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
I presented this (pdf, in french) in a short talk for a differential topology course.
I also dabbled with (pdf, in french) the approximation of compact hypersurfaces. I wasn’t able to get a constructive result in time, so I left it as a very rough draft. [I posted a much improved follow up in April.] In the document, I sketch a proof of the following.
Theorem. Let be a compact hypersuface of . There exists a sequence of polynomials defined on a compact of such that for sufficiently large, is a hypersurface and
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.
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