A twistor construction of Kaehler submanifolds of a quaternionic Kaehler manifold

Abstract. A class of minimal almost complex submanifolds of a Riemannian manifold M˜ 4n with a parallel quaternionic structure Q, in particular of a 4-dimensional oriented Riemannian manifold, is studied. A notion of Kähler submanifold is defined. Any Kähler submanifold is pluriminimal. In the case of a quaternionic Kähler manifold M˜ 4n of non zero scalar curvature, in particular, when M˜ 4 is an Einstein, non Ricci-flat, anti-self-dual 4-manifold, we give a twistor construction of Kähler submanifolds M2n of maximal possible dimension 2n.More precisely,we prove that any such Kähler submanifold M2n of M˜ 4n is the projection of a holomorphic Legendrian submanifold L2n ? Z of the twistor space (Z,H) of M˜ 4n , considered as a complex contact manifold with the natural holomorphic contact structure H ? TZ. Any Legendrian submanifold of the twistor space Z is defined by a generating holomorphic function. This is a natural generalization of Bryant’s construction of superminimal surfaces in S4 = HP1. Mathematics Subject Classification (1991). Primary: 53C40; Secondary: 53C55