Affine extension of geodesics confined to a compact set with finite auxiliary length, and the logarithmic speed bound

(The rescaled velocity.) A nonconstant geodesic has no zero velocity at any parameter: a zero-velocity datum launches the constant solution, and ODE uniqueness would make the entire maximal solution constant. Therefore its $h$-speed is positive. Let $\gamma :J\to M$ be a nonconstant geodesic with affine parameter $\lambda$ and let $\mathrm{\ell }$ be $h$-arclength, so $d\mathrm{\ell }\mathrm{/}d\lambda =q>0$. Write $v:=d\gamma \mathrm{/}d\mathrm{\ell }$ for the $h$-unit velocity of $\gamma$ and $u:=d\gamma \mathrm{/}d\lambda =qv$ for the affine velocity. Since $\gamma$ is a geodesic, ${\mathrm{\nabla }}_{u}u=0$, hence ${\mathrm{\nabla }}_{v}v=\sigma v$ for the function $\sigma :=-d\log q\mathrm{/}d\mathrm{\ell }$: indeed ${\mathrm{\nabla }}_{qv}(qv)=q^2{\mathrm{\nabla }}_{v}v+qv(q)v$, so ${\mathrm{\nabla }}_{v}v=-\frac{v(q)}{q}v$ and $v(q)=dq\mathrm{/}d\mathrm{\ell }$ because $q$ is a function of $\mathrm{\ell }$ along the curve.

(Derivative of the $h$-speed.) Covariant differentiation along the reparametrized curve gives ${\mathrm{\nabla }}_{v}u=q^{-1}{\mathrm{\nabla }}_{u}u=0$; equivalently ${\mathrm{\nabla }}_v(qv)=q^{\mathrm{\prime }}v+q{\mathrm{\nabla }}_{v}v=0$ by step 1.1. Thus the tensor product rule gives $\frac{d}{d\mathrm{\ell }}h(u,u)=({\mathrm{\nabla }}_{v}h)(u,u)+2h({\mathrm{\nabla }}_{v}u,u)=({\mathrm{\nabla }}_{v}h)(u,u)$. Since $h(u,u)=q^2$, one obtains $2q{q}^{\mathrm{\prime }}=({\mathrm{\nabla }}_{v}h)(u,u)$.

(The bound.) Let $K$ be compact. By [L2] there is $\kappa (K)<\mathrm{\infty }$ with $\mathrm{\mid }({\mathrm{\nabla }}_{Z}h)(X,Y)\mathrm{\mid }\le \kappa (K)\mathrm{\mid }X{\mathrm{\mid }}_h\mathrm{\mid }Y{\mathrm{\mid }}_h\mathrm{\mid }Z{\mathrm{\mid }}_h$ for all points of $K$ and all tangent vectors there. On the parameters with $\gamma (\mathrm{\ell })\in K$: $\mathrm{\mid }({\mathrm{\nabla }}_{v}h)(u,u)\mathrm{\mid }\le \kappa (K)\mathrm{\mid }u{\mathrm{\mid }}_h^2\mathrm{\mid }v{\mathrm{\mid }}_h=\kappa (K)q^2$, so by step 2.1, $2q\mathrm{\mid }{q}^{\mathrm{\prime }}\mathrm{\mid }\le \kappa (K)q^2$, that is, $\mathrm{\mid }d\log q\mathrm{/}d\mathrm{\ell }\mathrm{\mid }\le \kappa (K)\mathrm{/}2$ there. This proves (a) with $c(K):=\kappa (K)\mathrm{/}2$.

(Bounds on the affine data up to the endpoint.) In the situation of clause (b), let $L:={\int }_{{\lambda }_0}^{\beta }\mathrm{\mid }{\gamma }^{\mathrm{\prime }}{\mathrm{\mid }}_{h}d\lambda <\mathrm{\infty }$ be the finite tail length and put $\mathrm{\ell }(\lambda ):={\int }_{{\lambda }_0}^{\lambda }\mathrm{\mid }{\gamma }^{\mathrm{\prime }}{\mathrm{\mid }}_{h}dt$, a strictly increasing smooth map $[{\lambda }_0,\beta )\to [0,L)$ with smooth inverse, so $\mathrm{\ell }\to L$ as $\lambda \to \beta$. By clause (a) applied on the tail, with $\gamma ([{\lambda }_0,\beta ))\subseteq K$, one has $q(\mathrm{\ell })\le q(0)e^{c(K)L}=:M$ and $q(\mathrm{\ell })\ge q(0)e^{-c(K)L}=:m>0$ for all $\mathrm{\ell }\in [0,L)$.

(A compact set of bounded affine data.) The set $B:=\{(x,w)\in TM:x\in K,\mathrm{\mid }w{\mathrm{\mid }}_h\le M\}$ is compact. Indeed take finitely many compact chart pieces covering $K$, each contained in a tangent-bundle trivialization; such pieces are obtained by shrinking coordinate balls and taking a finite subcover of $K$. On each piece [L2] bounds the Euclidean velocity coordinates by $M\mathrm{/}\sqrt{m_j}$, and the condition $h_x(w,w)\le M^2$ cuts out a closed subset of the product of that compact piece and a closed Euclidean ball. Each resulting set is compact, and their finite union is $B$. Step 4.1 puts every affine datum $(\gamma (\lambda ),{\gamma }^{\mathrm{\prime }}(\lambda ))$ on the tail in $B$. No sequence or subsequence of velocities is selected.

(Uniform ODE continuation.) Apply [L1] to the smooth first-order geodesic vector field in tangent-bundle charts. At every datum in $B$ it supplies a neighborhood of initial data and a positive common existence time in that chart. Take a finite subcover of $B$ by all such neighborhoods and let $\delta >0$ be smaller than their finitely many times. Every datum in $B$ therefore launches a geodesic for at least time $\delta$. Fix ${\lambda }_{\ast }\in ({\lambda }_0,\beta )$ with $\beta -{\lambda }_{\ast }<\delta \mathrm{/}2$. The solution launched from $(\gamma ({\lambda }_{\ast }),{\gamma }^{\mathrm{\prime }}({\lambda }_{\ast }))$ exists past $\beta$ and agrees with $\gamma$ on their overlap by uniqueness. Gluing it to the earlier part extends the affine geodesic beyond $\beta$, contradicting maximality and proving (b). This uses only a finite subcover and finite minima, so no choice principle is needed.