Ivancevic_Applied-Diff-Geom

434 Applied Differential Geometry: A Modern Introductionthe complex structure are related. It gives M a Kähler structure, and hasmany implications.In particular, on C 2 , the Kähler form can be written asω = − i 2(dz1 ∧ dz 1 + dz 2 ∧ dz 2)= dx1 ∧ dy 1 + dx 2 ∧ dy 2 ,where z n = x n + i y n . In general, the Kähler form can be written incoordinatesω = g ij dz i ∧ dz j ,where g ij is a Hermitian metric, the real part of which is the Kähler metric.Locally, a Kähler form can be written as i∂ ¯∂f, where f is a function calleda Kähler potential. The Kähler form is a real (1, 1)−complex form. TheKähler potential is a real–valued function f on a Kähler manifold for whichthe Kähler form ω can be written as ω = i∂ ¯∂f, where,∂ = ∂ zk dz k and ¯∂ = ∂¯zk d¯z k .In local coordinates, the fact that dJ = 0 for a Kähler manifold MimpliesThis implies thatdJ = (∂ + ¯∂)ig i¯j dz i ∧ d¯z¯j = 0.∂ z lg i¯j = ∂ z ig l¯j (3.240)and similary with z and ¯z interchanged. From this we see that locally wecan express g i¯j asg i¯j =∂2 φ∂z i ∂¯z¯j .That is, ω = i∂ ¯∂φ, where φ is a locally defined function in the patch whoselocal coordinates we are using, which is known as the Kähler potential.If ω on M is a Kähler form, the conditions (3.240) imply that thereare numerous cancellations in (3.240). so that the only nonzero Christoffelsymbols (of the standard Levi–Civita connection) in complex coordinatesare those of the form Γ l jkholomorphic. Specifically,and Γ¯l¯j¯k, with all indices holomorphic or anti–Γ l jk = g l¯s ∂ z j g k¯s and Γ¯l¯j¯k = g¯ls ∂¯z¯jg¯ks .