The Twistor Method

Abstract

Let N be an oriented 2n—dimensional Riemannian manifold and also let SO(N) → N denote the SO(2n)—principal bundle of oriented orthonormal frames over N. The associated fibre bundle

$$ SO(N){{ \times }_{{SO(2n)}}}SO(2n)/U(n) = SO(N)/U(n) $$

is called the orthogonal twistor bundle over N. The fibre at x ∈ N parametrizes the set of all orientation-preserving orthogonal complex structures of the vector space T_xN. T= SO(N)/U(n) can be made into an almost complex manifold. In fact there are 2^γ, γ = n(n−1)/2, many natural almost complex structures on T. (See §2 for the description.) And one attempts to study minimal surfaces in N in terms of complex curves in T.