Locally conformal almost generalized $f$-cosymplectic manifolds

TL;DR

This paper introduces a new class of geometric structures called locally conformal almost generalized f-cosymplectic manifolds, exploring their properties, integrability conditions, and dimensional behavior with explicit examples.

Contribution

It defines and studies a novel class of almost contact metric structures with a closed Lee form and a smooth function, unifying and extending previous structures in the field.

Findings

In dimension 3, the Lee form may have transverse components.

In higher dimensions, the Lee form must be proportional to the contact form.

Explicit examples are provided in dimensions 3 and 5.

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 $(\phi, \xi, \eta, g)$ 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.