Kähler–Ricci 曲率 (用 Futaki 老师的记号) 定义为
$$
R_{i\bar j} = g^{k \bar \ell} R_{i\bar\ell \bar j k} = {{R_i}^k}_{\bar j k} = - {R_{i\bar j k}}^k = {R^k}_{ki\bar j} \in C^\infty (\operatorname{End}(E)\otimes \wedge^2 M).
$$
相比之下, Riemann–Ricci 曲率为
$$
R_{ij} = {{R_i}^k}_{jk}.
$$
此时的 Bianchi 恒等式为
$$
{{R_i}^k}_{\bar j k} + {R_{i\bar j k}}^k + {{R_{ik}}^k}_{\bar j} = 0.
$$
$R_{i\bar j}$ 由 $g_{k\bar\ell}$ 及其导数表示:
$$
R_{i\bar j} = - g^{k\bar\ell} \Big(
\frac{\partial^2}{\partial z^i \partial \bar z^j}
- g^{p\bar q}\frac{\partial g_{k\bar q}}{\partial z^i}
\frac{\partial g_{p\bar \ell}}{\partial \bar z^j}
\Big)
= - \frac{\partial^2}{\partial z^i \partial \bar z^j}
\log\det (g_{k\bar\ell}).
$$
与陈类的关系
$$
\sqrt{-1} R_{i\bar j} dz^i \wedge d\bar z^j = \sqrt{-1}\operatorname{tr}R = 2\pi c_1(M).
$$
Ricci 流
Ricci 流的方程为
$$
\frac{\partial g_{i\bar j}}{\partial t} = - R_{i \bar j}.
$$