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 counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a counterexample for some . The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different 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 steady state flows on possess closed flowlines[1] based on similar results for Beltrami flows on the Weinstein conjecture.[2]
^Etnyre, J.; Ghrist, R. (1997). "Contact Topology and Hydrodynamics". arXiv:dg-ga/9708011.
^Hofer, H. (1993). "Pseudoholomorphic curves in symplectizations with applications to the Weinstein conjecture in dimension three". Inventiones Mathematicae. 114 (3): 515–564. Bibcode:1993InMat.114..515H. doi:10.1007/BF01232679. ISSN 0020-9910. S2CID 123618375.
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