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In this paper we obtain several curvature properties of the twistor and reflector spaces of a paraquaternionic Kähler manifold and prove the existence of both positive and negative mixed 3-Sasakian structures in a principal SO(2, 1)-bundle over a paraquaternionic Kähler manifold. [ABSTRACT FROM AUTHOR]

Abstract: We prove that the twistor lifts of certain twistor holomorphic surfaces in four-dimensional manifolds are weakly stable harmonic sections. As a corollary, if ambient spaces are self-dual Einstein manifolds with nonnegative scalar curvature, then the twistor lifts of twistor holomorphic surfaces are weakly stable. Moreover, for certain surfaces in four-dimensional hyperkähler manifolds, we show that the surfaces are twistor holomorphic if their twistor lifts are weakly stable harmonic sections. In particular, we characterize twistor holomorphic surfaces in four-dimensional Euclidean space by weak stability of the twistor lifts. [Copyright &y& Elsevier]

Abstract: In this note, we consider surfaces in self-dual Einstein manifolds whose twistor lifts are harmonic sections. In particular, we state the stability of the twistor lifts as harmonic sections and determine such surfaces of genus zero. This paper is a short survey of our previous results in Hasegawa (2007, 2009, 2011) . [Copyright &y& Elsevier]