= Radial Riccati equation for distance spheres
{title2=$A'+A^2+\mathcal R_{\partial_r}=0$}
Before the <cut locus>, define $A(X)=\nabla_X\partial_r$ on the tangent space of a distance sphere. This is the negative of the <shape operator> with outward normal under the convention $S=-\nabla\nu$. A nonsingular matrix $Y$ of radial <Jacobi fields> satisfies $Y''+\mathcal R_{\partial_r}Y=0$ and $A=Y'Y^{-1}$, yielding the displayed matrix <Riccati equation>. Taking its trace gives $h'+\operatorname{tr}(A^2)+\operatorname{Ric}(\partial_r,\partial_r)=0$, where $h=\operatorname{tr}A=\partial_r\log\det Y$. This is the local differential identity behind the <Bishop-Gromov inequality>.
Back to article page
