Harmonicity on cosymplectic manifolds * † C˘ at˘ alin Gherghe ‡ Abstract We prove that an (ϕ, J )-holomorphic maps from a compact cosym- plectic manifold to a K¨ ahler manifold is not only a harmonic map but also an energy minimizer on its homotopy class. We also prove a converse result. 1 Introduction Combining both global and local aspects and borrowing both from Rieman- nian geometry and from analysis, the theory of harmonic maps between Rie- mannian manifolds has developed in many diverse branches. In particular, there is now a whole battery of deep and interesting results about harmonic maps to or from complex manifolds and K¨ ahler spaces. Within contact geometry, there are several classes of manifolds that can be considered as odd-dimensional analogs of K¨ ahler spaces, the most important ones being Sasakian and cosymplectic spaces. The theory of harmonic maps on smooth manifolds endowed with some special structures has its origin in the paper of Lichnerowicz [5], in which he considered holomorphic maps between K¨ ahler manifolds. In general the construction of energy minimizing maps is much more difficult than to find harmonic ones. The main purpose of this paper is to show that structure-preserving maps on cosymplectic manifolds minimize the energy of maps. We prove that an (ϕ, J )-holomorphic map from a compact * 2000 Mathematical Subject Clasification: 53C25, 53C43, 58E20 † Key words and phrases:harmonic maps, cosymplectic manifolds, energy functional, holomorphic maps ‡ Supported by CEEX, 2-CEx 06-11-22/25.07.2006 1