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A geometric obstruction to the contact type property

A geometric obstruction to the contact type property Math. Z. 228, 451–487 (1998) c Springer-Verlag 1998 Kai Cieliebak Harvard University, Department of Mathematics, Science Center 322, 1 Oxford Street, Cam- bridge, MA 02138, USA (e-mail: cielieba@math.harvard.edu) Received 1 September 1996; in final form 10 April 1997 1 Introduction Prompted by the existence results for periodic orbits on energy surfaces, in 1979 A. Weinstein introduced the following concept ([We2]): Let (M; !) be a symplectic manifold, i.e. a manifold M of even dimension 2n with a nondegenerate closed 2-form !. A hypersurface S  M (throughout this paper all hypersurfaces are assumed to be smooth without boundary) is said to be of contact type if there exists a 1-form  on S such that (i) d = !j , and n−1 (ii)  ^ (d) is a volume form on S. This condition has proved extremely fruitful, mainly for the following two properties of a hypersurface S of contact type: 1. (Stability): There exists a diffeomorphism  :[−; ]  S ! W onto a tubular neighborhood W of S in M , (f0g S)= S, such that all hypersurfaces S = (fg S) are conformally symplectomorphic to S, i.e. (S ;!) is symplectomorphic to (S; r!) for http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

A geometric obstruction to the contact type property

Cieliebak, Kai
Mathematische Zeitschrift , Volume 228 (3) – Jul 1, 1998
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References (11)

Publisher
Springer Journals
Copyright
Copyright © 1998 by Springer-Verlag Berlin Heidelberg
Subject
Legacy
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/PL00004626
Publisher site
See Article on Publisher Site

Abstract

Math. Z. 228, 451–487 (1998) c Springer-Verlag 1998 Kai Cieliebak Harvard University, Department of Mathematics, Science Center 322, 1 Oxford Street, Cam- bridge, MA 02138, USA (e-mail: cielieba@math.harvard.edu) Received 1 September 1996; in final form 10 April 1997 1 Introduction Prompted by the existence results for periodic orbits on energy surfaces, in 1979 A. Weinstein introduced the following concept ([We2]): Let (M; !) be a symplectic manifold, i.e. a manifold M of even dimension 2n with a nondegenerate closed 2-form !. A hypersurface S  M (throughout this paper all hypersurfaces are assumed to be smooth without boundary) is said to be of contact type if there exists a 1-form  on S such that (i) d = !j , and n−1 (ii)  ^ (d) is a volume form on S. This condition has proved extremely fruitful, mainly for the following two properties of a hypersurface S of contact type: 1. (Stability): There exists a diffeomorphism  :[−; ]  S ! W onto a tubular neighborhood W of S in M , (f0g S)= S, such that all hypersurfaces S = (fg S) are conformally symplectomorphic to S, i.e. (S ;!) is symplectomorphic to (S; r!) for

Journal

Mathematische Zeitschrift – Springer Journals

Published: Jul 1, 1998

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