### On some Properties of Riemannian Manifolds with Locally Conformal Almost Cosymplectic Structures

#### Abstract

Let *M* be a 2*m*+1-dimensional Riemannian manifold and let ∇ be the Levi-Civita connection and ξ be the Reeb vector field, η the Reeb covector field and *X* be the structure vector field satisfying a certain property on *M*. In this paper the following properties are proved:

(*i*) ξ and *X* define a 3-covariant vanishing structure;

(*ii*) the Jacobi bracket corresponding to ξ vanishes;

(*iii*) the harmonic operator acting on *X*^{b} gives

Δ*X*^{b}=*f** ||X*||^{2}* X*^{b},

which proves that *X*^{b} is an eigenfunction of Δ, having *f*||X||^{2} as eigenvalue;

(*iv*) the 2-form Ω and the Reeb covector η define a Pfaffian transformation, i.e.

*L _{X}* Ω=0,

*L*η=0;

_{X}(

*v*) ∇

^{2}

*X*defines the Ricci tensor;

(

*vi*) one has

∇

*,*

_{X}X = f X*f =*scalar,

which show that

*X*is an affine geodesic vector field;

(

*vii*) the triple (

*X,*ξ,φ

*X*) is an involutive 3-distribution on

*M*, in the sense of Cartan.

#### Full Text:

PDFDOI: https://doi.org/10.1478/C1A0401005

Copyright (c) 2015 AAPP | Physical, Mathematical, and Natural Sciences