Compact leaves of the foliation defined by the kernel of a $T^2$-invariant presymplectic form

TL;DR

This paper studies the structure of foliations induced by exact presymplectic forms on manifolds with torus symmetries, revealing conditions under which leaves are toroidal, with implications for geometric and dynamical systems.

Contribution

It proves the existence of at least two torus-shaped leaves in certain presymplectic foliations with $T^2$-symmetry, extending previous results to higher codimensions.

Findings

Existence of at least two torus leaves when $r=2$ under specific symmetry conditions.

Generalization of results for higher codimension cases ($r \\geq 1$).

Conditions linking $T^2$-symmetry and presymplectic form properties to leaf topology.

Abstract

We investigate the foliation defined by the kernel of an exact presymplectic form $d\alpha$ 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 $T^2$-action which preserves $d\alpha$ and satisfies that the function $\alpha(Z_2)$ is constant, where $Z_1$, $Z_2$ are the infinitesimal generators of the $T^2$-action. We also give its generalization for r $\geq$ 1.