Journal of Symplectic Geometry

Volume 16 (2018)

Number 5

Locally conformally symplectic bundles

Pages: 1377 – 1408



Alexandra Otiman (Institute of Mathematics “Simion Stoilow” of the Romanian Academy, Bucharest, Romania; and Faculty of Mathematics and Computer Science, University of Bucharest, Romania)


A locally conformally symplectic (LCS) form is an almost symplectic form $\omega$ such that a closed one-form $\theta$ exists with $d \omega = \theta \wedge \omega$. A fiber bundle with LCS fiber $(F, \omega, \theta)$ is called LCS if the transition maps are diffeomorphisms of $F$ preserving $\omega$ (and hence $\theta$). In this paper, we find conditions for the total space of an LCS fiber bundle to admit an LCS form which restricts to the LCS form of the fibers. This is done by using the coupling form introduced by Sternberg and Weinstein, [GLS], in the symplectic case. The construction is related to an adapted Hamiltonian action called twisted Hamiltonian which we study in detail. Moreover, we give examples of such actions and discuss compatibility properties with respect to LCS reduction of LCS fiber bundles. We end with a glimpse towards the locally conformally Kähler case.

Received 10 November 2015

Accepted 7 February 2018

Published 26 February 2019