= Riccati factorization of the Riemannian index form
{c}
{title2=$S=A\prime A^{-1}$}
In a parallel orthonormal frame along a <geodesic>, write the <Jacobi equation> as $A''+KA=0$, with $A(0)=0$, $A'(0)=I$ and symmetric $K$. In the absence of <conjugate points> on $(0,b]$, $A$ is invertible there. The conserved quantity $A^TA'-A'^TA=0$ makes $S=A'A^{-1}$ symmetric, and $S'+S^2+K=0$. For a field $u$ vanishing at both endpoints,
$$
I(u,u)=\int_0^b|u'-Su|^2\,dt.
$$
Indeed the difference of the integrands is $(u^TSu)'$; its boundary value at zero vanishes because $S=t^{-1}I+O(t)$ and $u=O(t)$. Equality forces $u=Ac$ and then $A(b)c=0$, so $u=0$. This proves strict positivity and the minimizing property of the <Jacobi field> with prescribed endpoint values.
// Target: riemannian-geometry.bigb
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