Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 433M, we can build a form in Ω 1,1 (M) — that is, a form of type (1, 1) in thefollowing way:ω = ig i¯j dz i ⊗ d¯z¯j − ig¯ji d¯z¯j ⊗ dz i .By the symmetry of g, we can write this asω = ig i¯j dz i ∧ d¯z j .Now, if ω is closed, that is, if dJ = 0, then ω is called a Kähler form and Mis called a Kähler manifold. At first sight, this Kählerity condition mightnot seem too restrictive. However, it leads to remarkable simplifications inthe resulting differential geometry on M.A Kähler structure on a complex manifold M combines a Riemannianmetric on the underlying real manifold with the complex structure. Sucha structure brings together geometry and complex analysis, and the mainexamples come from algebraic geometry. When M has n complex dimensions,then it has 2n real dimensions. A Kähler structure is related to theunitary group U(n), which embeds in SO(2n) as the orthogonal matricesthat preserve the almost complex structure (multiplication by i). In a coordinatechart, the complex structure of M defines a multiplication by i andthe metric defines orthogonality for tangent vectors. On a Kähler manifold,these two notions (and their derivatives) are related.A Kähler manifold is a complex manifold for which the exterior derivativeof the fundamental form ω associated with the given Hermitian metricvanishes, so dω = 0. In other words, it is a complex manifold with a Kählerstructure. It has a Kähler form, so it is also a symplectic manifold. It hasa Kähler metric, so it is also a Riemannian manifold.The simplest example of a Kähler manifold is a Riemann surface, whichis a complex manifold of dimension 1. In this case, the imaginary part ofany Hermitian metric must be a closed form since all 2−forms are closedon a real 2D manifold.In other words, a Kähler form is a closed two–form ω on a complexmanifold M which is also the negative imaginary part of a Hermitian metrich = g−iw. In this case, M is called a Kähler manifold and g, the real part ofthe Hermitian metric, is called a Kähler metric. The Kähler form combinesthe metric and the complex structure, g(M, Y ) = ω(M, JY ),where ω isthe almost complex structure induced by multiplication by i. Since theKähler form comes from a Hermitian metric, it is preserved by ω, sinceh(M, Y ) = h(JX, JY ). The equation dω = 0 implies that the metric and