92 results on '"Twistor space"'
Search Results
2. The Adjunction Inequality for Weyl-Harmonic Maps
- Author
-
Ream Robert
- Subjects
almost-complex manifolds ,twistor space ,weyl geometry ,32q60 ,53c28 ,53c43 ,Mathematics ,QA1-939 - Abstract
In this paper we study an analog of minimal surfaces called Weyl-minimal surfaces in conformal manifolds with a Weyl connection (M4, c, D). We show that there is an Eells-Salamon type correspondence between nonvertical 𝒥-holomorphic curves in the weightless twistor space and branched Weyl-minimal surfaces. When (M, c, J) is conformally almost-Hermitian, there is a canonical Weyl connection. We show that for the canonical Weyl connection, branched Weyl-minimal surfaces satisfy the adjunction inequality
- Published
- 2020
- Full Text
- View/download PDF
3. Geodesic rigidity of conformal connections on surfaces
- Author
-
Thomas Mettler
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Geodesic ,General Mathematics ,Conformal map ,Twistor space ,Projective structures ,Mathematics - Algebraic Geometry ,symbols.namesake ,Geodesic rigidity ,Euler characteristic ,Conformal connections ,FOS: Mathematics ,Primary 53A20 ,Secondary 53C24 ,53C28 ,Algebraic Geometry (math.AG) ,Mathematics ,Quantitative Biology::Biomolecules ,Surface (topology) ,Manifold ,Differential Geometry (math.DG) ,Metric (mathematics) ,symbols ,Conformal connection ,Mathematics::Differential Geometry - Abstract
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., Comment: 16 pages, exposition improved, references added
- Published
- 2021
4. Twistor spaces on foliated manifolds
- Author
-
Robert Wolak and Rouzbeh Mohseni
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Dynamical Systems ,General Mathematics ,Twistor theory ,High Energy Physics::Theory ,Differential Geometry (math.DG) ,Normal bundle ,FOS: Mathematics ,Foliation (geology) ,Twistor space ,Mathematics::Differential Geometry ,53C12, 53C28 ,Mathematics::Symplectic Geometry ,Orbifold ,Mathematics - Abstract
The theory of twistors on foliated manifolds is developed. We construct the twistor space of the normal bundle of a foliation. It is demonstrated that the classical constructions of the twistor theory lead to foliated objects and permit to formulate and prove foliated versions of some well-known results on holomorphic mappings. Since any orbifold can be understood as the leaf space of a suitably defined Riemannian foliation we obtain orbifold versions of the classical results as a simple consequence of the results on foliated mappings.
- Published
- 2021
5. Energy of sections of the Deligne–Hitchin twistor space
- Author
-
Markus Roeser, Florian Beck, and Sebastian Heller
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Twistor methods in differential geometry ,General Mathematics ,Holomorphic function ,Computer Science::Digital Libraries ,01 natural sciences ,Twistor theory ,Mathematics::Algebraic Geometry ,Line bundle ,0103 physical sciences ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,ddc:510 ,Relationships between algebraic curves and integrable systems ,Mathematics::Symplectic Geometry ,Hyper-Kähler and quaternionic Kähler geometry ,Mathematics ,Energy functional ,Meromorphic function ,Mathematics::Complex Variables ,Vector bundles on curves and their moduli ,010102 general mathematics ,Differential geometric aspects of harmonic maps ,Dewey Decimal Classification::500 | Naturwissenschaften::510 | Mathematik ,Moduli space ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Computer Science::Mathematical Software ,Twistor space ,010307 mathematical physics ,Mathematics::Differential Geometry - Abstract
We study a natural functional on the space of holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface, generalizing the energy of equivariant harmonic maps corresponding to twistor lines. We give a link to a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. Moreover, we prove that for a certain class of real holomorphic sections of the Deligne-Hitchin moduli space, the functional is basically given by the Willmore energy of corresponding (equivariant) conformal map to the 3-sphere. As an application we use the functional to distinguish new components of real holomorphic sections of the Deligne-Hitchin moduli space from the space of twistor lines., 33 pages
- Published
- 2021
- Full Text
- View/download PDF
6. On the existence of compact scalar-flat Kähler surfaces
- Author
-
M. PONTECORVO
- Subjects
kähler metric ,twistor space ,Mathematics ,QA1-939 - Abstract
A compact compler surface with non-trivial canonical bundle and a Kähler metric ofzero scalar curvature must be a ruled surface. It is also known that not every ruled surface can admit such extremal Kähler metrics. In this paper we review recentjoint work with Kim and LeBrun in which deforma- tion theory of pairs of singular complex spaces it is used to show that any ruled surface (M, J) has blow-ups (M, J) which admit Kähler metrics of zero scalar curvature.
- Published
- 1995
7. Deligne pairings and families of rank one local systems on algebraic curves
- Author
-
Gerard Freixas i Montplet, Richard Wentworth, Institut de Mathématiques de Jussieu - Paris Rive Gauche (IMJ-PRG (UMR_7586)), Université Paris Diderot - Paris 7 (UPD7)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), and Department of Mathematics, University of Maryland
- Subjects
Pure mathematics ,Holomorphic function ,58J52 ,01 natural sciences ,symbols.namesake ,Mathematics::Algebraic Geometry ,Line bundle ,Mathematics::K-Theory and Homology ,Analytic torsion ,0101 mathematics ,Connection (algebraic framework) ,[MATH]Mathematics [math] ,Mathematics ,Meromorphic function ,Algebra and Number Theory ,Mathematics::Complex Variables ,Riemann surface ,010102 general mathematics ,14C40 ,16. Peace & justice ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,symbols ,Twistor space ,Geometry and Topology ,Algebraic curve ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Analysis - Abstract
For smooth families $\mathcal{X} \to S$ of projective algebraic curves and holomorphic line bundles $\mathcal{L, M} \to X$ equipped with flat relative connections, we prove the existence of a canonical and functorial “intersection” connection on the Deligne pairing $\langle \mathcal{L, M} \rangle \to S$. This generalizes the construction of Deligne in the case of Chern connections of hermitian structures on $\mathcal{L}$ and $\mathcal{M}$. A relationship is found with the holomorphic extension of analytic torsion, and in the case of trivial fibrations we show that the Deligne isomorphism is flat with respect to the connections we construct. Finally, we give an application to the construction of a meromorphic connection on the hyperholomorphic line bundle over the twistor space of rank one flat connections on a Riemann surface.
- Published
- 2020
8. Quaternion-Kähler manifolds near maximal fixed point sets of $S^{1}$-symmetries
- Author
-
Aleksandra Borówka
- Subjects
Pure mathematics ,Applied Mathematics ,010102 general mathematics ,Connection (principal bundle) ,Kähler manifold ,Fixed point ,Submanifold ,01 natural sciences ,Line bundle ,0103 physical sciences ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Hyperkähler manifold ,Distribution (differential geometry) ,Mathematics - Abstract
Using quaternionic Feix–Kaledin construction, we provide a local classification of quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry with the fixed point set submanifold S of maximal possible dimension. For any real-analytic Kähler manifold S equipped with a line bundle with a real-analytic unitary connection with curvature proportional to the Kähler form, we explicitly construct a holomorphic contact distribution on the twistor space obtained by the quaternionic Feix–Kaledin construction from these data. Conversely, we show that quaternion-Kähler metrics with a rotating $$S^1$$S1-symmetry induce on the fixed point set of maximal dimension a Kähler metric together with a unitary connection on a holomorphic line bundle with curvature proportional to the Kähler form and the two constructions are inverse to each other. Moreover, we study the case when S is compact, showing that in this case the quaternion-Kähler geometry is determined by the Kähler metric on the fixed point set (of maximal possible dimension) and by the contact line bundle along the corresponding submanifold on the twistor space. Finally, we relate the results to the c-map construction showing that the family of quaternion-Kähler manifolds obtained from a fixed Kähler metric on S by varying the line bundle and the hyperkähler manifold obtained by hyperkähler Feix–Kaledin construction from S are related by hyperkähler/quaternion-Kähler correspondence.
- Published
- 2020
9. A note on Lagrangian submanifolds of twistor spaces and their relation to superminimal surfaces
- Author
-
Reinier Storm
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Complex Variables ,Circle bundle ,010102 general mathematics ,Structure (category theory) ,Fibration ,Surface (topology) ,Submanifold ,01 natural sciences ,Hermitian matrix ,Twistor theory ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,53C28, 53C42 ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics - Abstract
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 Kahler structure as well as the nearly Kahler structure. Moreover, it is shown that these Lagrangian submanifolds are minimal submanifolds.
- Published
- 2019
10. An inclusive immersion into a quaternionic manifold and its invariants
- Author
-
Kazuyuki Hasegawa
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Holomorphic function ,01 natural sciences ,Twistor theory ,53C26 ,53C28 ,Quaternionic representation ,0103 physical sciences ,Immersion (mathematics) ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,Algebraic curve ,0101 mathematics ,Invariant (mathematics) ,Quaternionic projective space ,Mathematics - Abstract
金沢大学人間社会研究域学校教育系 / Institute of Human and Social science, Teacher Education, We introduce a quaternionic invariant for an inclusive immersion into a quaternionic manifold, which is a quaternionic object corresponding to the Willmore functional. The lower bound of this invariant is given by topological invariant and the equality case can be characterized in terms of the natural twistor lift. When the ambient manifold is the quaternionic projective space and the natural twistor lift is holomorphic, we obtain a relation between the quaternionic invariant and the degree of the image of the natural twistor lift as an algebraic curve. Moreover the first variation formula for the invariant is obtained. As an application of the formula, if the natural twistor lift is a harmonic section, then the surface is a stationary point under any variations such that the induced complex structures do not vary. © 2017, Springer-Verlag Berlin Heidelberg., Embargo Period 12 months
- Published
- 2017
11. G 2-holonomy metrics connected with a 3-Sasakian manifold.
- Author
-
Bazaĭkin, Ya. V. and Malkovich, E. G.
- Subjects
- *
HOLONOMY groups , *RIEMANNIAN manifolds , *SASAKIAN manifolds , *FUNCTION spaces , *MATHEMATICS - Abstract
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]
- Published
- 2008
- Full Text
- View/download PDF
12. Non-Abelian Hodge Theory and Related Topics
- Author
-
Pengfei Huang, Département de Mathématiques [Nice], Université Nice Sophia Antipolis (... - 2019) (UNS), Université Côte d'Azur (UCA)-Université Côte d'Azur (UCA), and COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Université Côte d'Azur (UCA)
- Subjects
Pure mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,01 natural sciences ,Stratification (mathematics) ,oper ,Mathematics - Algebraic Geometry ,stratification ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,conformal limit ,0101 mathematics ,Abelian group ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Hodge theory ,010102 general mathematics ,Hitchin section ,Moduli space ,twistor space ,non-Abelian Hodge theory ,14D20, 14D21, 32G20, 53C07, 57N80 ,Higgs boson ,moduli space ,$\lambda$-connection ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Analysis - Abstract
International audience; This paper is a survey aimed on the introduction of non-Abelian Hodge theory that gives the correspondence between flat bundles and Higgs bundles. We will also introduce some topics arising from this theory, especially some recent developments on the study of the relevant moduli spaces together with some interesting open problems.
- Published
- 2019
13. A family of integrable perturbed Kepler systems
- Author
-
Anatol Odzijewicz, E. Wawreniuk, and Aneta Sliżewska
- Subjects
Hamiltonian mechanics ,Integrable system ,010102 general mathematics ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Mathematical Physics (math-ph) ,53D20, 53D05, 70F15, 70F05, 70E20 ,Poisson distribution ,01 natural sciences ,Kepler ,Jacobi elliptic functions ,symbols.namesake ,0103 physical sciences ,symbols ,Twistor space ,010307 mathematical physics ,0101 mathematics ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
In the framework of the Poisson geometry of twistor space we consider a family of perturbed 3-dimensional Kepler systems. We show that Hamilton equations of this systems are integrated by quadratures. Their solutions for some subcases are given explicitly in terms of Jacobi elliptic functions., 26 pages
- Published
- 2019
14. Projective geometry and the quaternionic Feix-Kaledin construction
- Author
-
David M. J. Calderbank and Aleksandra Borówka
- Subjects
Mathematics - Differential Geometry ,Connection (fibred manifold) ,Pure mathematics ,Mathematics(all) ,General Mathematics ,Curvature ,01 natural sciences ,symbols.namesake ,FOS: Mathematics ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Complex line ,Riemann surface ,Applied Mathematics ,010102 general mathematics ,Submanifold ,Manifold ,Differential Geometry (math.DG) ,symbols ,Twistor space ,Mathematics::Differential Geometry ,Complex manifold ,53A20, 53B10, 53C26, 53C28, 32L25 - Abstract
Starting from a complex manifold S with a real-analytic c-projective structure whose curvature has type (1,1), and a complex line bundle L with a connection whose curvature has type (1,1), we construct the twistor space Z of a quaternionic manifold M with a quaternionic circle action which contains S as a totally complex submanifold fixed by the action. This extends a construction of hypercomplex manifolds, including hyperkaehler metrics on cotangent bundles, obtained independently by B. Feix and D. Kaledin. When S is a Riemann surface, M is a self-dual conformal 4-manifold, and the quotient of M by the circle action is an Einstein-Weyl manifold with an asymptotically hyperbolic end, and our construction coincides with a construction presented by the first author in a previous paper. The extension also applies to quaternionic Kaehler manifolds with circle actions, as studied by A. Haydys and N. Hitchin., 28 pages, (v2) added material on Swann bundles, quaternionic Kaehler metrics and the Haydys-Hitchin correspondence, (v3) refereed version, restructured content, to appear in TAMS
- Published
- 2019
15. Real Holomorphic Sections of the Deligne–Hitchin Twistor Space
- Author
-
Markus Röser, Indranil Biswas, Sebastian Heller, and Ecole Internationale des Sciences du Traitement de l'Information (EISTI)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph] ,Complex system ,Holomorphic function ,01 natural sciences ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,0103 physical sciences ,FOS: Mathematics ,Compact Riemann surface ,0101 mathematics ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,53C26, 53C28, 14H60 ,Mathematics::Complex Variables ,010102 general mathematics ,Harmonic map ,Statistical and Nonlinear Physics ,Moduli space ,Differential Geometry (math.DG) ,Twistor space ,010307 mathematical physics - Abstract
We study the holomorphic sections of the Deligne-Hitchin moduli space of a compact Riemann surface that are invariant under the natural anti-holomorphic involutions of the moduli space. Their relationships with the harmonic maps are established. As a bi-product, a question of Simpson on such sections, posed in \cite{Si2}, is answered., Comment: Final version; to appear in Communications in Mathematical Physics
- Published
- 2019
16. On deformations of the dispersionless Hirota equation
- Author
-
Wojciech Kryński
- Subjects
Mathematics - Differential Geometry ,Class (set theory) ,Integrable system ,010308 nuclear & particles physics ,010102 general mathematics ,General Physics and Astronomy ,01 natural sciences ,Dispersionless equation ,Twistor theory ,Nonlinear Sciences::Exactly Solvable and Integrable Systems ,Differential Geometry (math.DG) ,Simple (abstract algebra) ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,Geometry and Topology ,0101 mathematics ,Mathematical Physics ,Mathematical physics ,Mathematics - Abstract
The class of hyper-CR Einstein–Weyl structures on R 3 can be described in terms of the solutions to the dispersionless Hirota equation. In the present paper we show that simple geometric constructions on the associated twistor space lead to deformations of the Hirota equation that have been introduced recently by B. Kruglikov and A. Panasyuk. Our method produces also the hyper-CR equation and can be applied to other geometric structures related to different twistor constructions.
- Published
- 2017
17. Pure Connection Formulation, Twistors and the Chase for a Twistor Action for General Relativity
- Author
-
Yannick Herfray, Laboratoire de Physique de l'ENS Lyon ( Phys-ENS ), École normale supérieure - Lyon ( ENS Lyon ) -Université Claude Bernard Lyon 1 ( UCBL ), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Physique de l'ENS Lyon (Phys-ENS), École normale supérieure de Lyon (ENS de Lyon)-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université de Lyon-Centre National de la Recherche Scientifique (CNRS), and École normale supérieure - Lyon (ENS Lyon)-Université Claude Bernard Lyon 1 (UCBL)
- Subjects
High Energy Physics - Theory ,effective Lagrangian: chiral ,General relativity ,FOS: Physical sciences ,integrability ,01 natural sciences ,[ PHYS.HTHE ] Physics [physics]/High Energy Physics - Theory [hep-th] ,Twistor theory ,High Energy Physics::Theory ,twistor ,SU(2) theory ,0103 physical sciences ,Euclidean geometry ,general relativity ,010306 general physics ,Mathematical Physics ,Mathematics ,Mathematical physics ,[PHYS.HTHE]Physics [physics]/High Energy Physics - Theory [hep-th] ,010308 nuclear & particles physics ,Complex line ,Graviton ,Statistical and Nonlinear Physics ,Hermitian matrix ,Connection (mathematics) ,High Energy Physics - Theory (hep-th) ,SU(2) ,gravitation ,Twistor space ,Mathematics::Differential Geometry - Abstract
This paper establishes the relation between traditional results from (euclidean) twistor theory and chiral formulations of General Relativity (GR), especially the pure connection formulation. Starting from a $SU(2)$-connection only we show how to construct natural complex data on twistor space, mainly an almost Hermitian structure and a connection on some complex line bundle. Only when this almost Hermitian structure is integrable is the connection related to an anti-self-dual-Einstein metric and makes contact with the usual results. This leads to a new proof of the non-linear-graviton theorem. Finally we discuss what new strategies this "connection approach" to twistors suggests for constructing a twistor action for gravity. In appendix we also review all known chiral Lagrangians for GR., This is the version published in J.Math.Phys. As compare to the previous version, some paragraph were rewritten to make the text easier to read and some typos corrected
- Published
- 2017
18. Twistor Spaces and Compact Manifolds Admitting Both Kähler and Non-Kähler Structures
- Author
-
Ljudmila Kamenova
- Subjects
Twistor theory ,Pure mathematics ,Simply connected space ,Differentiable manifold ,Twistor space ,Geometry and Topology ,Quaternion ,Surface (topology) ,Mathematical Physics ,Hyperkähler manifold ,Manifold ,Mathematics - Abstract
In this expository paper we review some twistor techniques and recall the problem of finding compact differentiable manifolds that can carry both Kahler and non-Kahler complex structures. Such examples were constructed independently by Atiyah, Blanchard and Calabi in the 1950’s. In the 1980’s Tsanov gave an example of a simply connected manifold that admits both Kahler and non-Kahler complex structures - the twistor space of a $K3$ surface. Here we show that the quaternion twistor space of a hyperkahler manifold has the same property.
- Published
- 2017
19. Quantum Riemannian geometry of phase space and nonassociativity
- Author
-
Shahn Majid and Edwin J. Beggs
- Subjects
Differential form ,General Mathematics ,58B32 ,FOS: Physical sciences ,General Relativity and Quantum Cosmology (gr-qc) ,Riemannian geometry ,Quantum mechanics ,01 natural sciences ,General Relativity and Quantum Cosmology ,symbols.namesake ,Poisson bracket ,Mathematics - Quantum Algebra ,0103 physical sciences ,FOS: Mathematics ,Quantum Algebra (math.QA) ,0101 mathematics ,Connection (algebraic framework) ,Mathematical physics ,Mathematics ,lcsh:Mathematics ,010102 general mathematics ,Quantum gravity ,Noncommutative geometry ,Order (ring theory) ,lcsh:QA1-939 ,Poisson geometry ,81R50 ,Differential geometry ,symbols ,Twistor space ,83C57 ,010307 mathematical physics - Abstract
Noncommutative or `quantum' differential geometry has emerged in recent years as a process for quantizing not only a classical space into a noncommutative algebra (as familiar in quantum mechanics) but also differential forms, bundles and Riemannian structures at this level. The data for the algebra quantisation is a classical Poisson bracket, the data for the quantum differential forms is a Poisson-compatible connection it was recently shown that after this, classical data such as classical bundles, metrics etc. all become quantised in a canonical `functorial' way at least to 1st order in deformation theory. There are, however, fresh compatibility conditions between the classical Riemannian and the Poisson structures as well as new physics such as nonassociativity at 2nd order. We give an introduction to this theory and some details for the case of CP${}^n$ where the commutation relations have the canonical form $[w^i,\bar w^j]=\mathrm{i}\lambda\delta_{ij}$ similar to the proposal of Penrose for quantum twistor space. Our work provides a canonical but ultimately nonassociative differential calculus on this algebra and quantises the metric and Levi-Civita connection at lowest order in $\lambda$., Comment: 14 pages latex
- Published
- 2017
20. Hessian of the natural Hermitian form on twistor spaces
- Author
-
Noël Le Du, Christophe Mourougane, Guillaume Deschamps, Laboratoire de mathématiques de Brest ( LM ), Université de Brest ( UBO ) -Institut Brestois du Numérique et des Mathématiques ( IBNM ), Université de Brest ( UBO ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de mathématiques de Brest (LM), Université de Brest (UBO)-Institut Brestois du Numérique et des Mathématiques (IBNM), Université de Brest (UBO)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,[ MATH.MATH-CV ] Mathematics [math]/Complex Variables [math.CV] ,Closed manifold ,53C28 ,53C26 ,32Q45 ,General Mathematics ,Invariant manifold ,twistor space ,01 natural sciences ,Pseudo-Riemannian manifold ,Twistor theory ,symbols.namesake ,Mathematics - Algebraic Geometry ,High Energy Physics::Theory ,FOS: Mathematics ,Hermitian manifold ,0101 mathematics ,Complex Variables (math.CV) ,hyperkähler manifold ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,4-dimensional Riemannian manifold ,Ricci curvature ,strong KT manifolds ,Mathematics ,quaternionic Kähler manifold ,Mathematics - Complex Variables ,010102 general mathematics ,Mathematical analysis ,Holonomy ,[MATH.MATH-CV]Mathematics [math]/Complex Variables [math.CV] ,Mathematics::Geometric Topology ,[ MATH.MATH-AG ] Mathematics [math]/Algebraic Geometry [math.AG] ,[ MATH.MATH-DG ] Mathematics [math]/Differential Geometry [math.DG] ,Differential Geometry (math.DG) ,[MATH.MATH-DG]Mathematics [math]/Differential Geometry [math.DG] ,symbols ,Twistor space ,[MATH.MATH-AG]Mathematics [math]/Algebraic Geometry [math.AG] ,Mathematics::Differential Geometry - Abstract
International audience; We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperkähler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic Kähler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperkähler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic Kähler manifold of non-positive scalar curvature. We check that no non-Kähler strong KT manifold occurs as such a twistor space.
- Published
- 2017
21. The first Chern class and conformal area for a twistor holomorphic immersion
- Author
-
Kazuyuki Hasegawa
- Subjects
Unit sphere ,Pure mathematics ,Twistor holomorphic surface ,Chern class ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Conformal area ,Twistor lift ,Physics::History of Physics ,Twistor theory ,Normal bundle ,Immersion (mathematics) ,Twistor space ,Mathematics::Symplectic Geometry ,Euler class ,First Chern class ,Mathematics - Abstract
We obtain an inequality involving the first Chern class of the normal bundle and the conformal area for a twistor holomorphic surface. Using this inequality, we can improve an inequality obtained by T. Friedrich for the Euler class of the normal bundle of a twistor holomorphic surface in the four-dimensional space form. Moreover, as a corollary, we see that the area of a superminimal surface in the unit sphere is an integer multiple of {Mathematical expression}, which is essentially proved by E. Calabi. © 2014 Mathematisches Seminar der Universität Hamburg and Springer-Verlag Berlin Heidelberg., in Press
- Published
- 2014
22. Homogeneity for a Class of Riemannian Quotient Manifolds
- Author
-
Joseph A. Wolf
- Subjects
Mathematics - Differential Geometry ,53C20, 53C26, 53C35, 22F30 ,General Mathematics ,Fibered knot ,Rank (differential topology) ,53C20 ,01 natural sciences ,Combinatorics ,0103 physical sciences ,FOS: Mathematics ,Order (group theory) ,0101 mathematics ,Mathematics ,010102 general mathematics ,Pure Mathematics ,53C35 ,53C26 ,math.DG ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Symmetric space ,Homogeneous space ,Isometry ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,Isometry group ,22F30 ,Analysis - Abstract
We study Riemannian coverings φ : M ˜ → Γ \ M ˜ where M ˜ is a normal homogeneous space G / K 1 fibered over another normal homogeneous space M = G / K and K is locally isomorphic to a nontrivial product K 1 × K 2 . The most familiar such fibrations π : M ˜ → M are the natural fibrations of Stiefel manifolds S O ( n 1 + n 2 ) / S O ( n 1 ) over Grassmann manifolds S O ( n 1 + n 2 ) / [ S O ( n 1 ) × S O ( n 2 ) ] and the twistor space bundles over quaternionic symmetric spaces (= quaternion-Kaehler symmetric spaces = Wolf spaces). The most familiar of these coverings φ : M ˜ → Γ \ M ˜ are the universal Riemannian coverings of spherical space forms. When M = G / K is reasonably well understood, in particular when G / K is a Riemannian symmetric space or when K is a connected subgroup of maximal rank in G, we show that the Homogeneity Conjecture holds for M ˜ . In other words we show that Γ \ M ˜ is homogeneous if and only if every γ ∈ Γ is an isometry of constant displacement. In order to find all the isometries of constant displacement on M ˜ we work out the full isometry group of M ˜ , extending Elie Cartan's determination of the full group of isometries of a Riemannian symmetric space. We also discuss some pseudo-Riemannian extensions of our results.
- Published
- 2016
23. Twistor spinors and extended conformal superalgebras
- Author
-
Ümit Ertem
- Subjects
High Energy Physics - Theory ,Mathematics - Differential Geometry ,Primary field ,Conformal anomaly ,General Physics and Astronomy ,FOS: Physical sciences ,01 natural sciences ,Twistor theory ,High Energy Physics::Theory ,General Relativity and Quantum Cosmology ,Conformal symmetry ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,Mathematical physics ,Quantitative Biology::Biomolecules ,Spinor ,Pure spinor ,Conformal field theory ,010102 general mathematics ,Mathematical Physics (math-ph) ,High Energy Physics - Theory (hep-th) ,Differential Geometry (math.DG) ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry - Abstract
We show that the first-order symmetry operators of twistor spinors can be constructed from conformal Killing-Yano forms in conformally-flat backgrounds. We express the conditions on conformal Killing-Yano forms to obtain mutually commuting symmetry operators of twistor spinors. Conformal superalgebras which consist of conformal Killing vectors and twistor spinors and play important roles in supersymmetric field theories in conformal backgrounds are extended to more general superalgebras by using the graded Lie algebra structure of conformal Killing-Yano forms and the symmetry operators of twistor spinors. The even part of the extended conformal superalgebra corresponds to conformal Killing-Yano forms and the odd part consists of twistor spinors., 16 pages, published version
- Published
- 2016
24. Twistor Spaces of Riemannian Manifolds with Even Clifford Structures
- Author
-
Charles Hadfield and Gerardo Arizmendi
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53C28 53C26, 53C35, 53C10, 53C15 ,010102 general mathematics ,Structure (category theory) ,Riemannian manifold ,01 natural sciences ,Twistor theory ,Differential geometry ,Differential Geometry (math.DG) ,Grassmannian ,0103 physical sciences ,FOS: Mathematics ,Twistor space ,010307 mathematical physics ,Geometry and Topology ,Mathematics::Differential Geometry ,0101 mathematics ,Mathematics::Symplectic Geometry ,Analysis ,Mathematics ,Spin-½ - Abstract
In this paper we introduce the twistor space of a Riemannian manifold with an even Clifford structure. This notion generalizes the twistor space of quaternion-Hermitian manifolds and weak-Spin(9) structures. We also construct almost complex structures on the twistor space for parallel even Clifford structures and check their integrability. Moreover, we prove that in some cases one can give K\"ahler and Nearly-K\"ahler metrics to these spaces., Comment: 11 pages, 4 tables
- Published
- 2016
25. Weyl-Euler-Lagrange equations on twistor space for tangent structure
- Author
-
Zeki Kasap
- Subjects
Partial differential equation ,Physics and Astronomy (miscellaneous) ,Kähler ,010308 nuclear & particles physics ,Differential equation ,010102 general mathematics ,Mathematical analysis ,Equations of motion ,Conformal map ,Twistor ,01 natural sciences ,Twistor theory ,Algebra ,mechanical system ,0103 physical sciences ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Kahler ,almost complex ,Lagrangian ,State space ,Twistor space ,0101 mathematics ,Dynamical system (definition) ,Mathematics - Abstract
Twistor spaces are certain complex three-manifolds, which are associated with special conformal Riemannian geometries on four-manifolds. Also, classical mechanic is one of the major subfields for mechanics of dynamical system. A dynamical system has a state determined by a collection of real numbers, or more generally by a set of points in an appropriate state space for classical mechanic. Euler-Lagrange equations are an efficient use of classical mechanics to solve problems using mathematical modeling. On the other hand, Weyl submitted a metric with a conformal transformation for unified theory of classical mechanic. This paper aims to introduce Euler-Lagrage partial differential equations (mathematical modeling, the equations of motion according to the time) for the movement of objects on twistor space and also to offer a general solution of differential equation system using the Maple software. Additionally, the implicit solution of the equation will be obtained as a result of a special selection of graphics to be drawn. © 2016 World Scientific Publishing Company.
- Published
- 2016
26. Hyperholomorphic connections on coherent sheaves and stability
- Author
-
Misha Verbitsky
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,hyperkahler manifold ,Holomorphic function ,Curvature ,twistor space ,53c55 ,14d21 ,Coherent sheaf ,53c05 ,Mathematics - Algebraic Geometry ,53c26 ,Mathematics::Algebraic Geometry ,53c07 ,53c28 ,coherent sheaf ,FOS: Mathematics ,QA1-939 ,Invariant (mathematics) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,stable bundle ,53c38 ,Differential Geometry (math.DG) ,Gravitational singularity ,Mathematics::Differential Geometry - Abstract
Let $M$ be a hyperkaehler manifold, and $F$ a torsion-free and reflexive coherent sheaf on $M$. Assume that $F$ (outside of its singularities) admits a connection with a curvature which is invariant under the standard SU(2)-action on 2-forms. If the curvature is square-integrable, then $F$ is stable and its singularities are hyperkaehler subvarieties in $M$. Such sheaves (called hyperholomorphic sheaves) are well understood. In the present paper, we study sheaves admitting a connection with SU(2)-invariant curvature which is not necessarily square-integrable. This situation arises often, for instance, when one deals with higher direct images of holomorphic bundles. We show that such sheaves are stable., Comment: 37 pages, version 11, reference updated, corrected many minor errors and typos found by the referee
- Published
- 2011
27. Surfaces in four-dimensional hyperkähler manifolds whose twistor lifts are harmonic sections
- Author
-
Kazuyuki Hasegawa
- Subjects
Pure mathematics ,Mean curvature ,Euclidean space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Holomorphic function ,Conformal map ,Mathematics::Geometric Topology ,Twistor theory ,Constant-mean-curvature surface ,Vector field ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
金沢大学人間社会研究域学校教育系, We determine surfaces of genus zero in self-dual Einstein manifolds whose twistor lifts are harmonic sections. We apply our main theorem to the case of four-dimensional hyperkähler manifolds. As a corollary, we prove that a surface of genus zero in four-dimensional Euclidean space is twistor holomorphic if its twistor lift is a harmonic section. In particular, if the mean curvature vector field is parallel with respect to the normal connection, then the surface is totally umbilic. Thus, our main theorem is a generalization of Hopf's theorem for a constant mean curvature surface of genus zero in threedimensional Euclidean space. Moreover, we can also see that a Lagrangian surface of genus zero in the complex Euclidean plane with conformal Maslov form is the Whitney sphere. © 2010 American Mathematical Society.
- Published
- 2011
28. Explicit construction of new Moishezon twistor spaces
- Author
-
Nobuhiro Honda
- Subjects
Surface (mathematics) ,Mathematics - Differential Geometry ,Pure mathematics ,Algebra and Number Theory ,Holomorphic function ,Space (mathematics) ,Twistor theory ,Algebra ,Mathematics - Algebraic Geometry ,Mathematics::Algebraic Geometry ,Differential Geometry (math.DG) ,32L25 ,Conic section ,FOS: Mathematics ,Twistor space ,Geometry and Topology ,Defining equation (physics) ,Mathematics::Differential Geometry ,Algebraic Geometry (math.AG) ,Analysis ,Resolution (algebra) ,Mathematics - Abstract
In this paper we explicitly construct Moishezon twistor spaces on nCP^2 for arbitrary n>1 which admit a holomorphic C*-action. When n=2, they coincide with Y. Poon's twistor spaces. When n=3, they coincide with the one studied by the author in math.DG/0403528. When n>3, they are new twistor spaces, to the best of the author's knowledge. By investigating the anticanonical system, we show that our twistor spaces are bimeromorphic to conic bundles over certain rational surfaces. The latter surfaces can be regarded as orbit spaces of the C*-action on the twistor spaces. Namely they are minitwistor spaces. We explicitly determine their defining equations in CP^4. It turns out that the structure of the minitwistor space is independent of n. Further we concrelely construct a CP^2-bundle over the resolution of this surface, and provide an explicit defining equation of the conic bundles. It shows that the number of irreducible components of the discriminant locus for the conic bundles increases as n does. Thus our twistor spaces have a lot of similarities with the famous LeBrun twistor spaces, where the minitwistor space CP^1 x CP^1 in LeBrun's case is replaced by our minitwistor spaces found in math.DG/0508088., 26 pages, 6 figures; V2 English slightly improved
- Published
- 2009
29. Twistor Geometry of Null Foliations in Complex Euclidean Space
- Author
-
Arman Taghavi-Chabert
- Subjects
Mathematics - Differential Geometry ,Quadric ,Dimension (graph theory) ,Holomorphic function ,FOS: Physical sciences ,Geometry ,General Relativity and Quantum Cosmology (gr-qc) ,01 natural sciences ,General Relativity and Quantum Cosmology ,Twistor theory ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,Mathematical Physics ,Mathematics ,010308 nuclear & particles physics ,Euclidean space ,010102 general mathematics ,Null (mathematics) ,Mathematical Physics (math-ph) ,Linear subspace ,Differential Geometry (math.DG) ,Twistor space ,Geometry and Topology ,Mathematics::Differential Geometry ,Analysis - Abstract
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 illustrated by means of two examples, one involving conformal Killing spinors, the other, conformal Killing-Yano $2$-forms. We focus on the odd-dimensional case, and we treat the even-dimensional case only tangentially for comparison.
- Published
- 2015
30. Compactified Twistor Fibration and Topology of Ward Unitons
- Author
-
Prim Plansangkate
- Subjects
High Energy Physics - Theory ,Mathematics - Differential Geometry ,Holomorphic function ,FOS: Physical sciences ,General Physics and Astronomy ,Vector bundle ,Topology ,01 natural sciences ,Twistor theory ,High Energy Physics::Theory ,0103 physical sciences ,FOS: Mathematics ,0101 mathematics ,010306 general physics ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Mathematics ,Chern class ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,Homotopy ,010102 general mathematics ,Fibration ,High Energy Physics - Theory (hep-th) ,Differential Geometry (math.DG) ,Twistor correspondence ,Twistor space ,Geometry and Topology ,Mathematics::Differential Geometry ,Exactly Solvable and Integrable Systems (nlin.SI) - Abstract
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.
- Published
- 2015
31. Geometry of some twistor spaces of algebraic dimension one
- Author
-
Nobuhiro Honda
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Ruled surface ,Dimension (graph theory) ,53A30 ,Connected sum ,K3 surface ,Twistor theory ,Mathematics - Algebraic Geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,QA1-939 ,Twistor space ,Geometry and Topology ,Projective plane ,Mathematics::Differential Geometry ,Algebraic number ,Algebraic Geometry (math.AG) ,Mathematics - Abstract
It is shown that there exists a twistor space on the $n$-fold connected sum of complex projective planes $n\mathbb{CP}^2$, whose algebraic dimension is one and whose general fiber of the algebraic reduction is birational to an elliptic ruled surface or a K3 surface. The former kind of twistor spaces are constructed over $n\mathbb{CP}^2$ for any $n\ge 5$, while the latter kind of example is constructed over $5\mathbb{CP}^2$. Both of these seem to be the first such example on $n\mathbb{CP}^2$. The algebraic reduction in these examples is induced by the anti-canonical system of the twistor spaces. It is also shown that the former kind of twistor spaces contain a pair of non-normal Hopf surfaces., 29 pages, 5 figures
- Published
- 2015
32. Generalized twistor spaces for hyperkähler manifolds
- Author
-
Rebecca Glover and Justin Sawon
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,General Mathematics ,Structure (category theory) ,Manifold ,Twistor theory ,Generalized complex structure ,Metric (mathematics) ,14J28, 53C26, 53C28, 53D18 ,Twistor space ,Mathematics::Differential Geometry ,Complex manifold ,Mathematics::Symplectic Geometry ,Mathematics - Abstract
Let M be a hyperk\"ahler manifold. The S^2-family of complex structures compatible with the hyperk\"ahler metric can be assembled into a single complex structure on Z=MxS^2; the resulting complex manifold is known as the twistor space of M. We describe the analogous construction for generalized complex structures in the sense of Hitchin. Specifically, we exhibit a natural S^2xS^2-family of generalized complex structures compatible with the hyperk\"ahler metric, and assemble them into a single generalized complex structure on X=MxS^2xS^2. We call the resulting generalized complex manifold the generalized twistor space of M., Comment: 24 pages
- Published
- 2015
- Full Text
- View/download PDF
33. On the moduli space of superminimal surfaces in spheres
- Author
-
Luis E. Fernandez
- Subjects
Modular equation ,Complex projective space ,lcsh:Mathematics ,Mathematical analysis ,lcsh:QA1-939 ,Moduli space ,Moduli of algebraic curves ,Mathematics (miscellaneous) ,Mathematics::Algebraic Geometry ,Projective space ,SPHERES ,Twistor space ,Quaternionic projective space ,Mathematics - Abstract
Using a birational correspondence between the twistor space ofS2nand projective space, we describe, up to birational equivalence, the moduli space of superminimal surfaces inS2nof degreedas curves of degreedin projective space satisfying a certain differential system. Using this approach, we show that the moduli space of linearly full maps is nonempty for sufficiently large degree and we show that the dimension of this moduli space forn=3and genus0is greater than or equal to2d+9. We also give a direct, simple proof of the connectedness of the moduli space of superminimal surfaces inS2nof degreed.
- Published
- 2003
34. New constructions of twistor lifts for harmonic maps
- Author
-
John C. Wood and Martin Svensson
- Subjects
Mathematics - Differential Geometry ,58E20 ,Explicit formulae ,General Mathematics ,Riemann surface ,Mathematical analysis ,Holomorphic function ,Harmonic map ,53C43 ,53C43 (Primary) 58E20 (Secondary) ,Twistor theory ,symbols.namesake ,math.DG ,Differential Geometry (math.DG) ,Symmetric space ,Simply connected space ,FOS: Mathematics ,symbols ,Twistor space ,Mathematics::Differential Geometry ,Mathematics - Abstract
We show that given a harmonic map $\varphi$ from a Riemann surface to a classical compact simply connected inner symmetric space, there is a $J_2$-holomorphic twistor lift of $\varphi$ (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., Comment: Some minor changes and a correction of Example 8.2
- Published
- 2014
35. Quantum cohomology of twistor spaces and their Lagrangian submanifolds
- Author
-
Jonathan David Evans
- Subjects
Mathematics - Differential Geometry ,Algebra and Number Theory ,Chern class ,53D12, 53D35, 53D40, 53D45, 32L25 ,Homology (mathematics) ,Submanifold ,Mathematics::Geometric Topology ,Twistor theory ,Differential Geometry (math.DG) ,Cup product ,Mathematics - Symplectic Geometry ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,Twistor space ,Geometry and Topology ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Analysis ,Eigenvalues and eigenvectors ,Mathematics ,Quantum cohomology ,Mathematical physics - Abstract
We compute the classical and quantum cohomology rings of the twistor spaces of 6-dimensional hyperbolic manifolds and the eigenvalues of quantum multiplication by the first Chern class. Given a half-dimensional totally geodesic submanifold we associate, after Reznikov, a monotone Lagrangian submanifold of the twistor space. In the case of a 3-dimensional totally geodesic submanifold of a hyperbolic 6-manifold we compute the obstruction term $m_0$ in the Fukaya-Floer $A_{\infty}$-algebra of a Reznikov Lagrangian and calculate the Lagrangian quantum homology. There is a well-known correspondence between the possible values of $m_0$ for a Lagrangian with nonvanishing Lagrangian quantum homology and eigenvalues for the action of $c_1$ on quantum cohomology by quantum cup product. Reznikov's Lagrangians account for most of these eigenvalues but there are four exotic eigenvalues we cannot account for., 44 pages; added an assumption on Stiefel-Whitney classes of the hyperbolic manifold. To appear in Journal of Differential Geometry
- Published
- 2014
36. Нормалност на туисторното пространство на 5-мерно многообразие с неприводима SО(3)-структура
- Author
-
Johann Davidov
- Subjects
Pure mathematics ,Group (mathematics) ,twistor spaces ,General Mathematics ,010102 general mathematics ,Structure (category theory) ,Space (mathematics) ,almost contact metric structures ,01 natural sciences ,5-manifold ,Manifold ,Algebra ,0103 physical sciences ,Metric (mathematics) ,Twistor space ,Mathematics::Differential Geometry ,010307 mathematical physics ,0101 mathematics ,Department of Analysis, Geometry and Topology ,irreducible S0(3)-structures ,Mathematics ,Rotation group SO - Abstract
[Davidov Johann; Давидов Йохан] A manifold with an irreducible SО(3)-structure is a 5-manifold M whose structure group can be reduced to the group SО(3), non-standardly imbedded in SО(5). The study of such manifolds has been initiated by M. Bobieński and P. Nurowski who, in particular, have shown that one can define four СR-structures on a twistor-like 7-dimensional space associated to M. In the present paper it is observed that these CR-structures are induced by almost contact metric structures. The purpose of the paper is to study the problem of normality of these structures. The main result gives necessary and sufficient condition for normality in geometric terms of the base manifold M. Examples illustrating this result are presented at the end of the paper. 2000 Mathematics Subject Classification: 53C28; 53D15; 53B15.
- Published
- 2014
37. Twistors and bi_hermitian surfaces of non-K\'ahler type
- Author
-
Massimiliano Pontecorvo, Akira Fujiki, Fujiki, A, and Pontecorvo, Massimiliano
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Betti number ,Mathematical analysis ,Type (model theory) ,twistor space ,Hermitian matrix ,Twistor theory ,bi-Hermitian metric ,Twistor space ,Geometry and Topology ,Mathematics::Differential Geometry ,non-K\"ahler surface ,53C15, 53C28 ,Mathematical Physics ,Analysis ,Mathematics - Abstract
The aim of this work is to give a twistor presentation of recent results about bi-Hermitian metrics on compact complex surfaces with odd first Betti number.
- Published
- 2014
38. Quaternion geometries on the twistor space of the six-sphere
- Author
-
Andrew Swann and Francisco Martín Cabrera
- Subjects
Mathematics - Differential Geometry ,Applied Mathematics ,Mathematical analysis ,Twistor theory ,53C26 (Primary), 53C10, 53C30 (Secondary) ,Differential Geometry (math.DG) ,Homogeneous space ,Torsion (algebra) ,FOS: Mathematics ,Twistor space ,Mathematics::Differential Geometry ,Quaternion ,Mathematics ,Mathematical physics - Abstract
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., Comment: 8 pages
- Published
- 2014
39. Rational curves and special metrics on twistor spaces
- Author
-
Misha Verbitsky
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,rational connected variety ,Space (mathematics) ,twistor space ,K3 surface ,Twistor theory ,High Energy Physics::Theory ,Mathematics - Algebraic Geometry ,FOS: Mathematics ,Hermitian manifold ,32Q15 ,SKT metric ,Complex Variables (math.CV) ,Algebraic Geometry (math.AG) ,Mathematics::Symplectic Geometry ,Mathematics ,Mathematics - Complex Variables ,Flag (linear algebra) ,non-Kähler manifold ,Riemannian manifold ,pluriclosed metric ,Moduli space ,53C26 ,53C28 ,Differential Geometry (math.DG) ,Moishezon variety ,Twistor space ,Geometry and Topology ,Mathematics::Differential Geometry ,Complex manifold - Abstract
A Hermitian metric $\omega$ on a complex manifold is called SKT or pluriclosed if $dd^c\omega=0$. Let M be a twistor space of a compact, anti-selfdual Riemannian manifold, admitting a pluriclosed Hermitian metric. We prove that in this case M is K\"ahler, hence isomorphic to $\C P^3$ or a flag space. This result is obtained from rational connectedness of the twistor space, due to F. Campana. As an aside, we prove that the moduli space of rational curves on the twistor space of a K3 surface is Stein., Comment: 12 pages
- Published
- 2012
40. On the hyperkaehler/quaternion Kaehler correspondence
- Author
-
Nigel Hitchin
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Mathematics::Complex Variables ,Connection (principal bundle) ,Statistical and Nonlinear Physics ,Kähler manifold ,Higgs bundle ,Twistor theory ,Mathematics::Algebraic Geometry ,Line bundle ,Differential Geometry (math.DG) ,Bundle ,FOS: Mathematics ,53C26, 53C28 ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Symplectic geometry ,Mathematics - Abstract
A hyperkaehler manifold with a circle action fixing just one complex structure admits a natural a hyperholomorphic line bundle. This forms the basis for the construction of a corresponding quaternionic Kaehler manifold in the work of A.Haydys. We construct in this paper the corresponding holomorphic line bundle on twistor space and compute many examples, including monopole and Higgs bundle moduli spaces. We also show that the bundle on twistor space has a natural meromorphic connection which realizes it as the quantum line bundle for the hyperkaehler family of holomorphic symplectic structures. Finally we give a twistor version of the HK/QK correspondence., 35 pages
- Published
- 2012
41. Pseudoholomorphic curves on nearly Kahler manifolds
- Author
-
Misha Verbitsky
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Almost complex manifold ,Sesquilinear form ,Statistical and Nonlinear Physics ,Kähler manifold ,Type (model theory) ,Mathematics::Geometric Topology ,Manifold ,Moduli space ,Differential Geometry (math.DG) ,Mathematics - Symplectic Geometry ,FOS: Mathematics ,Symplectic Geometry (math.SG) ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Mathematical Physics ,Differential (mathematics) ,Mathematics - Abstract
Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kahler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G_2-invariant) almost complex structure., 6 pages
- Published
- 2012
42. Twistor geometry of a pair of second order ODEs
- Author
-
Paul Tod, Stephen Casey, and Maciej Dunajski
- Subjects
Mathematics - Differential Geometry ,High Energy Physics - Theory ,Nonlinear Sciences - Exactly Solvable and Integrable Systems ,General relativity ,Scalar (mathematics) ,FOS: Physical sciences ,Statistical and Nonlinear Physics ,Conformal map ,General Relativity and Quantum Cosmology (gr-qc) ,Mathematical Physics (math-ph) ,General Relativity and Quantum Cosmology ,Twistor theory ,Differential Geometry (math.DG) ,High Energy Physics - Theory (hep-th) ,Conformal symmetry ,Variational principle ,FOS: Mathematics ,Twistor correspondence ,Twistor space ,Mathematics::Differential Geometry ,Exactly Solvable and Integrable Systems (nlin.SI) ,Mathematical Physics ,Mathematics ,Mathematical physics - Abstract
We discuss the twistor correspondence between path geometries in three dimensions with vanishing Wilczynski invariants and anti-self-dual conformal structures of signature $(2, 2)$. We show how to reconstruct a system of ODEs with vanishing invariants for a given conformal structure, highlighting the Ricci-flat case in particular. Using this framework, we give a new derivation of the Wilczynski invariants for a system of ODEs whose solution space is endowed with a conformal structure. We explain how to reconstruct the conformal structure directly from the integral curves, and present new examples of systems of ODEs with point symmetry algebra of dimension four and greater which give rise to anti--self--dual structures with conformal symmetry algebra of the same dimension. Some of these examples are $(2, 2)$ analogues of plane wave space--times in General Relativity. Finally we discuss a variational principle for twistor curves arising from the Finsler structures with scalar flag curvature., Final version to appear in the Communications in Mathematical Physics. The procedure of recovering a system of torsion-fee ODEs from the heavenly equation has been clarified. The proof of Prop 7.1 has been expanded. Dedicated to Mike Eastwood on the occasion of his 60th birthday
- Published
- 2012
43. Conformal Field Theories in Six-Dimensional Twistor Space
- Author
-
R.A. Reid-Edwards, Lionel Mason, and Arman Taghavi-Chabert
- Subjects
Integral transforms ,High Energy Physics - Theory ,Mathematics - Differential Geometry ,General Physics and Astronomy ,FOS: Physical sciences ,Penrose transform ,01 natural sciences ,Conformal group ,Integral geometry ,Twistor theory ,Physics and Astronomy (all) ,High Energy Physics::Theory ,0103 physical sciences ,FOS: Mathematics ,Conformal field theory ,Mathematical Physics ,Geometry and Topology ,010306 general physics ,Mathematics ,Mathematical physics ,Pure spinor ,010308 nuclear & particles physics ,Mathematical analysis ,Mathematical Physics (math-ph) ,Amplituhedron ,High Energy Physics - Theory (hep-th) ,Differential Geometry (math.DG) ,Twistor space ,Mathematics::Differential Geometry - Abstract
This article gives a study of the higher-dimensional Penrose transform between conformally invariant massless fields on space-time and cohomology classes on twistor space, where twistor space is defined to be the space of projective pure spinors of the conformal group. We focus on the 6-dimensional case in which twistor space is the six-quadric Q in CP^7 with a view to applications to the self-dual (0,2)-theory. We show how spinor-helicity momentum eigenstates have canonically defined distributional representatives on twistor space (a story that we extend to arbitrary dimension). These give an elementary proof of the surjectivity of the Penrose transform. We give a direct construction of the twistor transform between the two different representations of massless fields on twistor space (H^2 and H^3) in which the H^3s arise as obstructions to extending the H^2s off Q into CP^7. We also develop the theory of Sparling's `\Xi-transform', the analogous totally real split signature story based now on real integral geometry where cohomology no longer plays a role. We extend Sparling's \Xi-transform to all helicities and homogeneities on twistor space and show that it maps kernels and cokernels of conformally invariant powers of the ultrahyperbolic wave operator on twistor space to conformally invariant massless fields on space-time. This is proved by developing the 6-dimensional analogue of the half-Fourier transform between functions on twistor space and momentum space. We give a treatment of the elementary conformally invariant \Phi^3 amplitude on twistor space and finish with a discussion of conformal field theories in twistor space., Comment: 37 pages, 3 figures, in v2 a number of inaccuracies are removed and the discussion is improved; in v3, one reference added;v4 extra acknowledgement
- Published
- 2011
44. Twistor Theory for co-CR quaternionic manifolds and related structures
- Author
-
Stefano Marchiafava and Radu Pantilie
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,53C28, 53C26 ,General Mathematics ,Mathematical analysis ,Riemannian geometry ,Space (mathematics) ,Mathematics::Geometric Topology ,Twistor theory ,symbols.namesake ,Differential Geometry (math.DG) ,Quaternionic representation ,Ricci-flat manifold ,FOS: Mathematics ,symbols ,Differential topology ,Twistor space ,Mathematics::Differential Geometry ,Mathematics::Symplectic Geometry ,Hyperkähler manifold ,Mathematics - Abstract
In a general and non metrical framework, we introduce the class of co-CR quaternionic manifolds, which contains the class of quaternionic manifolds, whilst in dimension three it particularizes to give the Einstein-Weyl spaces. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic-Kaehler manifolds., 19 pages
- Published
- 2011
45. A note on small deformations of balanced manifolds
- Author
-
Shing-Tung Yau and Jixiang Fu
- Subjects
Mathematics - Differential Geometry ,Lemma (mathematics) ,Pure mathematics ,Differential Geometry (math.DG) ,FOS: Mathematics ,Twistor space ,General Medicine ,Mathematics::Differential Geometry ,Mathematics::Geometric Topology ,Mathematics::Symplectic Geometry ,Manifold ,Mathematics - Abstract
In this note we prove that, under a weak condition, small deformations of a compact balanced manifold are also balanced. This condition is satisfied on the twistor space over a compact self-dual four manifold., A revised version. The title has been changed
- Published
- 2011
46. On Discrete Differential Geometry in Twistor Space
- Author
-
George Shapiro
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Quadric ,Integrable system ,53A30 ,Cross-ratio ,General Physics and Astronomy ,Mathematics::Algebraic Geometry ,Differential geometry ,Differential Geometry (math.DG) ,FOS: Mathematics ,Twistor space ,Geometry and Topology ,Discrete differential geometry ,Plucker ,Quaternionic projective space ,Mathematical Physics ,Mathematics - Abstract
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 Pl\"ucker quadric, realized as the space of two-spheres in $S^4.$ We develop the geometry of the Pl\"ucker 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 discrete principal contact element nets for the Pl\"ucker quadric and prove several elementary results. Employing a second real real structure, we show that these results generalize previous results by Bobenko and Suris $(2007)$ on discrete differential geometry in the Lie quadric., Comment: 32 pages, 11 figures
- Published
- 2011
47. Autodual Connection in the Fourier Transform of a Higgs Bundle
- Author
-
Juhani Bonsdorff
- Subjects
14F10 ,General Mathematics ,hyper-Kahler ,Topology ,twistor space ,Higgs bundle ,symbols.namesake ,32C38 ,D-module ,autodual connection ,Mathematics ,hyperholomorphic ,Applied Mathematics ,Connection (principal bundle) ,Nahm transform ,14H60 ,Algebra ,53C26 ,Fourier transform ,53C07 ,53C28 ,symbols ,Twistor space - Published
- 2010
48. A CR twistor space of a G2-manifold
- Author
-
Misha Verbitsky
- Subjects
Mathematics - Differential Geometry ,Pure mathematics ,Integrable system ,Twistor space ,High Energy Physics::Theory ,Mathematics::Algebraic Geometry ,FOS: Mathematics ,Tangent vector ,Complex Variables (math.CV) ,Mathematics::Symplectic Geometry ,Mathematics ,Mathematics - Complex Variables ,G2-manifold ,Special holonomy ,Holonomy ,CR-manifold ,G2 manifold ,Computational Theory and Mathematics ,Differential Geometry (math.DG) ,Bundle ,Calibration ,CR manifold ,Geometry and Topology ,Mathematics::Differential Geometry ,Unit (ring theory) ,Analysis - Abstract
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., 14 pages, v. 2.0, a few misprints corrected, final version before the publication
- Published
- 2010
49. Connections on contact manifolds and contact twistor space
- Author
-
Luigi Vezzoni
- Subjects
Pure mathematics ,Integrable system ,General Mathematics ,Mathematical analysis ,Structure (category theory) ,Curvature ,Manifold ,Connection (mathematics) ,Twistor space ,Mathematics::Differential Geometry ,Algebra over a field ,Mathematics::Symplectic Geometry ,Mathematics ,Symplectic geometry - Abstract
In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and that any Sasakian structure induces a canonical contact connection. Furthermore (as in the symplectic case), any contact connection induces an almost CR structure on the contact twistor space which is integrable if and only if the curvature of the connection is of Ricci-type.
- Published
- 2010
50. Another infinite tri-Sasaki family and marginal deformations
- Author
-
Osvaldo Santillan
- Subjects
Pure mathematics ,General Mathematics ,Horizon ,Supergravity ,Mathematical analysis ,Holonomy ,General Physics and Astronomy ,Fibered knot ,Differential geometry ,Isometry ,Twistor space ,Mathematics::Differential Geometry ,Orbifold ,Mathematics - Abstract
Several Einstein–Sasaki seven-metrics appearing in the physical literature are fibred over four-dimensional Kähler–Einstein metrics. Instead we consider here the natural Kähler–Einstein metrics defined over the twistor space $Z$ of any quaternion Kähler four-space, together with the corresponding Einstein–Sasaki metrics. We work out an explicit expression for these metrics and we prove that they are indeed tri-Sasaki. Moreover, we present a squashed version of them which is of weak $G2$ holonomy. We focus in examples with three commuting Killing vectors and we extend them to supergravity backgrounds with T3 isometry, some of them with $AdS4 × X7$ near horizon limit and some others without this property. We would like to emphasize that there is an underlying linear structure describing these spaces. We also consider the effect of the $SL(2,R)$ solution-generating technique presented by Maldacena and Lunin to these backgrounds and we find some rotating membrane configurations reproducing the $E–S$ logarithmic behaviour.
- Published
- 2008
Catalog
Discovery Service for Jio Institute Digital Library
For full access to our library's resources, please sign in.