Characterization of a cut point

Set-up and immediate consequences. [F3, F4, F5, F6, given] Put $A:=A_p(v)=\{t>0:d_g(p,\gamma (t))=t\}$, so that $c=\sup A$ by [F3]. By [F4] the set $A$ is an initial interval and, if $c<+\mathrm{\infty }$, then $c\in A$ and $d_g(p,\gamma (c))=c$. For every $t\ge 0$ the segment $\gamma {\mathrm{\mid }}_{[0,t]}$ is a unit-speed curve of length $t$ joining $p$ to $\gamma (t)$, so [F5] gives $d_g(p,\gamma (t))\le t$; consequently every $t>c$ satisfies $t\mathrm{\notin }A$ and hence $d_g(p,\gamma (t))<t$, because $c$ is an upper bound for $A$.

Converse for a conjugate instant. [F5, F6, F13, step 1.1] Let $t>0$ and suppose $\gamma (0)=p$ and $\gamma (t)$ are conjugate along $\gamma {\mathrm{\mid }}_{[0,t]}$. For every $b>t$ the geodesic $\gamma {\mathrm{\mid }}_{[0,b]}$ is nonconstant, and its restriction to $[0,t]$ exhibits the conjugate pair $\gamma (0)$, $\gamma (t)$; [F13] therefore supplies a piecewise smooth curve on $[0,b]$ with the same endpoints and strictly smaller length than $L(\gamma {\mathrm{\mid }}_{[0,b]})=b$. Its length is an admissible competitor for the distance, so $d_g(p,\gamma (b))<b$; thus no $b>t$ lies in $A$, and since $c=\sup A$ we get $c\le t$.

Converse for two distinct minimizing geodesics. [F5, F6, F8, F14, step 1.1] Let $t>0$ and let $\sigma :[0,t]\to M$ be a unit-speed minimizing geodesic with $\sigma (0)=p$, $\sigma (t)=\gamma (t)$ and ${\sigma }^{\mathrm{\prime }}(0)\mathrm{\ne }v$; in particular $d_g(p,\gamma (t))=t$. Suppose for contradiction that $b\in A$ for some $b>t$, so that $d_g(p,\gamma (b))=b$. Define the concatenation $C:[0,b]\to M$ by $C(s)=\sigma (s)$ for $0\le s\le t$ and $C(s)=\gamma (s)$ for $t\le s\le b$; it is continuous and piecewise smooth, and since both pieces have unit speed, [F5] gives $L(C)=L(\sigma )+L(\gamma {\mathrm{\mid }}_{[t,b]})=t+(b-t)=b=d_g(p,\gamma (b))$, so $C$ minimizes length among piecewise $C^1$ curves with the same endpoints. By [F8] the arclength representative $\bar{C}$ of $C$ is a unit-speed affinely parametrized geodesic; the arclength function of $C$ is the identity because $C$ has unit speed on each piece, so $\bar{C}=C$ and $C$ is a unit-speed geodesic on $[0,b]$, in particular differentiable at $t$. Its one-sided derivatives at $t$ are ${\sigma }^{\mathrm{\prime }}(t)$ and ${\gamma }^{\mathrm{\prime }}(t)$, so ${\sigma }^{\mathrm{\prime }}(t)={\gamma }^{\mathrm{\prime }}(t)$: the geodesics $\sigma$ and $\gamma$ agree in value and velocity at the parameter value $t$. Translating the parameter to $t$ by $r\mapsto t+r$ (which preserves the geodesic equation), the uniqueness in [F14] makes the two agree on the whole interval $[0,t]$, so ${\sigma }^{\mathrm{\prime }}(0)={\gamma }^{\mathrm{\prime }}(0)=v$, a contradiction. Hence no $b>t$ lies in $A$, and $c=\sup A\le t$.

Forward: minimizing geodesics to times just beyond $c$. [F1, F2, F3, F4, F5, F6, F15, step 1.1] Now assume $c<+\mathrm{\infty }$. Since $c>0$, fix $K$ with $c-1\mathrm{/}k>0$ for every $k\ge K$ and put $t_k:=c+1\mathrm{/}k$ and ${\mathrm{\ell }}_k:=d_g(p,\gamma (t_k))$ for $k\ge K$. The set-up above gives $t_k\mathrm{\notin }A$, so ${\mathrm{\ell }}_k<t_k$; and by [F1], applied to the pair $p,\gamma (t_k)$, there is $w_k\in T_{p}M$ with ${\exp }_p(w_k)=\gamma (t_k)$, $\mathrm{\mid }{w}_k{\mathrm{\mid }}_g={\mathrm{\ell }}_k$, the curve $s\mapsto {\exp }_p(s{w}_k)$ on $[0,1]$ having length ${\mathrm{\ell }}_k$. Since $d_g(p,\gamma (c))=c$ and $d_g(\gamma (c),\gamma (t_k))\le t_k-c=1\mathrm{/}k$ by [F5], the triangle inequality [F6] gives ${\mathrm{\ell }}_k\ge c-1\mathrm{/}k$; with ${\mathrm{\ell }}_k\le t_k=c+1\mathrm{/}k$ this shows ${\mathrm{\ell }}_k\to c$, so ${\mathrm{\ell }}_k>0$ and $u_k:=w_k\mathrm{/}{\mathrm{\ell }}_k$ is defined. By [F15], $\mathrm{\mid }{u}_k{\mathrm{\mid }}_g={\mathrm{\ell }}_k^{-1}\mathrm{\mid }{w}_k{\mathrm{\mid }}_g=1$, so $u_k\in S_{p}M$, and ${\exp }_p({\mathrm{\ell }}_k{u}_k)={\exp }_p(w_k)=\gamma (t_k)$ because ${\mathrm{\ell }}_k{u}_k=w_k$.

Passing to a limiting direction. [F7, step 2.3] By [F7] the unit sphere $S_{p}M$ is sequentially compact, so there are a strictly increasing index map $k$ and a unit vector $u\in S_{p}M$ with $u_{k_j}\to u$ in the norm metric of $T_{p}M$.

The limiting geodesic reaches $\gamma (c)$. [F2, F5, F14, F15, step 2.3, step 3.1] Define $\sigma :[0,c]\to M$ by $\sigma (s)={\exp }_p(su)$; this is defined on all of $[0,c]$ by [F2] and is a unit-speed geodesic because $\mathrm{\mid }u{\mathrm{\mid }}_g=1$. From ${\mathrm{\ell }}_{k_j}\to c$ and $u_{k_j}\to u$ we get ${\mathrm{\ell }}_{k_j}{u}_{k_j}\to cu$ in $T_{p}M$, because $\mathrm{\mid }{\mathrm{\ell }}_{k_j}{u}_{k_j}-cu{\mathrm{\mid }}_g\le \mathrm{\mid }{\mathrm{\ell }}_{k_j}-c\mathrm{\mid }+c\mathrm{\mid }{u}_{k_j}-u{\mathrm{\mid }}_g$ by [F15]; the map $(s,w)\mapsto {\exp }_p(sw)$ is continuous on $\mathbb{R}\times T_{p}M$ by [F14], and ${\exp }_p({\mathrm{\ell }}_{k_j}{u}_{k_j})=\gamma (t_{k_j})\to \gamma (c)$ because $\gamma$ is a geodesic, hence continuous. Therefore $\sigma (c)={\exp }_p(cu)=\gamma (c)$.

The limiting geodesic is minimizing. [F5, F6, step 1.1, step 4.1] The curve $\sigma {\mathrm{\mid }}_{[0,c]}$ is a unit-speed geodesic on a parameter interval of length $c$, so $L(\sigma {\mathrm{\mid }}_{[0,c]})=c$ by [F5], and it joins $p$ to $\gamma (c)$, where $d_g(p,\gamma (c))=c$ by step 1.1. Its length equals the distance between its endpoints, so it is a minimizing geodesic.

Forward alternatives. [F9, F10, F11, F12, F14, F15, step 2.3, step 3.1, step 4.1, step 5.1] If $u\mathrm{\ne }v$, then $\sigma$ and $\gamma {\mathrm{\mid }}_{[0,c]}$ are unit-speed geodesics from $p$ to $\gamma (c)$ with different initial velocities, hence distinct, and both minimize by step 5.1; this is alternative 2. If $u=v$, then for every $j$ the vectors $w_{k_j}={\mathrm{\ell }}_{k_j}{u}_{k_j}$ and $z_{k_j}:=t_{k_j}v$ satisfy ${\exp }_p(w_{k_j})=\gamma (t_{k_j})={\exp }_p(z_{k_j})$, both converge to $cv=cu$, and they are distinct because $\mathrm{\mid }{w}_{k_j}{\mathrm{\mid }}_g={\mathrm{\ell }}_{k_j}$ and $\mathrm{\mid }{z}_{k_j}{\mathrm{\mid }}_g=t_{k_j}\mathrm{\mid }v{\mathrm{\mid }}_g=t_{k_j}$ differ; hence ${\exp }_p$ is not injective on any neighbourhood of $cv$. By [F11] a local diffeomorphism at $cv$ would be injective on some neighbourhood, so ${\exp }_p$ fails to be a local diffeomorphism at $cv$; by [F10], applied to the nonzero vector $cv$, the point ${\exp }_p(cv)=\gamma (c)$ is conjugate to $p$ along $s\mapsto {\exp }_p(scv)$ on $[0,1]$. The affine bijection $\varphi :[0,1]\to [0,c]$, $\varphi (s)=cs$, presents the map $\eta (s):={\exp }_p(scv)$ satisfies $\eta =\gamma {\mathrm{\mid }}_{[0,c]}\circ \varphi$, so [F12] transfers the conjugacy to $\gamma (0)$, $\gamma (c)$ along $\gamma {\mathrm{\mid }}_{[0,c]}$, which is alternative 1.

Assembly, least-instant form and boundary audit. [A1, F2, F3, F4, step 1.1, step 2.1, step 2.2, step 6.1] Step 6.1 proves (a); steps 2.1 and 2.2 prove the two clauses of (b). If $c<+\mathrm{\infty }$, step 6.1 shows that at least one of the two events occurs at $t=c$, while (b) shows that no such event occurs at any $t<c$; hence $c$ is the least positive instant of either kind. The audit: in dimension zero there is no unit tangent vector $v$, so there is no instance; in dimension one $S_{p}M$ has two points and all the arguments above apply verbatim (the sequential compactness of [F7] and the uniqueness of [F14] are dimension-free); $\gamma$ is nonconstant because $\mathrm{\mid }v{\mathrm{\mid }}_g=1$, so constant geodesics are excluded and the multiplicity-bound statement of [F9] is not needed; the case $c=+\mathrm{\infty }$ is allowed in (b) only as the impossible conclusion $c\le t$, so when $c=+\mathrm{\infty }$ neither event of (b) occurs; the compactness used in step 3.1 is supplied by [F7], and exactly the declared ${\mathrm{A}\mathrm{C}}_{\omega }$ is inherited through the Hopf–Rinow, cut-time, conjugacy and length-minimization interfaces [A1]. No converse beyond (b) is claimed: conjugacy or a second minimizer at an instant $t>c$ is not excluded, and nothing is asserted about how many minimizing geodesics exist. $\mathrm{\square }$