Seifert conjecture
In mathematics, the Seifert conjecture states that every nonsingular, continuous vector field on the 3-sphere has a closed orbit. It is named after Herbert Seifert. In a 1950 paper, Seifert asked if such a vector field exists, but did not phrase non-existence as a conjecture. He also established the conjecture for perturbations of the Hopf fibration.
The conjecture was disproven in 1974 by Paul Schweitzer, who exhibited a <math>C^1</math> counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a <math>C^{2+\delta}</math> counterexample for some <math>\delta > 0</math>. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different <math>C^\infty</math> counterexample. Later this construction was shown to have real analytic and piecewise linear versions. In 1997 for the particular case of incompressible fluids it was shown that all <math>C^\omega</math> steady state flows on <math>S^3</math> possess closed flowlines<ref>Template:Cite arXiv</ref> based on similar results for Beltrami flows on the Weinstein conjecture.<ref>Template:Cite journal</ref>
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