorphic to under the map .
Note that so that is injective. The derivative of at a point is , which is also injective. By lemma 1, it follows that there exists a neighborhood of such that is a diffeomorphim onto its image. By the compacity of , there exists such that and we obtain that is a diffeomorphism.
We fix some , where is such that , defined in lemma 2, is a diffeomorphism. Using the notations of the same lemma, we let
Thus is a signed distance function, negative in the interior . It can be extended smoothly to all in such a way that and . We then have and is easily seen to be a regular value.
1.3.2 Jordan-Brouwer from (1) or (2)
We need a first step in the proof of the Jordan-Brouwer theorem.
The set has at most two connected components.
Sketch of proof.
Let be any connected component of . Then is non-empty and closed in . Using local diffeomorphisms around the points of (the Jordan-Brouwer theorem was only used to ensure that the diffeomorphism is global), it can be verified that is open in . Therefore , from the connectedness of . Taking any and using again a local diffeomorphism around , we see that any connected component intersects one of the side of around and the result follows.
Now, if with having regular value , remark that
and that both sets in the disjoint union are non-empty (because has regular value ). From lemma 3, it follows that has precisely two connected components and it can be verified that one of them must be bounded using the compacity of .
If otherwise is orientable, the map , such that is a normal unit vector to , can be defined globally to be smooth. This implies that lemma 2 holds, which gives that for some function having regular value and in turn that the Jordan-Brouwer theorem holds.
2. Proof of theorem 1
We assume that is a compact and connected hypersurface and that is such that defined above is smooth. In the non-connected case, has a finite number of connected components and the method of proof is unchanged.
Let be such that . It is well known from approximation theory that for all , there exists a polynomial such that both and . In particular, with and sufficiently small, we have and for all . It is left to show that is diffeomorphic to .
Consider the smooth map
defined using the notations of lemma 2.
To see that it is injective, suppose that there are two points . Then . If , then by the mean value theorem there exists a point , with in between and , such that . This is impossible. Hence and is injective.
For surjectivity, let and recall that can be chosen so that and . Since and , we have and . It follows from the intermediate value theorem that there exists such that .
Finally, we show that is injective, for all . It will follows from lemma 1 that is a diffeomorphism. Recall that , for , can be seen as an endomorphism of .
For , let be defined as . Then for sufficiently small and any , we have that is invertible and
The invertibility of follows from and the uniform boundedness of on .
For the actual calculation of , let and notice that with as in lemma 2, we have Since is linear, we have and
A straightforward calculation yields
and we find that .
Hence by lemma 4 and with , is injective if does not contain . This follows from , and when is again sufficiently small.
3. Constructive approximation
To be continued… (some problems have yet to be solved).
Figure 1. Approximation of , where , using Bernstein polynomials of degrees , and . This figure is recuperated from a previous post.
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