Tangential jacobi fields are affine multiples of the velocity

Put $T=\dot{\gamma }$. By [F1] and [F2], on the interior of $I$, $D_t(fT)=f^{\mathrm{\prime }}T,D_t^2(fT)=f^{\mathrm{\prime }\mathrm{\prime }}T\mathrm{.}$ By [F4], $R(fT,T)T=0$. Hence the left side of the Jacobi equation for $J=fT$ is exactly $f^{\mathrm{\prime }\mathrm{\prime }}T$.

By [F5], the speed is constant. If it were zero, positive definiteness would give $T=0$ throughout. In a chart around each image point, the coordinate derivative is then zero, so the curve is locally constant; because $I$ is an interval it is connected, and $\gamma$ would be constant globally. This contradicts the hypothesis. Thus $T(t)\mathrm{\ne }0$ throughout $I$, and [F3] with step 1.1 shows that $J$ is Jacobi exactly when $f^{\mathrm{\prime }\mathrm{\prime }}(t)=0$ at every interior point. If an endpoint is included, the same equation there follows by continuity and the one-sided derivative convention.

Suppose $J$ is Jacobi. By step 2.1, $f^{\mathrm{\prime }\mathrm{\prime }}=0$ on the interior. The function $f^{\mathrm{\prime }}$ is continuous on $I$ and has zero derivative throughout its interior, so [F6] makes $f^{\mathrm{\prime }}$ a constant $a$ on all of $I$. The continuous function $h(t)=f(t)-at$ has zero derivative on the interior; a second application of [F6] gives a constant $b$ with $f(t)=at+b$ for every $t\in I$.

Conversely, if $f(t)=at+b$ on $I$, then $f^{\mathrm{\prime }\mathrm{\prime }}=0$ throughout its interior. Step 2.1 therefore gives that $J=fT$ is Jacobi, with the endpoint equation following by the same one-sided continuity argument.

If $\gamma$ is constant, then $T=0$ and $fT$ is the zero field for every smooth $f$; [F3] makes it Jacobi. This explains why the iff assertion is restricted to nonconstant geodesics. The argument makes no selections and uses no choice axiom.