Compatible Complex Structures on Twistor Spaces

Let (M, g) be an oriented 4-dimensional Riemannian manifold (not necessarily compact). Due to the Hodge-star operator ⋆, we have a decomposition of the bivector bundle ∧2 TM = ∧+ ⊕ ∧− . Here ∧± is the eigen-subbundle for the eigenvalue ±1 of ⋆. The metric g on M induces a metric, denoted by < , >, on the bundle ∧2 TM . Let π : Z = S (∧+) −→ M be the sphere bundle; the fiber over a point m ∈ M parameterizes the complex structures on the tangent space TmM compatible with the orientation and the metric g. It is the twistor space of the manifold (M, g). Since the structural group of Z is SO(3) ⊂ Aut(CP ), we can thus put the complex structure of CP 1 on each fiber. On the other hand, the Levi-Civita connection on (M, g) induces a splitting of the tangent bundle TZ into the direct sum of the horizontal and vertical distributions: TZ = H ⊕ V . Therefore, the twistor space Z admits a natural metric g̃ defined by its restrictions to H and V : we endow V with the Fubini-Study metric and H ≃ πTM with the pullback of the metric g. In this article we study some aspects of almost complex structures on (Z, g̃) which are Hermitian and extend the complex structure of the fibers. These structures will be called compatible almost complex structures on (Z, g̃). In particular, the integrability of two such structures means that the metric g̃ is bihermitian [Pon97], [AGG99]. To each morphism respecting the twistor fibration