リーマン曲率テンソル
リッチテンソルの定義より
R j i = ∑ a R a j i a = ∑ a ( ∂ { a j i } ∂ x a − ∂ { a a i } ∂ x j + ∑ b { a a b } { b j i } − ∑ b { a j b } { b a i } ) = ∑ a ( ∂ { a j i } ∂ x a − ∑ b { a j b } { b a i } ) + ∑ a ( − ∂ { a a i } ∂ x j + ∑ b { a a b } { b j i } ) {\displaystyle {\begin{aligned}R_{ji}&=\sum _{a}R_{aji}{}^{a}=\sum _{a}\left({\frac {\partial \left\{{{a} \atop {ji}}\right\}}{\partial x^{a}}}-{\frac {\partial \left\{{{a} \atop {ai}}\right\}}{\partial x^{j}}}+\sum _{b}\left\{{{a} \atop {ab}}\right\}\left\{{{b} \atop {ji}}\right\}-\sum _{b}\left\{{{a} \atop {jb}}\right\}\left\{{{b} \atop {ai}}\right\}\right)\\&=\sum _{a}\left({\frac {\partial \left\{{{a} \atop {ji}}\right\}}{\partial x^{a}}}-\sum _{b}\left\{{{a} \atop {jb}}\right\}\left\{{{b} \atop {ai}}\right\}\right)+\sum _{a}\left(-{\frac {\partial \left\{{{a} \atop {ai}}\right\}}{\partial x^{j}}}+\sum _{b}\left\{{{a} \atop {ab}}\right\}\left\{{{b} \atop {ji}}\right\}\right)\end{aligned}}}
∂ 2 u a ∂ x j ∂ x k = ∑ i ∂ u a ∂ x i { i j k } {\displaystyle {\frac {\partial ^{2}u^{a}}{\partial x^{j}\partial x^{k}}}=\sum _{i}{\frac {\partial u^{a}}{\partial x^{i}}}\left\{{{i} \atop {jk}}\right\}}
が得られる。両辺偏微分を行うと
∂ 3 u a ∂ x l ∂ x j ∂ x k = ∑ i ∂ 2 u a ∂ x l ∂ x i { i j k } + ∑ i ∂ u a ∂ x i ∂ { i j k } ∂ x l = ∑ b , i ∂ u a ∂ x b { b l i } { i j k } + ∑ i ∂ u a ∂ x i ∂ { i j k } ∂ x l {\displaystyle {\frac {\partial ^{3}u^{a}}{\partial x^{l}\partial x^{j}\partial x^{k}}}=\sum _{i}{\frac {\partial ^{2}u^{a}}{\partial x^{l}\partial x^{i}}}\left\{{{i} \atop {jk}}\right\}+\sum _{i}{\frac {\partial u^{a}}{\partial x^{i}}}{\frac {\partial \left\{{{i} \atop {jk}}\right\}}{\partial x^{l}}}=\sum _{b,i}{\frac {\partial u^{a}}{\partial x^{b}}}\left\{{{b} \atop {l\,i}}\right\}\left\{{{i} \atop {jk}}\right\}+\sum _{i}{\frac {\partial u^{a}}{\partial x^{i}}}{\frac {\partial \left\{{{i} \atop {jk}}\right\}}{\partial x^{l}}}}