Twistor theory for co-CR quaternionic manifolds and related structures

Marchiafava, Stefano; Pantilie, Radu

doi:10.1007/s11856-013-0001-3

Cited by 17 publications

(46 citation statements)

References 23 publications

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“…(5) Let

U = {double-struckR}^{3} \otimes {double-struckR}^{m}

and let S 2 be embedded as the (isotropic) conic in the projectivization of the complexification of the Euclidean space

R^{3}

. By defining

σ false(ℓ false) = ℓ^{⊥} \otimes C^{m}

, for any

ℓ \in S^{2} (\subseteq double-struckC P^{2})

, we obtain a maximal Euclidean S 2 ‐structure whose automorphism group is

SO (3) \times SO (m)

(see for more details on these structures). More generally, we may define the tensor product of Euclidean twistorial structures.…”

Section: Euclidean Twistorial Structuresmentioning

confidence: 99%

“…The following definition specializes notions of ; compare, also, . Definition If the torsion of ∇ is totally antisymmetric, we say that

(E, ρ, σ, \nabla)

is a Riemannian almost twistorial structure (on M ) .…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

“…Thus, the examples of twistor spaces of fit into our setting. Also, all of the examples of (see in the cases with splitting normal bundle exact sequence for the twistor spheres) whose twistor spaces are generalized flag manifolds also provide concrete examples of Riemannian twistorial structures. Moreover, similarly to , if

m = {frakturh}^{⊥}

and ∇ is torsion free, we may construct a G ‐invariant holomorphic distribution

scriptF

on Z whose preimage by

d φ

is horizontal with respect to π, and thus, by Corollary and Proposition , provides a construction of minimal submanifolds of M .…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

“…(2) Let

k, p, q \in N ∖ {0}

k \leq p / 2

M = {Gr}_{p}^{+} false(p + q, double-struckR false)

with its Riemannian symmetric space structure, Z the Grassmannian of isotropic k ‐dimensional subspaces of

C^{p + q}

, and

Y = \{false(u, v false) \in Z \times M | u \subseteq v^{double-struckC}\}

. Note that this Riemannian twistorial structure is of Hermitian type if and only if

p = 2 k

, case considered in ; also, the case

k = 1

p = 3

was considered in (see ). (3) Let

k \in N ∖ {0}

and

p_{1}, \dots, p_{k}, n \in N

such that

0 < p_{1} < \dots < p_{k} < n

.…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

“…The case

k \leq 2

is covered by and , up to the twistoriality of the embeddings of M into SU( n ). Note that [, Example 4.7] (see [, Example 4.6]) also provides an example of a Riemannian reductive homogeneous space which is not a symmetric space and which is endowed with a Riemannian twistorial structure, as above.…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

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Harmonic Maps and Twistorial Structures

2019

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“…(5) Let

U = {double-struckR}^{3} \otimes {double-struckR}^{m}

and let S 2 be embedded as the (isotropic) conic in the projectivization of the complexification of the Euclidean space

R^{3}

. By defining

σ false(ℓ false) = ℓ^{⊥} \otimes C^{m}

, for any

ℓ \in S^{2} (\subseteq double-struckC P^{2})

, we obtain a maximal Euclidean S 2 ‐structure whose automorphism group is

SO (3) \times SO (m)

(see for more details on these structures). More generally, we may define the tensor product of Euclidean twistorial structures.…”

Section: Euclidean Twistorial Structuresmentioning

confidence: 99%

“…The following definition specializes notions of ; compare, also, . Definition If the torsion of ∇ is totally antisymmetric, we say that

(E, ρ, σ, \nabla)

is a Riemannian almost twistorial structure (on M ) .…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

m = {frakturh}^{⊥}

and ∇ is torsion free, we may construct a G ‐invariant holomorphic distribution

scriptF

on Z whose preimage by

d φ

is horizontal with respect to π, and thus, by Corollary and Proposition , provides a construction of minimal submanifolds of M .…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

“…(2) Let

k, p, q \in N ∖ {0}

k \leq p / 2

M = {Gr}_{p}^{+} false(p + q, double-struckR false)

with its Riemannian symmetric space structure, Z the Grassmannian of isotropic k ‐dimensional subspaces of

C^{p + q}

, and

Y = \{false(u, v false) \in Z \times M | u \subseteq v^{double-struckC}\}

. Note that this Riemannian twistorial structure is of Hermitian type if and only if

p = 2 k

, case considered in ; also, the case

k = 1

p = 3

was considered in (see ). (3) Let

k \in N ∖ {0}

and

p_{1}, \dots, p_{k}, n \in N

such that

0 < p_{1} < \dots < p_{k} < n

.…”

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

“…The case

k \leq 2

Section: Riemannian Twistorial Structuresmentioning

confidence: 99%

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Harmonic Maps and Twistorial Structures

2019

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Harmonic morphisms and moment maps on hyper-Kähler manifolds

Benyounes

Loubeau

Pantilie³

2016

manuscripta math.

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Abstract. We characterise the actions, by holomorphic isometries on a Kähler manifold with zero first Betti number, of an abelian Lie group of dim ≥ 2 , for which the moment map is horizontally weakly conformal (with respect to some Euclidean structure on the Lie algebra of the group).Furthermore, we study the hyper-Kähler moment map ϕ induced by an abelian Lie group T acting by triholomorphic isometries on a hyper-Kähler manifold M , with zero first Betti number, thus obtaining the following:• If dim T = 1 then ϕ is a harmonic morphism. Moreover, we illustrate this on the tangent bundle of the complex projective space equipped with the Calabi hyper-Kähler structure, and we obtain an explicit global formula for the map.• If dim T ≥ 2 and either ϕ has critical points, or M is nonflat and dim M = 4 dim T then ϕ cannot be horizontally weakly conformal.

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Twistor theory for CR quaternionic manifolds and related structures

Marchiafava

Pantilie²

2011

Monatsh Math

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In a general and nonmetrical framework, we introduce the class of CR quaternionic manifolds containing the class of quaternionic manifolds, whilst in dimension three it particularizes to, essentially, give the conformal manifolds. We show that these manifolds have a rich natural Twistor Theory and, along the way, we obtain a heaven space construction for quaternionic manifolds. © 2011 Springer-Verlag

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Twistor theory for co-CR quaternionic manifolds and related structures

Cited by 17 publications

References 23 publications

Harmonic Maps and Twistorial Structures

Harmonic Maps and Twistorial Structures

Harmonic morphisms and moment maps on hyper-Kähler manifolds

Twistor theory for CR quaternionic manifolds and related structures

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