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the_confoliation_programme [2017/08/29 17:05]
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the_confoliation_programme [2017/08/29 17:05] (Version actuelle)
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| ===== The confoliation programme ===== | ===== The confoliation programme ===== |
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| * In dimension 3 it is known, due to Eliashberg-Thurston, that any foliation can be C0-approximated by a positive and a negative contact structure. Is the same true in higher dimensions for a symplectic foliation? | * In dimension 3 it is known, due to Eliashberg-Thurston, that any foliation can be C0-approximated by a positive and a negative contact structure. Is the same true in higher dimensions for a symplectic foliation? |
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| * In dimension 3 tautness of the foliation implies tightness of the contact structure. In higher dimensions, being strong symplectic implies that any approximation (if it exists) is fillable and thus tight as well. | * In dimension 3 tautness of the foliation implies tightness of the contact structure. In higher dimensions, being strong symplectic implies that any approximation (if it exists) is fillable and thus tight as well. |
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| * **(A. Mori)** In higher dimensions one can provide an alternate definition of confoliation: We say that a pair (\alpha,\tau) is a twisted contact structure, with \alpha a 1-form and \tau a 2-form, if d\alpha+\tau is non-degenerate along ker(alpha). Then, we will say that \alpha is a confoliation if there is a linear family \tau_x, x in (0,1], with (\alpha,\tau_x) twisted contact. In particular, this implies that αΛ(dα)^n is non-negative. One particular example of such a confoliation is the symplectic foliation considered by Mitsumatsu in S5. | * **(A. Mori)** In higher dimensions one can provide an alternate definition of confoliation: We say that a pair (\alpha,\tau) is a twisted contact structure, with \alpha a 1-form and \tau a 2-form, if d\alpha+\tau is non-degenerate along ker(alpha). Then, we will say that \alpha is a confoliation if there is a linear family \tau_x, x in (0,1], with (\alpha,\tau_x) twisted contact. In particular, this implies that αΛ(dα)^n is non-negative. One particular example of such a confoliation is the symplectic foliation considered by Mitsumatsu in S5. |
