An order four automorphism of a Lie algebra gives rise to an integrable system discussed by Terng. We show that solutions of this system may be identified with certain vertically harmonic twistor lifts of conformal maps of surfaces in a Riemannian symmetric space.
Specialising to 4-dimensional target, we find that surfaces with holomorphic mean curvature in 4-dimensional spaces with constant sectional or holomorphic sectional curvatures constitute an integrable system as do Hamiltonian stationary Lagrangian surfaces in a 4-dimensional Hermitian symmetric space (this last being a result of Helein-Romon).
We introduce integrable complex structures on twistor spaces fibered over complex manifolds. We then show, in particular, that the twistor spaces associated with generalized Kahler, SKT and strong HKT manifolds all naturally admit complex structures. Moreover, in the strong HKT case we construct a metric and three compatible complex structures on the twistor space that have equal torsions.
We study the local geometry of 4-manifolds equipped with a \emph{para-Kähler-Einstein} (pKE) metric, a special type of split-signature pseudo-Riemannian metric, and their associated \emph{twistor distribution}, a rank 2 distribution on the 5-dimensional total space of the circle bundle of self-dual null 2-planes. For pKE metrics with nonvanishing Einstein constant this twistor distribution has exactly two integral leaves and is `maximally non-integrable' on their complement, a so-called (2,3,5)-distribution. Our main result establishes a simple correspondence between the anti-self-dual Weyl te…
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 $G/G_0 \hookrightarrow Σ(G/H)$. We begin the paper by an example: $G/H=\R^4$. We study also the structure of 4-symmetric bundles over Riemannian symmetric spaces.
Every almost Hermitian structure $(g,J)$ on a four-manifold $M$ determines a hypersurface $Σ_J$ in the (positive) twistor space of $(M,g)$ consisting of the complex structures anti-commuting with $J$. In this note we find the conditions under which $Σ_J$ is minimal with respect to a natural Riemannian metric on the twistor space in the cases when $J$ is integrable or symplectic. Several examples illustrating the obtained results are also discussed.
We describe complex twistor spaces over inner 3-symmetric spaces $G/H$, such that $H$ acts transitively on the fibre. Like in the symmetric case, these are flag manifolds $G/K$ where $K$ is the centralizer of a torus in $G$. Moreover, they carry an almost complex structure defined using the horizontal distribution of the normal connection on $G/H$, that coincides with the complex structure associated to a parabolic subgroup $P \subset G^{\mathbb C}$ if it is integrable. Conversely, starting from a complex flag manifold $G^{\mathbb C}/P$, there exists a natural fibration with complex fibres on …
We use the compactified twistor correspondence for the (2+1)-dimensional integrable chiral model to prove a conjecture of Ward. In particular, we construct the correspondence space of a compactified twistor fibration and use it to prove that the second Chern numbers of the holomorphic vector bundles, corresponding to the uniton solutions of the integrable chiral model, equal the third homotopy classes of the restricted extended solutions of the unitons. Therefore we deduce that the total energy of a time-dependent uniton is proportional to the second Chern number.
We show that given a harmonic map $φ$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $φ$ (or its negative) if and only if it is nilconformal. In the case of harmonic maps of finite uniton number, we give algebraic formulae in terms of holomorphic data which describes their extended solutions. In particular, this gives explicit formulae for the twistor lifts of all harmonic maps of finite uniton number from a surface to the above symmetric spaces.
A hypercomplex structure on a differentiable manifold consists of three integrable almost complex structures that satisfy quaternionic relations. If, in addition, there exists a metric on the manifold which is Hermitian with respect to the three structures, and such that the corresponding Hermitian forms are closed, the manifold is said to be hyperkaehler. In the paper "Non-Hermitian Yang-Mills connections", Kaledin and Verbitsky proved that the twistor space of a hyperkaehler manifold admits a balanced metric; these were first studied in the article "On the existence of special metrics in com…
Let M be a G2-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 G2 manifold. We interpret G2-instanton bundles as CR-holomorphic bundles on its twistor space.
The transversal twistor space of a foliation F of an even codimension is the bundle ZF of the complex structures of the fibers of the transversal bundle of F. On ZF, there exists a foliation F' by covering spaces of the leaves of F, and any Bott connection of F produces an ordered pair (I,J) of transversal almost complex structures of F'. The existence of a Bott connection which yields a structure I that is projectable to the space of leaves is equivalent to the fact that F is a transversally projective foliation. A Bott connection which yields a projectable structure J exists iff F is a trans…
We consider the unique Hermitian connection with totally skew-symmetric torsion on a Hermitian manifold. We prove that if the torsion is parallel and the holonomy is Sp(n)U(1), considered as a subgroup of U(2n) x U(1), then the manifold is locally isomorphic to the twistor space of a quaternionic Kaehler manifold with positive scalar curvature. If the manifold is complete, then it is globally isomorphic to such a twistor space.
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Plücker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Plücker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie geometry in $S^3.$ Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. We define discret…
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-Kähler symmetric space. We then consider orthogonal complex structures on the pseudo-sphere, only to prove such a structure cannot exist.
Motivated by generalized geometry (à la Hitchin), we discuss the integrability conditions for four natural almost complex structures on the product bundle ${\mathcal Z}\times {\mathcal Z}\to M$, where ${\mathcal Z}$ is the twistor space of a Riemannian 4-manifold $M$ endowed with a metric connection $D$ with skew-symmetric torsion. These structures are defined by means of the connection $D$ and four (Kähler) complex structures on the fibres of this bundle. Their integrability conditions are interpreted in terms of Weyl geometry and this is used to supply examples satisfying the conditions.
For $n \geq 1$, the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ of the conformal $2n$-sphere is biholomorphic to the Zariski closure, taken in the complex Grassmannian manifold $\mathbf{G}(n+1, 2n+2)$, of the set of graphs of skew-symmetric linear endomorphism of $\mathbb{C}^{n+1}$. We use this fact to describe a natural stratification of the twistor space $\mathfrak{Z}(\mathbb{S}^{2n})$ with $n \geq 3$, in terms of what we have called {\it generalised complex orthogonal Stiefel manifolds} of $\mathbb{C}^{n+1}$. In particular, the twistor space $\mathfrak{Z}(\mathbb{S}^{6})$ is biholomorphic…
We present a geometric construction and characterization of $2n$-dimensional split-signature conformal structures endowed with a twistor spinor with integrable kernel. The construction is regarded as a modification of the conformal Patterson--Walker metric construction for $n$-dimensional projective manifolds. The characterization is presented in terms of the twistor spinor and an integrability condition on the conformal Weyl curvature. We further derive complete description of Einstein metrics and infinitesimal conformal symmetries in terms of suitable projective data. Finally, we obtain an e…
Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space
Dalian University of Technology（大连理工大学）；Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）；Warsaw University（华沙大学）；Huzhou Normal University（湖州师范学院）
We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost para-quaternionic manifold constructed with the help of a para-quaternionic connection. We show that if there is an integrable structure it is independent on the para-quaternionic connection. In dimension four, we express the anti-self-duality condition in terms of the Riemannian Ricci forms with respect to the associated para-quaternionic structure.
The target space of a (4,0) supersymmetric two-dimensional sigma model with Wess-Zumino term has a connection with totally skew-symmetric torsion and holonomy contained in Sp(n)Sp(1) (resp. Sp(n)), QKT (resp. HKT)-spaces. We study the geometry of QKT, HKT manifold and their twistor spaces. We show that the Swann bundle of a QKT manifold admits a HKT structure with special symmetry if and only if the twistor space of the QKT manifold admits an almost hermitian structure with totally skew-symmetric Nijenhuis tensor, thus connecting two structures arising from quantum field theories and supersymm…
We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $Υ$ representing this reduction is found to be given by a ternary symmetric form with special properties. A 5-dimensional manifold $(M,g,Υ)$ with Riemannian metric $g$ and ternary form generated by such a tensor has a corresponding SO(3) structure, whose Gray-Hervella type classification is established using so(3)-valued connections with torsion. Structures with antisymmetric torsions, we call them the nearly integrable SO(3) structures, are studied in detail. In parti…
We show that a harmonic map from a Riemann surface into the exceptional symmetric space $G_2/{\mathrm SO}(4)$ has a $J_2$-holomorphic twistor lift into one of the three flag manifolds of $G_2$ if and only if it is `nilconformal', i.e., has nilpotent derivative. Then we find relationships with almost complex maps from a surface into the 6-sphere; this enables us to construct examples of nilconformal harmonic maps into $G_2/{\mathrm SO}(4)$ which are not of finite uniton number, and which have lifts into any of the three twistor spaces. Harmonic maps of finite uniton number are all nilconformal;…
Four-dimensional quaternion-Kahler metrics, or equivalently self-dual Einstein spaces M, are known to be encoded locally into one real function h subject to Przanowski's Heavenly equation. We elucidate the relation between this description and the usual twistor description for quaternion-Kahler spaces. In particular, we show that the same space M can be described by infinitely many different solutions h, associated to different complex (local) submanifolds on the twistor space, and therefore to different (local) integrable complex structures on M. We also study quaternion-Kahler deformations o…
In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for $m$-dimensional Riemannian manifolds endowed with a $(m-1)$-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to $7$-manifolds endowed with a $G_2$-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold $(M, g)$ has a VCP s…
Using the twistor correspondence, this article gives a one-to-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Cech coboundary of 0-cochains on an elliptic curve covered by two annuli. The classes admitting Kahler representatives are described; each such class contains a circle of Kahler metrics. This gives new local examples of scalar-flat Kahler surfaces and generalises work of Joyce who con…
We give a detailed account of the geometric correspondence between a smooth complex projective quadric hypersurface $\mathcal{Q}^n$ of dimension $n \geq 3$, and its twistor space $\mathbb{PT}$, defined to be the space of all linear subspaces of maximal dimension of $\mathcal{Q}^n$. Viewing complex Euclidean space $\mathbb{CE}^n$ as a dense open subset of $\mathcal{Q}^n$, we show how local foliations tangent to certain integrable holomorphic totally null distributions of maximal rank on $\mathbb{CE}^n$ can be constructed in terms of complex submanifolds of $\mathbb{PT}$. The construction is ill…
HyperCR Einstein--Weyl equations in 2+1 dimensions reduce to a pair of quasi-linear PDEs of hydrodynamic type. All solutions to this hydrodynamic system can be in principle constructed from a twistor correspondence, thus establishing the integrability. Simple examples of solutions including the hydrodynamic reductions yield new Einstein--Weyl structures.
In this paper, we study non integrable distributions in a Riemannian manifold with a semi-symmetric metric connection, a semi-symmetric non-metric connection and a statistical connection. We obtain the Gauss, Codazzi, and Ricci equations for non integrable distributions with respect to the semi-symmetric metric connection, the semi-symmetric non-metric connection and the statistical connection. As applications, we obtain Chen's inequalities for non integrable distributions of real space forms endowed with a semi-symmetric metric connection and a semi-symmetric non-metric connection. We give so…
Let $\widetilde{\cal J}(S^{2n})$ be the set of orthogonal complex structures on $TS^{2n}$. We show that the twistor space $\widetilde{\cal J}(S^{2n})$ is a Kaehler manifold. Then we show that an orthogonal almost complex structure $J_f$ on $S^{2n}$ is integrable if and only if the corresponding section $f\colon\; S^{2n}\to \widetilde{\cal J}(S^{2n}) $ is holomorphic. These shows there is no integrable orthogonal complex structure on the sphere $S^{2n}$ for $n>1$. We also show that there is no complex structure in a neighborhood of the space $\widetilde{\cal J}(S^{2n})$. The method is to study …
In this paper, the description of biharmonic map equation in terms of the Maurer-Cartan form for all smooth map of a compact Riemannian manifold into a Riemannian symmetric space $(G/K,h)$ induced from the bi-invariant Riemannian metric $h$ on $G$ is obtained. By this formula, all biharmonic curves into symmetric spaces are determined, and all the biharmonic maps of an open domain of ${\Bbb R}^2$ with the standard Riemannian metric into $(G/K,h)$ are determined.
We consider the twistor theory of nilconformal harmonic maps from a Riemann surface into the Cayley plane $\mathbf{O} P^2=F_4/\mathrm{Spin}(9)$. By exhibiting this symmetric space as a submanifold of the Grassmannian of $10$-dimensional subspaces of the fundamental representation of $F_4$, techniques and constructions similar to those used in earlier works on twistor constructions of nilconformal harmonic maps into classical Grassmannians can also be applied in this case. The originality of our approach lies on the use of the classification of nilpotent orbits in Lie algebras as described by D…
We consider the generalized Kahler structures (g,J_+,J_-) that arise on a hyperkahler manifold (M,g,I,J,K) when we choose J_+ and J_- from the twistor space of M. We find a relation between semichiral and arctic superfields which can be used to determine the generalized Kahler potential for hyperkahler manifolds whose description in projective superspace is fully understood. We use this relation to determine an S^2-family of generalized Kahler potentials for Euclidean space and for the Eguchi-Hanson geometry. Cotangent bundles of Hermitian symmetric spaces constitute a class of hyperkahler man…
We study the Hitchin component in the space of representations of the fundamental group of a Riemann surface into a split real simple Lie group in the rank 2 case. We prove that such representations are described by a conformal structure and class of Higgs bundle we call cyclic and we show cyclic Higgs bundles correspond to a form of the affine Toda equations. In each case we relate cyclic Higgs bundles to geometric structures on the surface. We elucidate the geometry of generic 2-plane distributions in 5 dimensions, relating it to a parabolic geometry associated to the split real form of $G_2…
We give holomorphic Chern-Simons-like action functionals on supertwistor space for self-dual supergravity theories in four dimensions, dealing with N=0,...,8 supersymmetries, the cases where different parts of the R-symmetry are gauged, and with or without a cosmological constant. The gauge group is formally the group of holomorphic Poisson transformations of supertwistor space where the form of the Poisson structure determines the amount of R-symmetry gauged and the value of the cosmological constant. We give a formulation in terms of a finite deformation of an integrable \dbar-operator on a …
It is well-known that sigma-models with symmetric target spaces are classically integrable. At the example of the model with target space the flag manifold U(3)/U(1)^3 -- a non-symmetric space -- we show that the introduction of torsion allows to cast the equations of motion in the form of a zero-curvature condition for a one-parametric family of connections, which can be a sign of integrability of the theory. We also elaborate on geometric aspects of the proposed model.
On $4$-symmetric symplectic spaces, invariant almost complex structures -- up to sign -- arise in pairs. We exhibit some $4$-symmetric symplectic spaces, with a pair of "natural" compatible (usually not positive) invariant almost complex structures, one of them being integrable and the other one being maximally non integrable (i.e. the image of its Nijenhuis tensor at any point is the whole tangent space at that point). The integrable one defines a pseudo-Kähler Einstein metric on the manifold, and the non integrable one is Ricci Hermitian (in the sense that the almost complex structure preser…
This work has its origins in an attempt to describe systematically the integrable geometries and gauge theories in dimensions one to four related to twistor theory. In each such dimension, there is a nondegenerate integrable geometric structure, governed by a nonlinear integrable differential equation, and each solution of this equation determines a background geometry on which, for any Lie group $G$, an integrable gauge theory is defined. In four dimensions, the geometry is selfdual conformal geometry and the gauge theory is selfdual Yang-Mills theory, while the lower-dimensional structures a…
We study supersymmetric harmonic maps from the point of view of integrable system. It is well known that harmonic maps from R^2 into a symmetric space are solutions of a integrable system . We show here that the superharmonic maps from R^{2|2} into a symmetric space are solutions of a integrable system, more precisely of a first elliptic integrable system in the sense of C.L. Terng and that we have a Weierstrass-type representation in terms of holomorphic potentials (as well as of meromorphic potentials). In the end of the paper we show that superprimitive maps from R^{2|2} into a 4-symmetric …
