On a class of exact locally conformal cosymlectic manifolds

I Mihai,R Rosca,L Verstraelen

doi:10.1155/s0161171296000373

Abstract

An almost cosymplectic manifold M is a (2m + 1)‐dimensional oriented Riemannian manifold endowed with a 2‐form Ω of rank 2m, a 1‐form η such that Ωm Λ η ≠ 0 and a vector field ξ satisfying iξΩ = 0 and η(ξ) = 1. Particular cases were considered in [3] and [6].Let (M, g) be an odd dimensional oriented Riemannian manifold carrying a globally defined vector field T such that the Riemannian connection is parallel with respect to T. It is shown that in this case M is a hyperbolic space form endowed with an exact locally conformal cosymplectic structure. Moreover T defines an infinitesimal homothety of the connection forms and a relative infinitesimal conformal transformation of the curvature forms.The existence of a structure conformal vector field C on M is proved and their properties are investigated. In the last section, we study the geometry of the tangent bundle of an exact locally conformal cosymplectic manifold.

Highlights

^ manifold endowed with a 2-form f2 of rank 2m, a 1-form r such that f2’" q 0 and a vector field satisfying if2 0 and q() 1
Any such a manifold M is a space form of curvature -2c and f is the energy function corresponding to a Hamiltonian vector field associated with T
By the above equations we may say hat the Lie vector field T defines on infinitesimal homothety of all the connection forms 0

Summary

INTRODUCTION

In the last decade a series of papers have been devoted to almost cosymplectic manifolds. In the present paper we consider an almost cosymplectic manifold M(,rl, ,g) carrying a globally defined vector field T whose dual form b(T) is denoted by co. Any such a manifold M is a space form of curvature -2c and f is the energy function corresponding to a Hamiltonian vector field associated with T (in the sense of [3]). A non flat manifold of dimension m > 2 is an elliptic or hyperbolic space-form if and only if every vector field on M is an exterior concurrent one ([ 17]). On the tangent bundle manifold TM, d, and/,, define the vertical differentiation and the vertical derivation operators respectively ([7]).

Any vector field T such that

Caea C

Setting now

Hence in the case under consideration one has