Dmitry Alekseevsky (University of Hull)
D.V.Alekseevsky@hull.ac.uk
The talk is based on a joint work with S. Marchiafava (Rome).
We give a short survey of quaternionic Kähler geometry, in particular, we recall definition of the twistor space Z of a quaternionic Kähler manifold M4n as some complex contact manifold (Z, θ) associated with M4n. Then we define a notion of a Kähler submanifold N2m of a quaternionic Kähler manifold M4n. Any Kähler submanifold N2m is a minimal submanifold . In the case n=m=1 Kähler submanifolds are precisely superminimal surfaces of M4n.
A simple construction of Kähler submanifolds N2n of maximal possible dimension 2n in a quaternionic Kähler manifold M4n with non-zero scalar curvature is given. More precisely, we prove that Kähler submanifolds N2n are the projections on M4n of Legendrian submanifolds L of Z. If u, qi, pi are Darboux coordinates on Z, such that the contact form θ= du -pdq, then a Legendrian submanifold is determined by a holomorphic function u =u(q). In the case when M=S4, this construction reduces to R. Bryant's construction of superminimal surfaces in the 4-sphere S4.