The α-planes can be generalized to a suitable class of curved complex space-times. By a complex space-time (M,g) we mean a four-dimensional Hausdorff manifold M with holomorphic metric g. Thus, with respect to a holomorphic coordinate basis xa, g is a 4×4 matrix of holomorphic functions of xa, and its determinant is nowhere-vanishing. Remarkably, g determines a unique holomorphic connection ∇, and a holomorphic curvature tensor Rabcd. Moreover, the Ricci tensor Rab becomes complex-valued, and the Weyl tensor Cabcd may be split into independent holomorphic tensors, i.e. its self-dual and anti-self-dual parts, respectively. With our two-spinor notation, one has
are called left-flat or self-dual. This definition only makes sense if space-time is complex or real Riemannian, since in this case no complex conjugation relates primed to unprimed spinors (i.e. the corresponding spin-spaces are no longer anti-isomorphic). Hence, for example, the self-dual Weyl spinor ψ~A′B′C′D′ may vanish without its anti-self-dual counterpart ψABCD having to vanish as well, or the converse may hold.
By definition, α-surfaces are complex two-surfaces S in a complex space-time (M, g) whose tangent vectors v have the two-spinor form, where λA is varying, and πA′ is a fixed primed spinor field on S. From this definition, the following properties can be derived:
We want to prove that, if (M,g) is anti-self-dual, it admits a three-complex- parameter family of self-dual α-surfaces. Indeed, given any point p ∈ M and a spinor μA′ at p, one can find a spinor field πA′ on M, satisfying the equation
Hence πA′ defines a holomorphic two-dimensional distribution, spanned by the vector fields of the form λAπA′, which is integrable. Thus, in particular, there exists a self-dual α-surface through p, with tangent vectors of the form λAμA′ at p. Since p is arbitrary, this argument may be repeated ∀p ∈ M. The space P of all self-dual α-surfaces in (M,g) is three-complex-dimensional, and is called twistor space of (M, g).