= Jacobi curvature operator
{c}
{title2=$\mathcal R_X(v)=R(v,X)X$}
For the <Riemann curvature tensor> convention $R(X,Y)Z=\nabla_X\nabla_YZ-\nabla_Y\nabla_XZ-\nabla_{[X,Y]}Z$, define $\mathcal R_X(v)=R(v,X)X$. Curvature pair interchange makes this operator self-adjoint. If $v\perp X$, then $\langle\mathcal R_Xv,v\rangle=K(v,X)|v|^2|X|^2$. The <Jacobi field> equation is $D_t^2J+\mathcal R_{\dot\gamma}J=0$. The operator $v\mapsto R(X,v)X$ is $-\mathcal R_X$ and is likewise self-adjoint; this sign distinction matters when testing curvature inequalities.
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