Locally conformal almost generalized $f$-cosymplectic manifolds

Abstract

This paper introduces a new class of geometric structures in almost contact metric geometry, which we call locally conformal almost generalized $f$-cosymplectic manifolds. These are almost contact metric structures $(\varphi ,\xi ,\eta ,g)$ equipped with a closed Lee form $\omega$ and a smooth function $f$ satisfying $$d\eta =\omega \wedge \eta ,d\Phi =2f\eta \wedge \Phi +2\omega \wedge \Phi ,$$ where $\Phi (\cdot ,\cdot )=g(\cdot ,\varphi \cdot )$ is the second fundamental form. We derive integrability conditions and prove a dimensional dichotomy: in dimension $3$, $\omega$ may admit transverse components, while in higher dimensions it must be proportional to $\eta$. This rigidity, which contrasts with even-dimensional conformal symplectic geometry, is established and illustrated by explicit examples in dimensions $3$ and $5$. The framework generalizes and unifies prior results on locally conformal almost cosymplectic and almost $f$-cosymplectic structures.

17 pages