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In this short note, we review the well-known result that there is no orthogonal complex structure on S 6 with respect to the round metric. [ABSTRACT FROM AUTHOR]

We show that Hopf hypersurfaces in complex projective space are constructed from half-dimensional totally complex submanifolds in complex 2-plane Grassmannian and Legendrian submanifolds in the twistor space. [ABSTRACT FROM AUTHOR]

A study of tensors on the quaternionic projective plane HP 2 arising from a stable 3-form on C 6 and an associated action of SU (3) is related to the existence of a holomorphic rank 3 vector bundle over CP 5 discovered by Horrocks. It also leads to the construction of SU (3) invariant Spin (7) structures on HP 2 , which are characterised in terms of associated 4-forms. [ABSTRACT FROM AUTHOR]

Abstract: Let M be a -manifold. We consider an almost CR-structure on the sphere bundle of unit tangent vectors on M, called the CR twistor space. This CR-structure is integrable if and only if M is a holonomy -manifold. We interpret -instanton bundles as CR-holomorphic bundles on its twistor space. [Copyright &y& Elsevier]

Abstract: We give a proof that the sphere does not admit an integrable orthogonal complex structure using simple differential geometric methods. This appears as a corollary of a general analogous result concerning pseudo-spheres. We study the twistor space of a pseudo-Riemannian manifold in both the holomorphic and pseudo-Riemannian directions. In particular, we construct the twistor space of a pseudo-sphere as a known pseudo-Kähler symmetric space. This leads to the explicit, unexpected computation of the exterior derivative of the Kähler form on the base manifold. [Copyright &y& Elsevier]

Abstract: A set of canonical paraHermitian connections on an almost paraHermitian manifold is defined. ParaHermitian version of the Apostolov–Gauduchon generalization of the Goldberg–Sachs theorem in General Relativity is given. It is proved that the Nijenhuis tensor of a Nearly paraKähler manifolds is parallel with respect to the canonical connection. Salamon''s twistor construction on quaternionic manifold is adapted to the paraquaternionic case. A hyper-paracomplex structure is constructed on Kodaira–Thurston (properly elliptic) surfaces as well as on the Inoe surfaces modeled on . A locally conformally flat hyper-paraKähler (hypersymplectic) structure with parallel Lee form on Kodaira–Thurston surfaces is obtained. Anti-self-dual non-Weyl flat neutral metric on Inoe surfaces modeled on is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKähler is constructed. [Copyright &y& Elsevier]

We shall describe the twistor space of a hyperka¨hler 4n-manifold with an isometric S1-action which is holomorphic for one of the complex structures, scales the corresponding holomorphic symplectic form and whose fixed point set has complex dimension n.We deduce that any hyperka¨hler metric on the cotangent bundle of a real-analytic Ka¨hler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Ka¨hler metric and for which the S1-action by scalar multiplication in the fibres is isometric is unique in a neighbourhood of the zero section. These metrics have been constructed independently by the author and Kaledin. [Copyright &y& Elsevier]

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian 4-manifold and particular Lagrangian submanifolds of the twistor space over the 4-manifold is proven. More explicitly, for every superminimal surface a submanifold of the twistor space is constructed which is Lagrangian for all the natural almost Hermitian structures on the twistor space. The twistor fibration restricted to the constructed Lagrangian gives a circle bundle over the superminimal surface. Conversely, if a submanifold of the twistor space is Lagrangian for all the natural almost Hermitian structures, then the Lagrangian projects to a superminimal surface and is contained in the Lagrangian constructed from this surface. In particular this produces many Lagrangian submanifolds of the twistor spaces C P 3 and F 1 , 2 (C 3) with respect to both the Kähler structure as well as the nearly Kähler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds. [ABSTRACT FROM AUTHOR]

Donaldson-Friedman constructed anti-self-dual classes on K 3 # 3 CP 2 ‾ using twistor space. We show that some of these conformal classes have almost-Kähler representatives. [ABSTRACT FROM AUTHOR]

Space-like surfaces and time-like surfaces with zero mean curvature vector in oriented neutral 4-manifolds are isotropic and compatible with the orientations of the spaces if and only if their lifts to the space-like and the time-like twistor spaces respectively are horizontal. In neutral Kähler surfaces and paraKähler surfaces, complex curves and paracomplex curves respectively are such surfaces and characterized by one additional condition. In neutral 4-dimensional space forms, the holomorphic quartic differentials defined on such surfaces vanish. There exist time-like surfaces with zero mean curvature vector and zero holomorphic quartic differential which are not compatible with the orientations of the spaces and the conformal Gauss maps of time-like surfaces of Willmore type and their analogues give such surfaces. [ABSTRACT FROM AUTHOR]

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]