Acta Applicandae Mathematicae 67: 91–115, 2001. © 2001 Kluwer Academic Publishers. Printed in the Netherlands. 91 Maps Interchanging f -Structures and their Harmonicity K. L. DUGGAL 1 , S. IANUS 2 and A. M. PASTORE 3 1 Department of Mathematics, University of Windsor, Windsor, Ontario N9B3P4, Canada. e-mail: yq8@uwindsor.ca 2 Department of Mathematics, University of Bucharest, Bucharest, 70109, Romania. e-mail: ianus@pompeiu.imar.ro 3 Department of Mathematics, University of Bari, Bari, 70125, Italy. e-mail: pastore@pascal.dm.uniba.it (Receiced: 21 January 2000) Abstract. We study some remarkable classes of metric f -structures on differentiable manifolds (namely, almost Hermitian, almost contact, almost S-structures and K-structures). We state and prove the necessary condition(s) for the existence of maps commuting such structures. The paper con- tains several new results, of geometric significance, on CR-integrable manifolds and the harmonicity of such maps. Mathematics Subject Classifications (2000): 53C25, 58E20. Key words: f -structures, harmonic maps, CR-manifolds. 0. Introduction The theory of contact manifolds has its roots in differential equations, optics and phase space of a dynamical system (for details see Arnold [1]). The two large classes of examples of contact manifolds are the principal circle bundles of the Boothby–Wang fibration, including the Hopf-fibration of the odd-dimensional sphere over complex projective space, and the tangent sphere bundles (for de- tails see Blair [6]). In 1963, Yano [39] introduced an f -structure on a C ∞ m- dimensional manifold M, defined by a nonvanishing tensor field ϕ of type (1, 1) which satisfies ϕ 3 + ϕ = 0 and has constant rank r . It is known that in this case r is even, r = 2n. Moreover, TM splits into two complementary subbundles Im ϕ and ker ϕ and the restriction of ϕ to Im ϕ determines a complex structure on such subbundle. An f -structure is a generalization of almost complex and almost contact structures according as r = m and r = m - 1, respectively. It is also known that the existence of an f -structure on M is equivalent to a reduction of the structure group to U(n)×O(s), where s = m-2n. An interesting case occurs when the subbundle Ker ϕ is parallelizable for which the structure group is U(n) ×I s , and we have an f -structure with parallelizable kernel, briefly denoted by f·pk-structure