Quaternionic structures on a manifold and subordinated structures

Alekseevsky, Dmitri V.; Marchiafava, Stefano

doi:10.1007/bf01759388

Cited by 99 publications

(211 citation statements)

References 15 publications

Supporting

Mentioning

210

Contrasting

Order By: Relevance

“…For the almost hypercomplex structure H = (J α ) there exists a unique linear connection ∇ H which preserves H, that is, ∇ H J α = 0, α = 1, 2, 3, and whose torsion tensor equals T H . ∇ H is called the Obata connection of H (see for example [AM,p. 37]) and it is known that the torsion of ∇ H vanishes, T H = 0, if and only if H is hypercomplex.…”

Section: Almost Quaternionic Structuresmentioning

confidence: 99%

Compatible complex structures on almost quaternionic manifolds

Alekseevsky¹,

Marchiafava

Pontecorvo

1999

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. On an almost quaternionic manifold (M 4n , Q) we study the integrability of almost complex structures which are compatible with the almost quaternionic structure Q. If n ≥ 2, we prove that the existence of two compatible complex structuresis an oriented conformal 4-manifold, we prove a maximum principle for the angle function I 1 , I 2 of two compatible complex structures and deduce an application to anti-self-dual manifolds.By considering the special class of Oproiu connections we prove the existence of a well defined almost complex structure J on the twistor space Z of an almost quaternionic manifold (M 4n , Q) and show that J is a complex structure if and only if Q is quaternionic. This is a natural generalization of the Penrose twistor constructions.

show abstract

Section: Almost Quaternionic Structuresmentioning

confidence: 99%

Compatible complex structures on almost quaternionic manifolds

Alekseevsky¹,

Marchiafava

Pontecorvo

1999

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

“…see [1]. For these reasons, in the latter case, there exists a distinguished class of linear connections compatible with the structure.…”

Section: Introductionmentioning

confidence: 99%

Geometry of almost Cliffordian manifolds: classes of subordinated connections

Hrdina¹,

Vašík²

2014

Turk J Math

View full text Add to dashboard Cite

An almost Clifford and an almost Cliffordian manifold is a G -structure based on the definition of Clifford algebras. An almost Clifford manifold based on O := Cl(s, t) is given by a reduction of the structure group GL(km, R) to GL(m, O) , where k = 2 s+t and m ∈ N . An almost Cliffordian manifold is given by a reduction of the structure group to GL(m, O)GL(1, O) . We prove that an almost Clifford manifold based on O is such that there exists a unique subordinated connection, while the case of an almost Cliffordian manifold based on O is more rich. A class of distinguished connections in this case is described explicitly.

show abstract

“…(1.1) J α = J β • J γ = −J γ • J β , J 2 α = −I for all cyclic permutations (α, β, γ) of (1, 2, 3) and I denotes the identity [1].…”

Section: Introductionmentioning

confidence: 99%

“…It consists of the given Hermitian metric g with respect to the three almost complex structures of H and the three Kähler forms associated with g by H [1].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Manev

2012

J. Geom.

View full text Add to dashboard Cite

Abstract. Almost hypercomplex manifolds with Hermitian and Norden metrics and more specially the corresponding quaternionic Kähler manifolds are considered. Some necessary and sufficient conditions the investigated manifolds be isotropic hyper-Kählerian and flat are found. It is proved that the quaternionic Kähler manifolds with the considered metric structure are Einstein for dimension at least 8. The class of the non-hyper-Kähler quaternionic Kähler manifold of the considered type is determined. ContentsIntroduction 1 1. Almost hypercomplex manifolds with NH-metric structure 2 2. Quaternionic Kähler manifolds with NH-metric structure 4 3. Quaternionic Kähler NH-manifolds in a classification of almost hypercomplex NH-manifolds 7 4. Non-hyper-Kähler quaternionic Kähler NH-manifolds 9 References 10 IntroductionIn this work * we continue the investigations on a manifold M with an almost hypercomplex structure H. We equip this almost hypercomplex manifold (M, H) with a metric structure G, generated by a pseudo-Riemannian metric g of neutral signature ([4], [5]).It is known, if g is a Hermitian metric on (M, H), the derived metric structure G is the known hyper-Hermitian structure. It consists of the given Hermitian metric g with respect to the three almost complex structures of H and the three Kähler forms associated with g by H [1].In our case the considered metric structure G has a different type of compatibility with H. The structure G is generated by a neutral metric g such that the one (resp., the other two) of the almost complex structures of H acts as an isometry (resp., act as anti-isometries) with respect to g in each tangent fibre. Let the almost complex structures of H act as isometries or anti-isometries with respect to the metric. Then the existence of an anti-isometry generates exactly the existence of one more antiisometry and an isometry. Thus, G contains the metric g and three (0,2)-tensors associated by H -a Kähler form and two metrics of the same type. The existence of such bilinear forms on an almost hypercomplex manifold is proved in [4]. The neutral metric g is Hermitian with respect to the one almost complex structure of H and g is an anti-Hermitian (i. e. a Norden) metric regarding the other two almost complex structures of H. For this reason we call the derived almost hypercomplex manifold (M, H, G) an almost hypercomplex manifold with NH-metric structure or an almost hypercomplex NH-manifold.If the three almost complex structures of H are parallel with respect to the LeviCivita connection ∇ of g then such hypercomplex NH-manifolds of Kähler type we call hyper-Kähler NH-manifolds, which are flat according to [5].In the first section we recall some facts about the almost hypercomplex NHmanifolds known from [1], [4], [5], [7].In the second section we introduce the corresponding quaternionic Kähler manifold of an almost hypercomplex manifold with NH-metric structure. We establish that the quaternionic Kähler NH-manifolds are Einstein for dimension 4n ≥ 8. For comparison, it is known that the ...

show abstract

Quaternionic structures on a manifold and subordinated structures

Cited by 99 publications

References 15 publications

Compatible complex structures on almost quaternionic manifolds

Compatible complex structures on almost quaternionic manifolds

Geometry of almost Cliffordian manifolds: classes of subordinated connections

Quaternionic Kähler manifolds with Hermitian and Norden metrics

Contact Info

Product

Resources

About