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In this paper, we provide new rigidity results for four-dimensional Riemannian manifolds and their twistor spaces. In particular, using the moving frame method, we prove that C P 3 is the only twistor space whose Bochner tensor is parallel; moreover, we classify Hermitian Ricci-parallel and locally symmetric twistor spaces and we show the nonexistence of conformally flat twistor spaces. We also generalize a result due to Atiyah, Hitchin and Singer concerning the self-duality of a Riemannian four-manifold. [ABSTRACT FROM AUTHOR]

A study is made of algebraic curves and surfaces in the flag manifold F = S U (3) / T 2 , and their configuration relative to the twistor projection π from F to the complex projective plane P 2 , defined with the help of an anti-holomorphic involution j . This is motivated by analogous studies of algebraic surfaces of low degree in the twistor space P 3 of the 4-dimensional sphere S 4 . Deformations of twistor fibers project to real surfaces in P 2 , whose metric geometry is investigated. Attention is then focussed on toric del Pezzo surfaces that are the simplest type of surfaces in F of bidegree (1 , 1) . These surfaces define orthogonal complex structures on specified dense open subsets of P 2 relative to its Fubini-Study metric. The discriminant loci of various surfaces of bidegree (1 , 1) are determined, and bounds given on the number of twistor fibers that are contained in more general algebraic surfaces in F . [ABSTRACT FROM AUTHOR]

Let X be a quasi-projective curve, compactified to (Y, D) with X = Y - D . We construct a Deligne–Hitchin twistor space out of moduli spaces of framed λ -connections of rank 2 over Y with logarithmic singularities and quasi-parabolic structure along D. To do this, one should divide by a Hecke-gauge groupoid. Tame harmonic bundles on X give preferred sections, and the relative tangent bundle along a preferred section has a mixed twistor structure with weights 0, 1, 2. The weight 2 piece corresponds to the deformations of the KMS structure including parabolic weights and the residues of the λ -connection. [ABSTRACT FROM AUTHOR]

A hyperkähler manifold M has a family of induced complex structures indexed by a two-dimensional sphere S 2 ≅ CP 1 . The twistor space of M is a complex manifold Tw (M) together with a natural holomorphic projection Tw (M) → CP 1 , whose fiber over each point of CP 1 is a copy of M with the corresponding induced complex structure. We remove one point from this sphere (corresponding to one fiber in the twistor space), and for the case of M = R 4 n , n ∈ N , equipped with the standard hyperkähler structure, we construct one quantization that replaces the family of Berezin–Toeplitz quantizations parametrized by S 2 - { p t } . We provide semiclassical asymptotics for this quantization. [ABSTRACT FROM AUTHOR]

We study the holomorphic vector bundles E over the twistor space Tw (M) of a compact simply connected hyperkähler manifold M. We give a characterization of the semistability condition for E in terms of its restrictions to the holomorphic sections of the holomorphic twistor projection π : Tw (M) ⟶ CP 1 . It is shown that if E admits a holomorphic connection, then E is holomorphically trivial and the holomorphic connection on E is trivial as well. For any irreducible vector bundle E on Tw (M) of prime rank, we prove that its restriction to the generic fibre of π is stable. On the other hand, for a K3 surface M, we construct examples of irreducible vector bundles of any composite rank on Tw (M) whose restriction to every fibre of π is non-stable. We have obtained a new method of constructing irreducible vector bundles on hyperkähler twistor spaces; this method is employed in constructing these examples. [ABSTRACT FROM AUTHOR]

Given a smooth complex projective variety M and a smooth closed curve X ⊂ M such that the homomorphism of fundamental groups π 1 (X) ⟶ π 1 (M) is surjective, we study the restriction map of Higgs bundles, namely from the Higgs bundles on M to those on X. In particular, we investigate the interplay between this restriction map and various types of branes contained in the moduli spaces of Higgs bundles on M and X. We also consider the setup where a finite group is acting on M via holomorphic automorphisms or anti-holomorphic involutions, and the curve X is preserved by this action. Branes are studied in this context. [ABSTRACT FROM AUTHOR]

Let E be an oriented vector bundle of rank 4 over a torus T 2 with a metric h and E ^ the twistor space associated with E. We will study the horizontality in E ^ with respect to the connection ∇ ^ induced by an h-connection ∇ . We will describe the subset of the fiber E ^ a of E ^ at a point a ∈ T 2 given by horizontality along normal polygonal curves in R 2 for an initial value at a in the case where the subset is finite. We will see that if E ^ has a partially horizontal section Ω , then there exists an h-connection ∇ ′ related to ∇ such that h, ∇ ′ and Ω give a Kähler structure of E. We will make analogous discussions for an oriented vector bundle of rank 4 over T 2 with a neutral metric. [ABSTRACT FROM AUTHOR]

In this paper, we show that if (X, g) is an oriented four-dimensional Einstein manifold which is self-dual or anti-self-dual then superminimal surfaces in X of appropriate spin enjoy the Calabi–Yau property, meaning that every immersed surface of this type from a bordered Riemann surface can be uniformly approximated by complete superminimal surfaces with Jordan boundaries. The proof uses the theory of twistor spaces and the Calabi–Yau property of holomorphic Legendrian curves in complex contact manifolds. [ABSTRACT FROM AUTHOR]

We construct new twistorial examples of non-Kähler almost Hermitian manifolds with Hermitian Ricci tensor by means of a natural almost Hermitian structures on the twistor space of an almost Hermitian four manifold. [ABSTRACT FROM AUTHOR]

A conformal map from a Riemann surface to the Euclidean four-space is explained in terms of its twistor lift. A local factorization of a differential of a conformal map is obtained. As an application, the factorization of a differential provides an upper bound of the area of a super-conformal map around a branch point. [ABSTRACT FROM AUTHOR]

We show that a conformal connection on a closed oriented surface $$\Sigma $$ of negative Euler characteristic preserves precisely one conformal structure and is furthermore uniquely determined by its unparametrised geodesics. As a corollary it follows that the unparametrised geodesics of a Riemannian metric on $$\Sigma $$ determine the metric up to constant rescaling. It is also shown that every conformal connection on the $$2$$ -sphere lies in a complex $$5$$ -manifold of conformal connections, all of which share the same unparametrised geodesics. [ABSTRACT FROM AUTHOR]

We explicitly describe all $${ SO }(7)$$-invariant almost quaternion-Hermitian structures on the twistor space of the six-sphere and determine the types of their intrinsic torsion. [ABSTRACT FROM AUTHOR]

We initiate the study of the generalized quaternionic manifolds by classifying the generalized quaternionic vector spaces, and by giving two classes of nonclassical examples of such manifolds. Thus, we show that any complex symplectic manifold is endowed with a natural (nonclassical) generalized quaternionic structure, and the same applies to the heaven space of any three-dimensional Einstein-Weyl space. In particular, on the product $$Z$$ of any complex symplectic manifold $$M$$ and the sphere, there exists a natural generalized complex structure, with respect to which $$Z$$ is the twistor space of $$M$$. [ABSTRACT FROM AUTHOR]

In the paper we investigate evolution of the relativistic particle (massive and massless) with spin defined by Hamiltonian containing the terms with momentum-spin-orbit coupling. We integrate the corresponding Hamiltonian equations in quadratures and express their solutions in terms of elliptic functions. [ABSTRACT FROM AUTHOR]

Let M be a hyperkähler manifold, and F a reflexive sheaf on M. Assume that F (away from its singularities) admits a connection ▿ with a curvature Θ which is invariant under the standard SU(2)-action on 2-forms. If Θ is square-integrable, such sheaf is called hyperholomorphic. Hyperholomorphic sheaves were studied at great length in [21]. Such sheaves are stable and their singular sets are hyperkähler subvarieties in M. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessary L-integrable. We show that such sheaves are polystable. [ABSTRACT FROM AUTHOR]

In this article, we show that the space of nodal rational curves, which is so called a Severi variety (of rational curves), on any non-singular projective surface is always equipped with a natural Einstein-Weyl structure, if the space is 3-dimensional. This is a generalization of the Einstein-Weyl structure on the space of smooth rational curves on a complex surface, given by Hitchin. As geometric objects naturally associated to Einstein-Weyl structure, we investigate null surfaces and geodesics on the Severi varieties. Also, we see that if the projective surface has an appropriate real structure, then the real locus of the Severi variety becomes a positive definite Einstein-Weyl manifold. Moreover, we construct various explicit examples of rational surfaces having 3-dimensional Severi varieties of rational curves. [ABSTRACT FROM AUTHOR]

We discuss the harmonic spheres conjecture that the space of harmonic maps of the Riemann sphere into the loop space of a compact Lie group G are related to the moduli space of Yang-Mills G-fields on the four-dimensional Euclidean space. [ABSTRACT FROM AUTHOR]

We describe a new construction of anti-self-dual metrics on four-manifolds. These metrics are characterized by the property that their twistor spaces project as affine line bundles over surfaces. To any affine bundle with the appropriate sheaf of local translations, we associate a solution of a second-order partial differential equations system D2 V = 0 on a five-dimensional manifold $${\mathbf{Y}}$$. The solution V and its differential completely determine an anti-self-dual conformal structure on an open set in { V = 0}. We show how our construction applies in the specific case of conformal structures for which the twistor space $${\mathcal{Z}}$$ has $${ \dim\left|-\frac{1}{2}K_\mathcal{Z}\right|\geq 2}$$, projecting thus over $${\mathbb C\mathbb P_2}$$ with twistor lines mapping onto plane conics. [ABSTRACT FROM AUTHOR]

We construct complete noncompact Riemannian metrics with G 2-holonomy on noncompact orbifolds that are ℝ3-bundles with the twistor space [InlineMediaObject not available: see fulltext.] as a spherical fiber. [ABSTRACT FROM AUTHOR]

We consider manifolds equipped with a foliation ℱ of codimension 4 q, and an almost quaternionic structure Q on the transversal bundle of ℱ. After discussing conditions of projectability and integrability of Q, we study the transversal twistor space Zℱ which, by definition, consists of the Q-compatible almost complex structures. We show that Zℱ can be endowed with a lifted foliation and two natural almost complex structures J1, J2 on the transversal bundle of . We establish the conditions which ensure the projectability of J1 and J2, and the integrability of J1 ( J2 is never integrable). [ABSTRACT FROM AUTHOR]

A search for invariants of second order ODE systems under the class of point transformations, which mix the parameter and the dependent variables, uncovers a torsion tensor generalizing part of the curvature tensor of an affine connection. We study the geometry of ODE systems for which this torsion vanishes. These are the ODE systems for which deformations of solutions fixing a point constitute a field of Segré varieties in the tangent bundle of the locally defined space of solutions. Conversely, a field of Segré varieties for which certain differential invariants vanish induces a torsion-free ODE system on the space of solutions to a natural PDE system. The geometry on the solution space is used to produce first integrals for torsion-free ODE systems, given as algebraic invariants of a curvature tensor involving up to fourth derivatives of the equations. In the generic case, there are enough first integrals to solve the equations explicitly in spite of the absence of symmetry. In the case of torsion-free ODE pairs, the field of Segré varieties is equivalent to a half-flat split signature conformal structure, and we characterize in terms of curvature those systems having an abundance of totally geodesic surfaces. [ABSTRACT FROM AUTHOR]

We study the algebraic dimension of twistor spaces of positive type over 4CP2. We show that such a twistor space is Moishezon if and only if its anti-canonical class is not nef. More precisely, we show the equivalence of being Moishezon with the existence of a smooth rational curve having negative intersection number with the anticanonical class. Furthermore, we give precise information on the dimension and base locus of the fundamental linear system |-1/2|. This implies, for example, dim|-1/2K| ≤ a(Z). We characterize those twistor spaces over 4CP2, which contain a pencil of divisors of degree one by the property dim|-1/2K| = 3. [ABSTRACT FROM AUTHOR]