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A characterization of the tight 3‐sphere II
Author(s) -
Hofer H.,
Wysocki K.,
Zehnder K.
Publication year - 1999
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199909)52:9<1139::aid-cpa5>3.0.co;2-l
Subject(s) - mathematics , orbit (dynamics) , characterization (materials science) , vector field , pure mathematics , product (mathematics) , manifold (fluid mechanics) , decomposition , field (mathematics) , geometry , physics , mechanical engineering , ecology , optics , engineering , biology , aerospace engineering
Recall that every closed, oriented, connected 3‐manifold M admits a contact form λ. We shall give conditions on a periodic orbit P 0 of the Reeb vector field defined by λ that allows the construction of an open book decomposition of M \ P 0 into embedded planes asymptotic to P 0 such that P 0 is the binding orbit of the decomposition. It follows that M is the tight 3‐sphere. As a by‐product, a new dynamical criterion for the tightness of a contact structure is derived in the proof. The results extend earlier results. © 1999 John Wiley & Sons, Inc.