About: Hyperkähler manifold

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In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979.

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rdfs:label	Hyperkähler manifold (en) Varietà iperkähler (it) 초켈러 다양체 (ko) 超ケーラー多様体 (ja)
rdfs:comment	In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the quaternionic relations . In particular, it is a hypercomplex manifold. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds. Hyperkähler manifolds were defined by Eugenio Calabi in 1979. (en) 미분기하학에서 초켈러 다양체(超Kähler多樣體, 영어: hyper-Kähler manifold)는 그 접공간이 사원수의 좌표를 가진 공간의 구조를 가지는 리만 다양체이다. (ko) 微分幾何学において、超ケーラー多様体(hyperkähler manifold)は、次元 4k次元のリーマン多様体で、がSp(k)を含んでいる場合を言う（ここに、Sp(k) はシンプレクティック群のコンパクトな形を表していて、-次元の四元数エルミート空間の四元数線型ユニタリ自己準同型の群と同一視される）。超ケーラー多様体は、ケーラー多様体の特別なクラスで、ケーラー多様体の四元数と考えることができる。超ケーラー多様体はみな、リッチ平坦であり、従って、Sp(k) はSU(2k)の部分群であることから容易に分かるように、カラビ・ヤウ多様体である。超ケーラー多様体は、エウジェニオ・カラビにより 1978年に定義された。 (ja) In geometria differenziale, una varietà iperkähler è una varietà riemanniana di dimensione e di contenuto in Sp(k) (qui Sp(k) denota una forma compatta del identificata con il gruppo di endomorfismi unitari quaternionico-lineari di uno spazio hermitiano quaternionico -dimensionale). Le varietà di iperkähler sono classi speciali di varietà di Kähler. Possono essere pensate come analoghi quaternionici delle varietà di Kähler. Tutte le varietà iperkähler hanno tensore di curvatura di Ricci nullo e sono quindi varietà di Calabi - Yau (questo può essere facilmente visto notando che Sp(k) è un sottogruppo del gruppo unitario speciale SU(2k)). (it)
dcterms:subject	Complex manifolds Differential geometry Structures on manifolds Riemannian manifolds Quaternions
Wikipage page ID	646120 (xsd:integer)
Wikipage revision ID	1117971708 (xsd:integer)
Link from a Wikipage to another Wikipage	Calabi–Yau Quaternion Quaternion-Kähler manifold Monopole moduli space Almost complex manifold Complex manifolds Holonomy group Riemann surface Differential geometry Structures on manifolds Compact space Neighbourhood (mathematics) Yang–Mills theory Quaternionic manifold Quaternionic vector space Riemannian manifolds Calabi conjecture Calabi–Yau manifold Communications in Mathematical Physics Complex projective space Physics Subgroup Symplectic group Torus Algebraic geometry Quaternions Eugenio Calabi Nigel Hitchin Parallel transport Gravitational instanton Hitchin's equations Quiver (mathematics) Riemannian manifold Hilbert scheme Hiraku Nakajima Hypercomplex manifold Hyperkähler quotient Advances in Mathematics K3 surface Eguchi–Hanson space Hodge structure Holonomy Differential geometry Special unitary group Sphere Kunihiko Kodaira Kähler manifold Metric tensor Shing-Tung Yau Twistor theory Vector bundle Gibbons–Hawking ansatz Nahm equations Asymptotically locally Euclidean Almost complex structure Ricci-flat
Link from a Wikipage to an external page	http://www.msri.org/people/staff/levy/files/Book59/64ogrady.pdf https://www.ams.org/mathscinet-getitem%3Fmr=2931873 http://www.numdam.org/item%3Fid=SB_1991-1992__34__137_0%7Ctitle=
sameAs	Hyperkähler manifold Hyperkähler manifold Hyperkähler manifold Hyperkähler manifold Hyperkähler manifold Hyperkähler manifold
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