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Locally conformal almost generalized $f$-cosymplectic manifolds

Fortuné Massamba, Jude Rosnick Bayeni Mitoueni

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 $(φ, ξ, η, g)$ 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.

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 -cosymplectic manifolds. These are almost contact metric structures equipped with a closed Lee form and a smooth function satisfying where is the second fundamental form. We derive integrability conditions and prove a dimensional dichotomy: in dimension , may admit transverse components, while in higher dimensions it must be proportional to . This rigidity, which contrasts with even-dimensional conformal symplectic geometry, is established and illustrated by explicit examples in dimensions and . The framework generalizes and unifies prior results on locally conformal almost cosymplectic and almost -cosymplectic structures.
Paper Structure (4 sections, 8 theorems, 86 equations)

This paper contains 4 sections, 8 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Generalized $f$-cosymplectic manifolds
  3. Locally conformal almost generalized $f$-cosymplectic manifolds
  4. Conclusion

Key Result

Lemma 2.1

Let $(M, \phi, \xi, \eta, g)$ be a $(2n+1)$-dimensional almost contact manifold, and let $\Phi$ be its second fundamental form. Consider the linear map Then $L$ is injective, that is, if $\alpha \wedge \Phi^{n-1} = 0$ on $\ker\eta$, then $\alpha = 0$.

Theorems & Definitions (16)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • Example 3.3
  • Theorem 3.4
  • Corollary 3.5
  • ...and 6 more