Locally conformal almost generalized $f$-cosymplectic manifolds

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 $(\phi, \xi, \eta, g)$ 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 $\Phi(\cdot, \cdot) = g(\cdot, \phi \cdot)$ 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.