Conjugate point

A conjugate point marks a nonzero endpoint-vanishing Jacobi variation and the loss of positive definiteness of the second variation.

For the energy functional of a Riemannian manifold, this variational condition yields conjugate points along a geodesic.

Assume the Strengthened Legendre condition $P>0$ and let $q(\phi )={\int }_a^b(P{\phi }^{\prime 2}+Q{\phi }^2)$. Then $q$ is strictly positive on all nonzero fixed-endpoint Lipschitz variations exactly when the normalized Jacobi equation solution with $w(a)=0$, $P(a)w^{\prime }(a)=1$ has no zero in $(a,b]$. A zero at $c$ gives a nonzero variation equal to $w$ up to $c$ and zero thereafter, with $q=0$. Conversely, if no zero occurs, the Jacobi identity for the second variation writes $q$ as a nonnegative square. Equality forces $\phi =Cw$, and the endpoint at $b$ forces $C=0$.

For continuous $P\ge 0,Q$, a scalar second variation $q=\int (P{\phi }^{\prime 2}+Q{\phi }^2)$ is strictly positive on nonzero fixed-endpoint Lipschitz variations if and only if every regularized form $q_{\eta }$ for $\eta >0$ has no conjugate point in $(a,b]$, and the original weak Jacobi boundary problem has no nonzero Lipschitz solution. The Jacobi positivity criterion applies to each regularized coefficient $P+\eta >0$. Letting $\eta$ decrease to zero gives $q\ge 0$. For a nonnegative quadratic form, $q(\phi )=0$ forces its mixed bilinear form with every test function to vanish, by considering $q(\phi +t\psi )$ for both signs of $t$. This is the weak Jacobi equation, so the kernel condition gives strict positivity. Necessity follows by the same argument in reverse. It distinguishes strict positivity from a uniform coercive estimate when the derivative coefficient degenerates.

Along a nonconstant affinely parametrized geodesic, the multiplicity of a conjugate point at positive time $t_0$ is the dimension of the space of Jacobi fields vanishing at both endpoints. The tangential component of such a Jacobi field is affine and vanishes at both endpoints, hence is zero. Uniqueness for the Jacobi field equation embeds this space into the $(n-1)$-dimensional normal space by $J\mapsto D_{t}J(0)$. Thus its dimension is at most $n-1$.

For a Jacobi field along a geodesic with nonpositive sectional curvature, the curvature convention with positive round-sphere curvature givesThus $|J{|}^2$ is nonnegative and convex. If it vanishes at both ends of a segment, convexity forces it to vanish everywhere. No nonzero Jacobi field can vanish at both ends, so the segment has no conjugate points. Only curvature of planes containing the geodesic tangent is needed, and completeness is unnecessary.