Compact leaves of the foliation defined by the kernel of a T²-invariant presymplectic form

Abstract

We investigate the foliation defined by the kernel of an exact presymplectic form dα of rank 2n on a (2n + r)-dimensional closed manifold M. For r = 2, we prove that the foliation has at least two leaves which are homeomorphic to a 2-dimensional torus, if M admits a locally free T²-action which preserves dα and satisfies that the function α(Z₂) is constant, where Z₁, Z₂ are the infinitesimal generators of the T²-action. We also give its generalization for r ≥1.