Knots and minimal flows on 3-manifolds
A flow \(\phi : \mathbb{R} \times M \to M\) on a smooth compact Riemannian 3-manifold \(M\) is minimal if the closure of each semitrajectory starting at an arbitrary point of \(M\) is equal to \(M\). The main theorem states that, for such a flow of class \(C^1\) and any two points \(p\) and \(q\) in \(M\) which are sufficiently close, the simple closed curve \(\Gamma\) formed by an arc of a trajectory of the flow connecting \(p\) and \(q\) and a geodesic arc orthogonal to the flow at \(p\) is a nontrivial prime knot in \(M\). This is motivated by the open problem of the existence of a minimal flow on the 3-sphere \(S^3\), more precisely to find an obstruction to the existence of such a flow (so one would like to show that \(\Gamma\) cannot be prime in the case of the 3-sphere). Note that the Seifert conjecture claims that any flow on \(S^3\) must have a compact trajectory, and that a minimal \(C^r\) flow on \(S^3\) would be a counterexample to this conjecture (such counterexamples are known for \(r = 1\) and 2). For the proof it is shown that the existence of a certain type of bidimensional foliation on the complement of a solid torus (a regular neighbourhood of \(\Gamma\)) implies that the solid torus is prime in \(M\). The minimality of the flow then makes it possible to use this result.
scientific article; zbMATH DE number 794445