Search Results for “seifert conjecture”

Source: 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

C

1

{\displaystyle C^{1}}

counterexample. Schweitzer's construction was then modified by Jenny Harrison in 1988 to make a

C

2
+
δ

{\displaystyle C^{2+\delta }}

counterexample for some

δ
>
0

{\displaystyle \delta >0}

. The existence of smoother counterexamples remained an open question until 1993 when Krystyna Kuperberg constructed a very different

C

∞

{\displaystyle C^{\infty }}

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

C

ω

{\displaystyle C^{\omega }}

steady state flows on

S

3

{\displaystyle S^{3}}

possess closed flowlines based on similar results for Beltrami flows on the Weinstein conjecture.