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 $(ϕ, ξ, η, g)$ equipped with a closed Lee form $ω$ and a smooth function $f$ satisfying $$ dη= ω\wedge η, \;\; dΦ= 2fη\wedge Φ+ 2ω\wedge Φ, $$ where $Φ(\cdot, \cdot) = g(\cdot, ϕ\cdot)$ is the second fundamental form. We derive integrability conditions and prove a dimensional dichotomy: in dimension $3$, $ω$ may admit transverse components, while in higher dimensions it must be proportional to $η$. 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.