Acta Applicandae Mathematicae 54: 121–134, 1998. © 1998 Kluwer Academic Publishers. Printed in the Netherlands. 121 On Transversally Holomorphic Maps of Kählerian Foliations ELISABETTA BARLETTA and SORIN DRAGOMIR Università degli Studi della Basilicata, Dipartimento di Matematica, 85100 Potenza, Italy (Received: 26 June 1997; in final form: 26 February 1998) Abstract. Any transversally holomorphic foliated map ϕ: (M, F ) → (M ′ , F ′ ) of Kählerian foli- ations with F harmonic, is shown to be a transversally harmonic map and an absolute minimum of the energy functional E T (ϕ) = 1 2  M ‖d T ϕ‖ 2 µ in its foliated homotopy class. Mathematics Subject Classifications (1991): Primary: 53C12; Secondary: 53C55. Key words: Kählerian foliation, transversally harmonic map, Vaisman manifold. 1. Introduction Let (M, F ) and (M ′ , F ′ ) be two foliated manifolds and ϕ : M → M ′ a foliated map. When F and F ′ are Riemannian foliations there is a concept of tension field τ(ϕ) ∈ Ŵ ∞ (ϕ −1 ν(F ′ )) associated with ϕ (and coinciding with the ordinary tension field (cf. [4, p. 107]) when M, M ′ are Riemannian manifolds and F , F ′ the trivial foliations by points). A calculation based on techniques in [9] leads us to the expression (7) of τ(ϕ). It is noteworthy that (7) contains explicitly the mean curvature κ ∈  1 B (F ) of the Riemannian foliation F . We show that ϕ is transver- sally harmonic if and only if ϕ satisfies the PDEs (8) (in which the principal part is the basic Laplacian  B of F , a strongly transversally elliptic operator, cf. [6, p. 91]). Our main result is that a transversally holomorphic foliated map ϕ of (M, F ) into (M ′ , F ′ ), with F , F ′ Kählerian and F harmonic, is a transversally harmonic map (cf. Theorem 1). As an application we look at holomorphic maps of Vaisman (or generalized Hopf) manifolds (in the sense of [3, p. 33]). 2. Transversally Harmonic Maps Let M be a real m-dimensional C ∞ manifold and F a codimension q foliation of M. Let T(F ) be the tangent bundle of F and ν(F ) = T(M)/T(F ) the normal (or transverse) bundle. Let π : T(M) → ν(F ) be the natural bundle map. We shall need the basic complex:  0 B (F ) d B −→  1 B (F ) d B −→ ··· d B −→  q B (F ) d B −→ 0 ACAP1314.tex; 16/10/1998; 11:40; p.1 VTEX(VAIDE) PIPS No.:183271 (acapkap:mathfam) v.1.15