Wronskian of two jacobi fields is constant

Given: The inherited assumption is ${\mathrm{A}\mathrm{C}}_{\omega }$; $I$ is an interval with nonempty interior; $\gamma$ is an affinely parametrized geodesic; and $J,K$ are Jacobi fields along it.

${\mathrm{A}\mathrm{C}}_{\omega }$ is the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)).

Each Jacobi field satisfies $D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma }=0,$ with the curvature sign fixed in Jacobi field.

The curvature four-tensor is $\mathrm{Rm}(X,Y,Z,W)=g(R(X,Y)Z,W)$ (Riemann curvature four-tensor).

Pair interchange holds: $\mathrm{Rm}(X,Y,Z,W)=-\mathrm{Rm}(Y,X,Z,W),\mathrm{Rm}(X,Y,Z,W)=-\mathrm{Rm}(X,Y,W,Z),\mathrm{Rm}(X,Y,Z,W)=\mathrm{Rm}(Z,W,X,Y)\mathrm{.}$ The symmetry theorem assumes and propagates ${\mathrm{A}\mathrm{C}}_{\omega }$ (Algebraic symmetries of the Riemann tensor).

The Levi-Civita connection is metric compatible, so along a curve

for sections $U,V$ of the pulled-back tangent bundle; in a local frame this is the metric-compatibility identity applied to the connection along $\gamma$ (Levi civita connection, Metric compatible connection on a riemannian vector bundle, Covariant derivative along a curve).

$D_t$ is the covariant derivative along the given curve; it is defined for intervals with nonempty interior, with one-sided values at included endpoints (Covariant derivative along a curve).

A continuous real function on an interval whose derivative vanishes at every interior point is constant on the whole interval (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$; consequently two such functions with the same derivative differ by a constant).

$g$ is a symmetric inner product on each tangent space (Riemannian metric and riemannian manifold).