Won best poster award at the CSSC.
Won best poster award at the CSSC.
When mixing coffee in your favorite mug, at every instant a particule is staying still.
Let be a vector field and denote by its stream. That is, and . A domain is said to be invariant (under the stream of ) if for all and . The curve is said to be a closed orbit of if there exists such that .
If is invariant and diffeomorphic to a closed ball of , then has a zero in .
If , then any closed orbit of encloses a zero of .
Proof of the theorem.
Suppose that for all and let . Since is uniformly continuous on , there exists such that implies . Also, by Brouwer’s fixed point theorem, there exists such that . This yields a closed orbit such that any two points on are at distance at most from each other. Since is closed, there must exist such that . Hence we find that , even though . This is impossible. Thus is not bounded away from zero and must have a zero in the compact .
Proof of the corollary.
When , the Jordan-Brouwer theorem implies that closed orbits separate the plane in two connected components, one of which is bounded. Schoenflies’ theorem, strengthening the above, ensures that the union of bounded component with the closed orbit is diffeomorphic to the closed disk. Invariance follows from the unicity of the solution to initial value problems when is .
This can be generalized as follows. For the sake of mixing things up, we state the result in topological terms.
Theorem (Particular case of the Poincaré-Hopf theorem).
Let be a compact submanifold of with non-zero Euler characteristic , and let be a smooth isotopy. Then for all , there exists distinct points such that
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.
A complete proof of the tubular neighborhood theorem for submanifolds of euclidean space is presented. I was unable to find an elementary version 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.