scholarly journals Twistor spaces on foliated manifolds

2021 ◽  
pp. 2150056
Author(s):  
Rouzbeh Mohseni ◽  
Robert A. Wolak
Keyword(s):  
Normal Bundle ◽  
Twistor Space ◽  
Riemannian Foliation ◽  
Holomorphic Mappings ◽  
Twistor Theory ◽  
Simple Consequence ◽  
Twistor Spaces ◽  
Leaf Space ◽  
Foliated Manifolds ◽  
Theory Lead

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.

scholarly journals A REMARK ON THE BRYLINSKI CONJECTURE FOR ORBIFOLDS

2011 ◽  
Vol 91 (1) ◽  
pp. 1-12 ◽  
Author(s):  
L. BAK ◽  
A. CZARNECKI
Keyword(s):  
Riemannian Foliation ◽  
De Rham Cohomology ◽  
De Rham ◽  
Leaf Space ◽  
Foliated Manifolds

AbstractThe paper presents a proof of the Brylinski conjecture for compact Kähler orbifolds. The result is a corollary of the foliated version of the Mathieu theorem on symplectic harmonic representations of de Rham cohomology classes. The proofs are based on the idea of representing an orbifold as the leaf space of a Riemannian foliation and on the correspondence between foliated and holonomy invariant objects for foliated manifolds.

scholarly journals Generalized almost even-Clifford manifolds and their twistor spaces

2021 ◽  
Vol 8 (1) ◽  
pp. 96-124
Author(s):  
Luis Fernando Hernández-Moguel ◽  
Rafael Herrera
Keyword(s):  
Twistor Space ◽  
Complex Structure ◽  
Twistor Spaces ◽  
Recent Interest ◽  
Generalized Complex Structure ◽  
Generalized Complex ◽  
Space Construction ◽  
Clifford Structures

Abstract Motivated by the recent interest in even-Clifford structures and in generalized complex and quaternionic geometries, we introduce the notion of generalized almost even-Clifford structure. We generalize the Arizmendi-Hadfield twistor space construction on even-Clifford manifolds to this setting and show that such a twistor space admits a generalized complex structure under certain conditions.

scholarly journals Riemannian foliations with Ehresmann connection

2018 ◽  
Vol 20 (4) ◽  
pp. 395-407
Author(s):  
N. I. Zhukova
Keyword(s):  
Linear Connection ◽  
Riemannian Foliation ◽  
Minimal Set ◽  
Foliated Manifold ◽  
Trivial Bundle ◽  
Riemannian Foliations ◽  
Ehresmann Connection ◽  
The Family ◽  
Leaf Space ◽  
Complete Riemannian Manifolds

It is shown that the structural theory of Molino for Riemannian foliations on compact manifolds and complete Riemannian manifolds may be generalized to a Riemannian foliations with Ehresmann connection. Within this generalization there are no restrictions on the codimension of the foliation and on the dimension of the foliated manifold. For a Riemannian foliation (M,F) with Ehresmann connection it is proved that the closure of any leaf forms a minimal set, the family of all such closures forms a singular Riemannian foliation (M,F¯¯¯¯). It is shown that in M there exists a connected open dense F¯¯¯¯-saturated subset M0 such that the induced foliation (M0,F¯¯¯¯|M0) is formed by fibers of a locally trivial bundle over some smooth Hausdorff manifold. The equivalence of some properties of Riemannian foliations (M,F) with Ehresmann connection is proved. In particular, it is shown that the structural Lie algebra of (M,F) is equal to zero if and only if the leaf space of (M,F) is naturally endowed with a smooth orbifold structure. Constructed examples show that for foliations with transversally linear connection and for conformal foliations the similar statements are not true in general.

scholarly journals Integrable Complex Structures on Twistor Spaces

2019 ◽  
Vol 70 (3) ◽  
pp. 937-963
Author(s):  
Steven Gindi
Keyword(s):  
Twistor Space ◽  
Complex Structures ◽  
Complex Manifolds ◽  
Twistor Spaces

Abstract 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.

scholarly journals Energy of sections of the Deligne–Hitchin twistor space

2020 ◽  
Author(s):  
Florian Beck ◽  
Sebastian Heller ◽  
Markus Röser
Keyword(s):  
Moduli Space ◽  
Compact Riemann Surface ◽  
Twistor Space ◽  
Harmonic Maps ◽  
Energy Functional ◽  
Conformal Map ◽  
Twistor Spaces ◽  
Holomorphic Sections ◽  
Meromorphic Connection ◽  
Willmore Energy

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 show that the energy is the residue of the pull-back along the section of a natural meromorphic connection on the hyperholomorphic line bundle recently constructed by Hitchin. As a byproduct, we show the existence of a hyper-Kähler potentials for new components of real holomorphic sections of twistor spaces of hyper-Kähler manifolds with rotating $$S^1$$ S 1 -action. Additionally, we prove that for a certain class of real holomorphic sections of the Deligne–Hitchin moduli space, the energy 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.

scholarly journals Complex Multiplication in Twistor Spaces

2020 ◽  
Author(s):  
D Huybrechts
Keyword(s):  
Twistor Space ◽  
Complex Multiplication ◽  
K3 Surface ◽  
Twistor Spaces ◽  
Arithmetic Properties ◽  
Twistor Construction

Abstract Despite the transcendental nature of the twistor construction, the algebraic fibres of the twistor space of a K3 surface share certain arithmetic properties. We prove that for a polarised K3 surface with complex multiplication, all algebraic fibres of its twistor space away from the equator have complex multiplication as well.

Twistors and fields with sources on worldlines

1985 ◽  
Vol 397 (1812) ◽  
pp. 143-155 ◽  
Keyword(s):  
Minkowski Space ◽  
Twistor Space ◽  
Rest Mass ◽  
Double Cover ◽  
Ruled Surface ◽  
Hausdorff Spaces ◽  
Twistor Spaces ◽  
Non Hausdorff Spaces ◽  
Massless Fields

It is shown that zero-rest-mass fields with sources on an analytic worldline are naturally defined on a double cover of some region of Minkowski space. Twistor spaces are constructed that correspond to such regions and these turn out to be non-Hausdorff spaces, obtained by identifying two copies of regions in ordinary twistor space, except on a ruled surface that corresponds to the worldline. It is shown that cohomology classes on the twistor space corresponds to sourced fields on Minkowski space, thus extending the twistor descrip­tion of massless fields.

scholarly journals Twistor spaces and the adiabatic limits of Dirac operators

2001 ◽  
Vol 164 ◽  
pp. 53-73 ◽  
Author(s):  
Masayoshi Nagase
Keyword(s):  
Dirac Operator ◽  
Spin Structure ◽  
Twistor Space ◽  
Dirac Operators ◽  
Twistor Spaces ◽  
Adiabatic Limits ◽  
Twisted Dirac Operator

We show that a (Spinq-style) twistor space admits a canonical Spin structure. The adiabatic limits of η-invariants of the associated Dirac operator and of an intrinsically twisted Dirac operator are then investigated.

scholarly journals Riemannian foliations with parallel curvature

1983 ◽  
Vol 90 ◽  
pp. 145-153
Author(s):  
Robert A. Blumenthal
Keyword(s):  
Tangent Bundle ◽  
Normal Bundle ◽  
Compact Manifold ◽  
Riemannian Metric ◽  
Riemannian Foliation ◽  
Riemannian Foliations ◽  
Smooth Compact Manifold

Let M be a smooth compact manifold and let be a smooth codimension q Riemannian foliation of M. Let T(M) be the tangent bundle of M and let E ⊂ T(M) be the subbundle tangent to . We may regard the normal bundle Q = T(M)/E of as a subbundle of T(M) satisfying T(M) = E ⊕ Q. Let g be a smooth Riemannian metric on Q invariant under the natural parallelism along the leaves of .

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