Affine 接続$\nabla$をもつ多様体$M$上で定義された三重$C^{\infty}$-線形写像
\[R\colon \mathcal{X}(M)\times\mathcal{X}(M)\times \mathcal{X}(M) \to \mathbb{R}; (X,Y,Z) \mapsto \nabla_{X}(\nabla_{Y}Z)- \nabla_{Y}(\nabla_{X}Z) – \nabla_{[X,Y]}Z\] を、接続$\nabla$の曲率テンソル場という。
\[
[X,Y]=-[Y,X] \]
\[X(fg)=(Xf)g+f(Xg)\] 特に,
\[X(1)=0 \] \[ [fX, gY ] = fg[X, Y ] + f(Xg)Y − g(Y f)X. \] 特に,
\[[fX, Y ] = f[X, Y ] − (Y f)X. \] \[\nabla_{ [fX, Y ] }(Z)=\nabla_{ f[X, Y ] }Z-\nabla_{ (Yf)X }(Z)\] \[\nabla_{ [X, Y ] }(fZ)=([X,Y]f)Z+f\nabla_{ [X, Y ] }(Z)\]
$X, Y,Z\in \mathcal{X}(M)$, $f,g,h \in C^{\infty}$,
\[R(fX,Y,Z)= fR(X,Y,Z)\] \[R(X,Y,Z)= -R(Y,X,Z)\] \[R(X,fY,Z)= fR(Y,X,Z)\] \[\nabla_{ [X, Y ] }(fZ)=([X,Y]f)Z+f\nabla_{ [X, Y ] }(Z)\] \[R(X,Y,fZ)=fR(X,Y,Z)\]
\begin{align}
R(fX,Y)Z&= \nabla_{fX}(\nabla_{Y}(Z))- \nabla_{Y}(\nabla_{fX}(Z))-\nabla_{ [fX, Y ] }(Z)\\
&= f\nabla_{X}(\nabla_{Y}(Z))-\nabla_{Y}(f\nabla_{X}(Z))-(\nabla_{ f[X, Y ] }Z-\nabla_{ (Yf)X }(Z))\\
&= f\nabla_{X}(\nabla_{Y}(Z))-((Yf)(\nabla_{X}(Z))+f\nabla_{Y}\nabla_{X}Z)-(\nabla_{ f[X, Y ] }Z-\nabla_{ (Yf)X }(Z))\\
&= f\nabla_{X}(\nabla_{Y}(Z))-((Yf)(\nabla_{X}(Z))+f\nabla_{Y}\nabla_{X}Z)-(f\nabla_{ [X, Y ] }Z-(Yf)\nabla_{ X }(Z))\\
&= f\nabla_{X}(\nabla_{Y}(Z))-(f\nabla_{Y}\nabla_{X}Z)-(f\nabla_{ [X, Y ] }Z)\\
&= f(\nabla_{X}(\nabla_{Y}(Z))-(\nabla_{Y}\nabla_{X}Z)-(\nabla_{ [X, Y ] }Z))\\
&= fR(X,Y,Z)
\end{align}
これで一つ目の等式がわかる。二つ目の等号は、\[
[X,Y]=-[Y,X]
\]からすぐわかる。三つ目の等号は、
\begin{align}
R(fX,Y)Z&= \nabla_{fX}(\nabla_{Y}(Z))- \nabla_{Y}(\nabla_{fX}(Z))-\nabla_{ [fX, Y ] }(Z)\\
&= \nabla_{fX}(\nabla_{Y}(Z))- \nabla_{Y}(\nabla_{fX}(Z))-\nabla_{ [fX, Y ] }(Z)\\
&= fR(X,Y,Z)
\end{align}
$ \nabla_{X}(\nabla_{Y}(fZ))= \nabla_{X}((Yf)Z+(\nabla_{fY}(Z)))= [X(Yf)Z+(Yf)\nabla_{X}Z]+[Xf(\nabla_{Y}(Z))+f\nabla_{X}(\nabla_{Y}(Z))]$
同様に,
$ \nabla_{Y}(\nabla_{X}(fZ))=\nabla_{Y}((Xf)Z+(\nabla_{fX}(Z)))= [Y(Xf)Z+(Xf)\nabla_{Y}Z]+[Yf(\nabla_{X}(Z))+f\nabla_{Y}(\nabla_{X}(Z))]$
\begin{align}
& \nabla_{X}(\nabla_{Y}(fZ))- \nabla_{Y}(\nabla_{X}(fZ))\\
&=X(Yf)Z+f\nabla_{X}(\nabla_{Y}(Z))-(Y(Xf)Z+f\nabla_{Y}(\nabla_{X}(Z))) \\
&=[X,Y](fZ)+f( \nabla_{X}(\nabla_{Y}Z)- \nabla_{Y}(\nabla_{X}Z))\\
&=\nabla_{ [X, Y ] }(fZ)-f\nabla_{ [X, Y ] }(Z)+f( \nabla_{X}(\nabla_{Y}Z)- \nabla_{Y}(\nabla_{X}Z)) \\
&=\nabla_{ [X, Y ] }(fZ)+f( \nabla_{X}(\nabla_{Y}Z)- \nabla_{Y}(\nabla_{X}Z)-\nabla_{ [X, Y ] }(Z))\\
&=\nabla_{ [X, Y ] }(fZ)+fR(X,Y,Z)
\end{align}
$f,g,h \in C^{\infty}$,
\[R(fX,gY,hZ)= fghR(X,Y,Z)\]