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

Abstract

Let M be a 2m+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
ΔX^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_X η=0;
(v) ∇²X 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,φX) is an involutive 3-distribution on M, in the sense of Cartan.