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We give a new proof that the sphere S^6 does not admit an integrable orthogonal complex structure, as in \cite{LeBrun}, following the methods from twistor theory. We present the twistor space of a pseudo-sphere S^{2n}_{2q}=SO_{2p+1,2q}/SO_{2p,2q} as a pseudo-K\"ahler symmetric space. We then consider orthogonal complex structures on the pseudo-sphere, only to prove such a structure cannot exist., Comment: Added the MSC's hoping Arxiv will "run" a better distribuition through Subj-class's. The article has 20 pages

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-Kahler representatives.

A complex Grassmann manifold G 2 ( C m + 2 ) of all 2-dimensional complex subspaces in C m + 2 has two nice geometric structures – the Kahler structure and the quaternionic Kahler structure. We study totally complex submanifolds of G 2 ( C m + 2 ) with respect to the quaternionic Kahler structure. We show that the projective cotangent bundle P ( T ⁎ C P m + 1 ) of a complex projective space C P m + 1 is a twistor space of the quaternionic Kahler manifold G 2 ( C m + 2 ) . Applying the twistor theory, we construct maximal totally complex submanifolds of G 2 ( C m + 2 ) from complex submanifolds of C P m + 1 . Then we obtain many interesting examples. In particular we classify maximal homogeneous totally complex submanifolds. We show the relationship between the geometry of complex submanifolds of C P m + 1 and that of totally complex submanifolds of G 2 ( C m + 2 ) .

Starting from a real analytic conformal Cartan connection on a real analytic surface $S$, we construct a complex surface $T$ containing a family of pairs of projective lines. Using the structure on $S$ we also construct a complex $3$-space $Z$, such that $Z$ is a twistor space of a self-dual conformal $4$-fold and $T$ is a quotient of $Z$ by a holomorphic local $\mathbb{C}^*$ action. We prove that $T$ is a minitwistor space of an asymptotically hyperbolic Einstein-Weyl space with $S$ as an asymptotic boundary.

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.

We classify 4-dimensional austere submanifolds in Euclidean space ruled by 2-planes. The algebraic possibilities for second fundamental forms of an austere 4-fold M were classified by Bryant, falling into three types which we label A, B, and C. We show that if M is 2-ruled of Type A, then the ruling map from M into the Grassmannian of 2-planes in R^n is holomorphic, and we give a construction for M starting with a holomorphic curve in an appropriate twistor space. If M is 2-ruled of Type B, then M is either a generalized helicoid in R^6 or the product of two classical helicoids in R^3. If M is 2-ruled of Type C, then M is either a one of the above, or a generalized helicoid in R^7. We also construct examples of 2-ruled austere hypersurfaces in R^5 with degenerate Gauss map., Comment: 20 pages

In the first part of this work we consider compact riemannian manifolds M with holonomy in Sp ( n ) Sp (1). We show that M admits a compatible complex structure if and only if the holonomy is in Sp ( n ), up to finite coverings. We also show that the sign of the Ricci curvature completely determines the algebraic dimension of the twistor space. In the second part, by way of contrast, we give two geometric constructions of simply- connected quaternionic manifolds with a compatible complex structure which is not hypercomplex. The first examples are non-compact and symmetric but we can show existence of compact quotients in some special dimension. The second one is compact and follows from general results of Joyce [6].

We shall describe the twistor space of a hyperkahler 4 n -manifold with an isometric S 1 -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 hyperkahler metric on the cotangent bundle of a real-analytic Kahler manifold which is compatible with the canonical holomorphic symplectic structure, extends the given Kahler metric and for which the S 1 -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.

金沢大学人間社会研究域学校教育系, 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) . © 2011 Elsevier B.V. All rights reserved.

We deal here with the geometry of the so-called twistor fibration Z → S 1 3 over the De Sitter 3-space, where the total space Z is a five-dimensional reductive homogeneous space with two canonical invariant almost CR structures. Fixed the normal metric on Z we study the harmonic map equation for smooth maps of Riemann surfaces into Z . A characterization of spacelike surfaces with harmonic twistor lifts to Z is obtained. Also it is shown that the harmonic map equation for twistor lifts can be formulated as the curvature vanishing of an S 1 -loop of connections i.e. harmonic twistor lifts exist within S 1 -families. Special harmonic maps such as holomorphic twistor lifts are also considered and some remarks concerning (compact) vacua of the twistor energy are given.

We investigate the cohomology groups H 1 ( Z , O ( L ⊗ m )), where Z is the twistor space of a compact quaternionic-Kahler manifold M , of dimension 4 k , for k ⩾ 1, L is the holomorphic contact line bundle on Z , and m ⩾ 0. The Penrose transform is used to prove a vanishing theorem for this cohomology group when M has negative scalar curvature. This theorem implies that if ( M , g ) is a compact quaternionic-Kahler manifold of dimension 4 k , for k ⩾ 1, then ( M , g ) has no nontrivial deformations through quaternionic-Kahler manifolds. The vanishing theorem is also used to show that the first Betti number of M is zero, when k > 1 and M has negative scalar curvature.

It is known that certain self-dual connections are defined on quaternionic Kahler manifolds. A.D.H.M-vanishing theorem H 1 ( P 3 , F (−2)) = 0 is generalized to anti-self-dual bundles on an arbitrary quaternionic Kahler manifolds with positive scalar curvature. The cohomology group is related to the solution space of quaternionic operators.

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 paraKahler 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 Sol 1 4 . A locally conformally flat hyper-paraKahler (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 Sol 1 4 is presented. An example of anti-self-dual neutral metric which is not locally conformally hyper-paraKahler is constructed.

Using the methods of twistor geometry we classify compact conformally flat hermitian surfaces. This is accomplished by first associating to the universal covering of such a surface an open set of a quadric in CP 3 (the twistor space of S4) and then applying some classical projective geometry.

Given a twistor space over a Hermitian symmetric space of compact type we construct a map onto a twistor space over another inner symmetric space of compact type. This map is holomorphic and preserves the superhorizontal distributions. We describe an application to harmonic maps.

Some anti-self-dual bundles on G 2 (C n +2 ) are classified by using a monad method on the twistor space. In the 1-instanton case, this monad enables us to describe the moduli space completely. It is shown that the moduli space is identified with an open cone over P (Λ 2 C n +2 ). This is a generalization of the 1-instanton moduli space on P 2* = G 2 (C 3 ).

We study the Einstein condition for a natural family of Riemannian metrics on the twistor space of partially complex structures of a fixed rank on the tangent spaces of a Riemannian manifold compatible with its metric. A generalization of the Einstein condition (discussed in the Besse book [Enstein Manifolds, Ergeb. Math. Grensgeb. (3), vol. 10, Springer, New York, 1987]) is also considered.

We construct explicit examples of 4-dimensional Riemannian metrics which admit precisely two independent hermitian structures with the same orientation. It is shown that metrics of this type exist on four-dimensional tori.

On compact balanced Hermitian manifolds we obtain obstructions to the existence of harmonic 1-forms, ∂ -harmonic (1,0)-forms and holomorphic (1,0)-forms in terms of the Ricci tensors with respect to the Riemannian curvature and the Hermitian curvature. Necessary and sufficient conditions the (1,0)-part of a harmonic 1-form to be holomorphic and vice versa, a real 1-form with a holomorphic (1,0)-part to be harmonic are found. The vanishing of the first Dolbeault cohomology groups of the twistor space of a compact irreducible hyper-Kahler manifold is shown.

In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space G / G 0 can be embedded into the twistor space of the corresponding symmetric space G / H . Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift J taking values in G / G 0 ↪ Σ ( G / H ) . We begin the paper with an example: G / H = R 4 . We also study the structure of 4-symmetric bundles over Riemannian symmetric spaces.

We provide a general criteria for the integrability of the almost para-quaternionic structure of an almost para-quaternionic manifold (M,P) of dimension bigger or equal to eight, in terms of the integrability of two or three sections of the defining rank three vector bundle P. We relate it with the integrability of the canonical almost complex structure of the twistor space and to the integrability of the canonical almost para-complex structure of the reflector space of (M,P). We show that (M, P) has plenty of locally defined, compatible, complex and para-complex structures, provided that P is para-quaternionic., 26 pages

Canonical twistor fibrations lead to Pfaffian systems by means of their superhorizontal distribution. The aim of this note is to identify explicitly the Pfaffian systems of five or less variables that arise in this way in terms of the classification given by A.Awane and M.Goze., Comment: 18 pages