The purpose of this appendix to [2] is to discuss the non-extension of the Riemannian Ricci curvature deformation theorems in [2] to Hermitian connections for Hermitian metrics. However the Riemannian results of [2] do extend to the Levi-Civita connection of a Hermitian metric. To avoid confusion, we first recall some standard definitions and facts.
Let (M, J) be a fixed complex manifold with a fixed complex structure o r. Let g be a Hermitian metric for (M, J), i.e., g(Jx, Jy)=g(x, y) for all x, y~TM. Since g is in particu!ar a Riemannian metric for M, g determines a unique Levi-Civita connection D with Dg=O and Tor(D)=0. (Here Tor(D)(X, Y)=DxY-DrX-[X, Y] for X, Y vector fields on M.) We denote by ric(g) and sc(g) respectively the Ricci curvature and scalar curvature determined by D and g. The fundamental 2-form ~b(g) of the Hermitian manifold (M, J, g) is given by q~(g) (x, y): =g(Jx, y). Ifd~(g) =0, then (M, or, g) is called Kfihler. It is well known that DJ=O iff (M, J, g) is K~ihler. We note also the elementary * This work was done under the program of the Sonderforschungsbereich 'Theoretische Mathematik' at the University of Bonn.