|
In mathematics, a Hermitian manifold is the complex analogue of a Riemannian manifold. Specifically, 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 an 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 almost complex structure. This form is always non-degenerate, with the suitable integrability condition (of it also being closed and thus a symplectic form) we get an almost Kähler structure. If both almost complex structure and fundamental form are integrable, we have a Kähler structure. ==Formal definition== A Hermitian metric on a complex vector bundle ''E'' over a smooth manifold ''M'' is a smoothly varying positive-definite Hermitian form on each fiber. Such a metric can be written as a smooth section : such that : for all ζ, η in ''E''''p'' and : for all nonzero ζ in ''E''''p''. A Hermitian manifold is a complex manifold with a Hermitian metric on its holomorphic tangent space. Likewise, an almost Hermitian manifold is an almost complex manifold with a Hermitian metric on its holomorphic tangent space. On a Hermitian manifold the metric can be written in local holomorphic coordinates (''z''α) as : where are the components of a positive-definite Hermitian matrix. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hermitian manifold」の詳細全文を読む スポンサード リンク
|