Every smooth nonzero vector field on the 3-sphere has at least one closed orbit.
The conjecture was proposed in 1950 and proved true for Hopf
maps. The conjecture was subsequently demonstrated to be false over 
(Schweitzer 1974), over 
(Harrison 1988), and finally false in general (Kuperberg
1994).
Seifert Conjecture
See also
Sphere, Vector FieldExplore with Wolfram|Alpha
References
Cipra, B. "Collaboration Closes in on Closed Geodesics." What's Happening in the Mathematical Sciences, Vol. 1. Providence, RI: Amer. Math. Soc., pp. 27-30, 1993.Kuperberg, K. "A Smooth Counterexample to the Seifert Conjecture." Ann. Math. 140, 723-732, 1994.Kuperberg, G. "A Volume-Preserving Counterexample to the Seifert Conjecture." Comment. Math. Helv. 71, 70-97, 1996a.Kuperberg, G. and Kuperberg, K. "Generalized Counterexamples to the Seifert Conjecture." Ann. Math. 143, 547-576, 1996b.Kuperberg, G. and Kuperberg, K. "Generalized Counterexamples to the Seifert Conjecture." Ann. Math. 144, 239-268, 1996c.Harrison, J. "
Counterexamples to the Seifert Conjecture." Topology 27, 249-278
1988.Rabinowitz, P. "Periodic Solutions of Hamiltonian Systems."
Comm. Pure Appl. Math. 31, 157-184, 1978.Sander, E. "Seifert
Conjecture Overthrown." http://www.geom.uiuc.edu/docs/forum/seifert/.Schweitzer,
P. A. "Counterexamples to the Seifert Conjecture and Opening Closed Leaves
of Foliations." Ann. Math. 100, 386-400, 1974.Seifert,
H. "Closed Integral Curves in 3-Space and Isotopic Two-Dimensional Deformations."
Proc. Amer. Math. Soc. 1, 287-302, 1950.Weinstein, A.
"Symplectic V-Manifolds, Periodic Orbits of Hamiltonian Systems, and the Volume
of Certain Riemannian Manifolds." Comm. Pure Appl. Math. 30, 265-271,
1977.Wilson, F. W. Jr. "On the Minimal Sets of Non-Singular
Vector Fields." Ann. Math. 84, 529-536, 1966.
Cite this as:
Weisstein, Eric W. "Seifert Conjecture." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SeifertConjecture.html