Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 435The curvature tensor also greatly simplifies. The only non–zero componentsof the Riemann tensor, when written in complex coordinates, havethe form R i¯jk¯l (up to index permutations consistent with symmetries of thecurvature tensor). And we haveas well as the Ricci tensorR i¯jk¯l = g i¯s ∂ z kΓ¯s¯j¯l ,Rīj = R¯kī¯kj = −∂ z j Γ¯kī¯k .Since the Kähler form ω is closed, it represents a cohomology class in thede Rham cohomology. On a compact manifold, it cannot be exact becauseω n /n! ≠ 0 is the volume form determined by the metric. In the special caseof a projective variety, the Kähler form represents an integral cohomologyclass. That is, it integrates to an integer on any 1D submanifold, i.e., analgebraic curve. The Kodaira Embedding Theorem says that if the Kählerform represents an integral cohomology class on a compact manifold, thenit must be a projective variety. There exist Kähler forms which are notprojective algebraic, but it is an open question whether or not any Kählermanifold can be deformed to a projective variety (in the compact case).A Kähler form satisfies Wirtinger’s inequality,|ω(M, Y )| ≤ |M ∧ Y | ,where the r.h.s is the volume of the parallelogram formed by the tangentvectors M and Y . Corresponding inequalities hold for the exterior powersof ω. Equality holds iff M and Y form a complex subspace. Therefore, thereis a calibration form, and the complex submanifolds of a Kähler manifoldare calibrated submanifolds. In particular, the complex submanifolds arelocally volume minimizing in a Kähler manifold. For example, the graphof a holomorphic function is a locally area–minimizing surface in C 2 = R 4 .Kähler identities is a collection of identities which hold on a Kählermanifold, also called the Hodge identities. Let ω be a Kähler form, d = ∂+ ¯∂be the exterior derivative, [A, B] = AB − BA be the commutator of twodifferential operators, and A ∗ denote the formal adjoint of A. The followingoperators also act on differential forms α on a Kähler manifold:L(α) = α ∧ ω, Λ(α) = L ∗ (α) = α⌋ω, d c = −JdJ,where J is the almost complex structure, J = −I, and ⌋ denotes the interior