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Seifert Conjecture


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 C^1 (Schweitzer 1974), over C^2 (Harrison 1988), and finally false in general (Kuperberg 1994).


See also

Sphere, Vector Field

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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. "C^2 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

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