proceedings of the american mathematical society Volume 123, Number 12, December 1995 ON CHARACTERISTICS OF CIRCLE-INVARIANT PRESYMPLECTIC FORMS AUGUSTIN BANYAGA AND PHILIPPE RUKIMBIRA (Communicated by Linda Keen) Abstract. We prove that a circle-invariant exact 2-form of rank 2n on a compact (2n+ 1 )-dimensional manifold admits two closed cha^cteristics. This solves a particular case of a generalized Weinstein conjecture. 1. Introduction A contact manifold is a couple (M, a), where M is a smooth manifold of dimension 2/1 + 1 > 3 and a is a contact form on M, that is, a is a 1-form such that a A (da)" is everywhere nonzero. The characteristic vector field of a is the unique vector field Ç on M such that i^a = 1 and içda = 0. Here, ig denotes the interior product operation. An integral curve of the vector field <* is called a characteristic of the contact manifold (M, a). In [WEI], Weinstein conjectured that there exists at least one closed character- istic on a compact (simply connected) contact manifold. Viterbo [VIT] proved this conjecture for compact contact manifolds (M, a) which are hypersurfaces of contact type in R2"+2 ; that is, if y.M -» R2n+2 is the embedding of M into R2"+2 and Q = Y."=i dx¡ A dy¡ is the standard symplectic form, then y*Q = da. Several generalizations of the results were subsequently obtained by Hof er-Viterbo [HVI], [HOV]. Their methods are analytical: Viterbo first treated this problem as a problem of finding periodic solutions of Hamilton equations on an energy surface and translated it into finding critical points of some action functional on some loop space. The subsequent approach of Hof er and Viterbo uses, in addition, the theory of pseudo-holomorphic curves [HOV], [HVI], [HOF]. Recently, Hofer [HOF] proved the Weinstein conjecture for S3 and for any compact 3-manifold M with n2(M) /0 or the given contact form is over- twisted (see Eliashberg [ELI] for the definition). Observe that the closed char- acteristic Hofer finds must be a contractible loop. Received by the editors June 14, 1994. 1991 Mathematics Subject Classification. Primary 58F05,58F22;Secondary 53C15,53C57. Key words and phrases. Presymplectic form, characteristics, generalized Weinstein conjecture. The first author was partially supported by NSF grant DMS 90-01861. © 1995 American Mathematical Society 3901 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use