Wiki. “Kähler–Ricci 曲率” [Kähler--Ricci曲率]

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 $$ \frac{\partial g_{i\bar j}}{\partial t} = - R_{i \bar j}. $$