Seifert conjecture says a nonsingular vector field on S^3 has at least one closed orbit. This conjecture is disproven in various smooth categories.
Now my question :
Can one give an alternative counter example to Seifert conjecture in the following way:
A polynomial vector field on $C^{2}$ defines a one dimensional singular holomorphic foliation. So we have a two dimenional real foliation of R^{4}-singularities. Is there an example of such foliation which is transverse to S^{3}\subset C^{2} and the corresponding one dimensional foliation of S^{3} disproves the Seifert conjecture?
This a good question.
A good place to start in answering this question is
J. Etnyre, R. Ghrist, Contact topology and hydrodynamics II: Solid tori:
http://www.math.upenn.edu/~ghrist/preprints/tubes.pdf
This paper has a good set of references to work on the Seifert conjecture. In addition, it is observed that, for flows of arbitrary regularity, there is "too much room" to have a topological forcing theory (p. 1). Hence, many constraints are required, many of which should be geometric in nature.
Another way to state Seifert's conjecture is given in
J.R. dos Santos Filho, M.F. da Silva, Global solvability for first order real linear partial differential operators, J. of Diff. Equations 247, 2009, 2688-2704:
http://ac.els-cdn.com/S0022039609003283/1-s2.0-S0022039609003283-main.pdf?_tid=d5a78cb6-f1ee-11e4-80da-00000aab0f26&acdnat=1430697061_b9c722303ae22bfc9ed738bd5f8d4377
Seifert conjecture: Every smooth vector field on the 3D sphere has a periodic orbit. This conjecture was proved to be false by P. Schwitzer for $C^1$ vector fields and K Kuperberg in the $C^{nifty}$. It has also been shown to be true for a real non-singular field on $\mathbb{R}^3$ such that (b.2) $\not\Rightarrow (a), p. 2690.
It shown that the Seifert conjecture is true for the restricted class of Reeb vector fields in
C. Abbas, H. Hofer, Holomorphic curves and global questions in contact geometry, 2006:
https://math.berkeley.edu/~simic/Home_page/Hofer_book.pdf
See Ch. 6, starting on page 113, on periodic orbits of the real vector field.
This a good question.
A good place to start in answering this question is
J. Etnyre, R. Ghrist, Contact topology and hydrodynamics II: Solid tori:
http://www.math.upenn.edu/~ghrist/preprints/tubes.pdf
This paper has a good set of references to work on the Seifert conjecture. In addition, it is observed that, for flows of arbitrary regularity, there is "too much room" to have a topological forcing theory (p. 1). Hence, many constraints are required, many of which should be geometric in nature.
Another way to state Seifert's conjecture is given in
J.R. dos Santos Filho, M.F. da Silva, Global solvability for first order real linear partial differential operators, J. of Diff. Equations 247, 2009, 2688-2704:
http://ac.els-cdn.com/S0022039609003283/1-s2.0-S0022039609003283-main.pdf?_tid=d5a78cb6-f1ee-11e4-80da-00000aab0f26&acdnat=1430697061_b9c722303ae22bfc9ed738bd5f8d4377
Seifert conjecture: Every smooth vector field on the 3D sphere has a periodic orbit. This conjecture was proved to be false by P. Schwitzer for $C^1$ vector fields and K Kuperberg in the $C^{nifty}$. It has also been shown to be true for a real non-singular field on $\mathbb{R}^3$ such that (b.2) $\not\Rightarrow (a), p. 2690.
It shown that the Seifert conjecture is true for the restricted class of Reeb vector fields in
C. Abbas, H. Hofer, Holomorphic curves and global questions in contact geometry, 2006:
https://math.berkeley.edu/~simic/Home_page/Hofer_book.pdf
See Ch. 6, starting on page 113, on periodic orbits of the real vector field.
I recommend you the paper of Marco Brunella entitled
"On holomorphic foliations in complex surfaces transverse to a sphere"
in Boletim da Sociedade Brasileira de Matemática - Bulletin Brazilian Mathematical Society 1995, Volume 26, Issue 2, pp 117-128,
that you can get from
http://www.emis.de/journals/em/docs/boletim/vol262/v26-2-a1-1995.pdf
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