Jacobi fields in constant sectional curvature

Given: The constant sectional curvature $K$, a nondegenerate interval $I$ containing $0$, and a supplied unit-speed affine geodesic $\gamma$ on $I$.

${\mathrm{A}\mathrm{C}}_{\omega }$ is countable choice. Its only role here is inherited through the constant-sectional-curvature and curvature-tensor interfaces; the coordinate computations and the unique extensions of supplied vectors use no further choice, and no full AC is assumed (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), Constant sectional curvature and space form, Curvature tensor of constant sectional curvature).

Constant sectional curvature $K$ means that every tangent two-plane has sectional curvature $K$. In dimensions zero and one this predicate is vacuous for every $K$ (Constant sectional curvature and space form).

Under this hypothesis, the curvature operator is $R(X,Y)Z=K(g(Y,Z)X-g(X,Z)Y)$ (Curvature tensor of constant sectional curvature).

A Jacobi field is a smooth field satisfying $D_t^{2}J+R(J,T)T=0$ throughout the interval (Jacobi field).

The affine geodesic equation gives $D_{t}T=0$, and unit speed gives $g(T,T)=1$ (Geodesic of an affine connection, given).

Levi-Civita metric compatibility, expressed in a local frame with metric matrix $H$ and connection matrix $B$, gives $H^{\mathrm{\prime }}=B^{\mathsf{T}}H+HB$. The along-curve frame formula is $D_t(eu)=e(u^{\mathrm{\prime }}+Bu)$, and covariant differentiation also satisfies $D_t(fV)=f^{\mathrm{\prime }}V+f{D}_{t}V$. Therefore, for fields $U,V$ along $\gamma$, $(g(U,V){)}^{\mathrm{\prime }}=g(D_{t}U,V)+g(U,D_{t}V)\mathrm{.}$ (Levi civita connection, Metric compatible connection on a riemannian vector bundle, Local frame formula for covariant differentiation along a curve, Covariant derivative along a curve)

Every supplied initial vector along $\gamma$ has a unique parallel extension to all of $I$; a field is parallel when $D_{t}E=0$ (Existence and uniqueness of parallel sections, Parallel section along a curve).

A continuous real function on an interval whose derivative vanishes at every interior point is constant on that 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).

Initial value and covariant derivative at $0$ determine a unique Jacobi field along all of $I$, including when $0$ is an included endpoint (Existence and uniqueness of jacobi fields from initial data).

Since $\gamma$ has unit speed it is nonconstant, and for every smooth $f:I\to \mathbb{R}$, $fT$ is Jacobi exactly when $f(t)=at+b$ for constants $a,b$ (Tangential jacobi fields are affine multiples of the velocity).