Jacobi fields in euclidean space

In the global Cartesian chart, [F1] gives the constant metric matrix $g_{ij}={\delta }_{ij}$, which is smooth, symmetric, and positive definite. All coordinate derivatives ${\mathrm{\partial }}_i{g}_{j\mathrm{\ell }}$ vanish, so [F3] gives ${\mathrm{\Gamma }}^k{}_{ij}=0$ identically.

Write an arbitrary smooth field as $J(t)={\sum }_i{j}_i(t){\mathrm{\partial }}_i{\mathrm{\mid }}_{\gamma (t)}$ using [F6]. By the pullback-connection definition in [F7] and the Christoffel definition in [F3], $D_t{\mathrm{\partial }}_i={\sum }_j{\dot{\gamma }}^j{\mathrm{\nabla }}_{{\mathrm{\partial }}_j}{\mathrm{\partial }}_i={\sum }_{j,k}{\dot{\gamma }}^j{\mathrm{\Gamma }}^k{}_{ji}{\mathrm{\partial }}_k=0,$ because step 1.1 gives every ${\mathrm{\Gamma }}^k{}_{ji}=0$. The product rule in [F7], applied twice, gives $D_{t}J={\sum }_i{j}_i^{\mathrm{\prime }}(t){\mathrm{\partial }}_i$ and $D_t^{2}J={\sum }_i{j}_i^{\mathrm{\prime }\mathrm{\prime }}(t){\mathrm{\partial }}_i$.

The coordinate functions of $\gamma$ are $x^i+t{v}^i$, hence ${\ddot{\gamma }}^i=0$; [F5] and step 1.1 show that $\gamma$ is an affinely parametrized geodesic. Substitution of $\mathrm{\Gamma }=0$ and its zero derivatives into [F4] gives every curvature component zero; [F2] then gives $R=0$. By step 1.2 and [F8], the Jacobi equation is therefore ${\sum }_i{j}_i^{\mathrm{\prime }\mathrm{\prime }}{\mathrm{\partial }}_i=0$, and [F2] implies $j_i^{\mathrm{\prime }\mathrm{\prime }}=0$ for every coordinate on the interior of $I$.

For each $i$, the function $j_i^{\mathrm{\prime }}$ is continuous on $I$ and has derivative $j_i^{\mathrm{\prime }\mathrm{\prime }}=0$ at every interior point, so [F9] makes it a constant $B_i$. By [F10], the continuous function $j_i(t)-B_{i}t$ has derivative $j_i^{\mathrm{\prime }}(t)-B_i=0$ on the interior; another application of [F9] makes it a constant $A_i$. Thus $j_i(t)=A_i+t{B}_i$ throughout $I$, including any endpoints by continuity, and the component vectors give $J(t)=A+tB$.

Conversely, for any constant $A,B\in {\mathbb{R}}^n$, the field $J(t)=A+tB$ is smooth. Steps 1.1–1.2 and [F7] give $D_{t}J=B$ in the constant coordinate frame, $D_t^{2}J=0$, and step 2.1 gives $R=0$; [F8] therefore makes $J$ Jacobi.

If $n=0$, the tangent spaces and field contain only zero, represented by the unique $A=B=0$; if $n=1$, the same component calculation is scalar. The case $v=0$ is included because step 2.1 still gives a constant geodesic and step 1.2 allows arbitrary smooth coefficient functions; $A=0$, $B=0$, or $J=0$ need no exclusion. At included time endpoints all derivatives are one-sided and the affine formula extends by continuity. Only finitely many coordinates are considered, and the proof makes no choice; singleton intervals are excluded by the nondegeneracy hypothesis. Steps 2.1, 3.1 and 3.2 prove both directions of the stated classification.