Some Further Properties of Kählerian Twistor Spinors

Abstract

AbstractOn a spin Kähler manifold M2m a new first integral QΨ of the Kählerian twistor equations is presented. If the scalar curvature has a critical point then QΨ vanishes. In case M2m is closed, this fact provides a simple geometrical obstruction for Kählerian twistor spinors and, consequently, some new vanishing theorems. Twistor spinors with QΨ ≠ 0 are investigated. Some invariants of the corresponding space of twistor spinors are constructed if the other basic first integral CΨ does not vanish, too. Each twistor spinor Ψ with CΨ ≠ 0 and QΨ ≠ 0 determines a foliation of M2m whose leaves are totally geodesic immersed Kähler manifolds. It is shown that Kählerian twistor spinors of type (r, s) can be interpreted as the minima of a certain functional. Some properties of Kählerian twistor spinors of the exceptional type r = (m + 2)/2 are proved.