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In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure.

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rdfs:label	Hermitian manifold (en) 에르미트 다양체 (ko) エルミート多様体 (ja)
rdfs:comment	미분기하학에서 에르미트 다양체(Hermite多樣體, 영어: Hermitian manifold)는 일종의 계량 텐서를 가진 복소다양체이다. 복소 기하학에서 리만 다양체에 대응되는 개념이다. 켈러 다양체와 칼라비-야우 다양체는 에르미트 다양체의 특수한 경우다. (ko) 数学におけるエルミート多様体（英語: Hermitian manifold）とはリーマン多様体の複素微分幾何における類似である。より正確には、エルミート多様体とは、各点の正則接空間にエルミート内積を持ち、それらが滑らかに変化する複素多様体のことを指す。また、エルミート多様体を複素構造を保つリーマン計量を持つ実多様体として定義することもできる。複素構造は、本質的には可積分条件をもつ概複素構造であり、この条件は多様体上にユニタリ構造（(U(n) structure))をもたらす。可積分条件を落とすと、概エルミート多様体を得る。任意の概エルミート多様体上に、計量と概複素構造にのみ依存する基本2形式(fundamental 2-form)と呼ばれる微分形式を定めることができる。基本2形式は常に非退化である。これが閉形式である（すなわちシンプレクティック形式である）という追加の可積分条件を課すことにより、概ケーラー構造(almost Kähler structure)を得る。もし概複素構造と基本2形式の両方が可積分であれば、ケーラー構造を持つ。 (ja) In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure. (en)
dcterms:subject	Complex manifolds Differential geometry Structures on manifolds Riemannian manifolds Riemannian geometry
Wikipage page ID	2640459 (xsd:integer)
Wikipage revision ID	1090262116 (xsd:integer)
Link from a Wikipage to another Wikipage	Hermitian form Definite bilinear form Almost complex manifold Complex manifolds Holomorphic vector bundle Holonomy group Volume form Differential geometry Structures on manifolds Mathematics Orthonormal Riemannian manifolds Levi-Civita connection Complex differential form Complex manifold Complex vector bundle G-structure Symplectic manifold Tangent space Wedge product Foundations of Differential Geometry Frame bundle Riemannian manifold Riemannian metric Riemannian volume form Hermitian matrix Riemannian geometry Holomorphic tangent bundle Differential geometry Inner product Kähler manifold Reduction of the structure group Smooth manifold Unitary group Principal bundle Nondegenerate form Kähler structure Symplectic form Almost complex structure Wiley Interscience Integrability condition Closed differential form Complexified Unitary frame bundle
sameAs	Hermitian manifold Hermitian manifold Hermitian manifold Hermitian manifold Hermitian manifold Hermitian manifold Hermitian manifold
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has abstract	In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a smoothly varying Hermitian inner product on each (holomorphic) tangent space. One can also define a Hermitian manifold as a real manifold with a Riemannian metric that preserves a complex structure. A complex structure is essentially an almost complex structure with an integrability condition, and this condition yields a unitary structure (U(n) structure) on the manifold. By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we can introduce a fundamental 2-form (or cosymplectic structure) that depends only on the chosen metric and the almost complex structure. This form is always non-degenerate. With the extra integrability condition that it is closed (i.e., it is a symplectic form), we get an almost Kähler structure. If both the almost complex structure and the fundamental form are integrable, then we have a Kähler structure. (en) 미분기하학에서 에르미트 다양체(Hermite多樣體, 영어: Hermitian manifold)는 일종의 계량 텐서를 가진 복소다양체이다. 복소 기하학에서 리만 다양체에 대응되는 개념이다. 켈러 다양체와 칼라비-야우 다양체는 에르미트 다양체의 특수한 경우다. (ko) 数学におけるエルミート多様体（英語: Hermitian manifold）とはリーマン多様体の複素微分幾何における類似である。より正確には、エルミート多様体とは、各点の正則接空間にエルミート内積を持ち、それらが滑らかに変化する複素多様体のことを指す。また、エルミート多様体を複素構造を保つリーマン計量を持つ実多様体として定義することもできる。複素構造は、本質的には可積分条件をもつ概複素構造であり、この条件は多様体上にユニタリ構造（(U(n) structure))をもたらす。可積分条件を落とすと、概エルミート多様体を得る。任意の概エルミート多様体上に、計量と概複素構造にのみ依存する基本2形式(fundamental 2-form)と呼ばれる微分形式を定めることができる。基本2形式は常に非退化である。これが閉形式である（すなわちシンプレクティック形式である）という追加の可積分条件を課すことにより、概ケーラー構造(almost Kähler structure)を得る。もし概複素構造と基本2形式の両方が可積分であれば、ケーラー構造を持つ。 (ja)
gold:hypernym	Analogue
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foaf:isPrimaryTopicOf	wikipedia-en:Hermitian_manifold
is Link from a Wikipage to another Wikipage of	List of differential geometry topics Rizza manifold List of manifolds Kaehler form

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