リーマン曲率テンソル
0 = ∂ 3 u a ∂ x l ∂ x j ∂ x k − ∂ 3 u a ∂ x j ∂ x l ∂ x k = ∂ u a ∂ x b [ ∂ { b j k } ∂ x l − ∂ { b l k } ∂ x j + ∑ i { b l i } { i j k } − ∑ i { b j i } { i l k } ] = ∂ u a ∂ x b R j k l b {\displaystyle {\begin{aligned}0&={\frac {\partial ^{3}u^{a}}{\partial x^{l}\partial x^{j}\partial x^{k}}}-{\frac {\partial ^{3}u^{a}}{\partial x^{j}\partial x^{l}\partial x^{k}}}\\&={\frac {\partial u^{a}}{\partial x^{b}}}\left[{\frac {\partial \left\{{{b} \atop {jk}}\right\}}{\partial x^{l}}}-{\frac {\partial \left\{{{b} \atop {lk}}\right\}}{\partial x^{j}}}+\sum _{i}\left\{{{b} \atop {l\,i}}\right\}\left\{{{i} \atop {jk}}\right\}-\sum _{i}\left\{{{b} \atop {ji}}\right\}\left\{{{i} \atop {lk}}\right\}\right]\\&={\frac {\partial u^{a}}{\partial x^{b}}}R_{jkl}{}^{b}\end{aligned}}}
したがって、 R j k l b = 0 {\displaystyle R_{jkl}{}^{b}=0} 。
R k j i h = − R j k i h {\displaystyle R_{kjih}=-R_{jkih}} (定義より)
R k j i h = − R k j h i {\displaystyle R_{kjih}=-R_{kjhi}} (後述)
R k j i h + R i k j h + R j i k h = 0 {\displaystyle R_{kjih}+R_{ikjh}+R_{jikh}=0} (定義より)