Short local geodesics in a CAT(1) space are geodesics, and closed local geodesics have length at least $2\pi$

Uniqueness of short geodesics. Let $p,q\in X$ with $d(p,q)<\pi$ and let $g,g^{\mathrm{\prime }}$ be geodesic segments from $p$ to $q$, parametrized linearly on $[0,d(p,q)]$. Fix $t\in [0,d(p,q)]$, put $r=g(t)$ and $r^{\mathrm{\prime }}=g^{\mathrm{\prime }}(t)$, and consider the geodesic triangle with vertices $p,q,r$ whose sides are the segment $g^{\mathrm{\prime }}$ from $p$ to $q$ and the two subarcs of $g$ from $p$ to $r$ and from $r$ to $q$; its side lengths are $d(p,q)$, $d(p,r)=t$ and $d(r,q)=d(p,q)-t$, so its perimeter is $2d(p,q)<2\pi$ and its spherical comparison triangle is degenerate, with $\bar{r}$ lying on the side $[\bar{p},\bar{q}]$ at distance $t$ from $\bar{p}$. The comparison point of $r^{\mathrm{\prime }}$ is the point of $[\bar{p},\bar{q}]$ at distance $t$ from $\bar{p}$, because $r^{\mathrm{\prime }}$ lies on the side $g^{\mathrm{\prime }}$ at that distance from $p$; by degeneracy this comparison point equals $\bar{r}$. The CAT(1) inequality applied to the pair $(r,r^{\mathrm{\prime }})$ of points of this triangle gives $d(r,r^{\mathrm{\prime }})\le d_S(\bar{r},\bar{r})=0$, so $r=r^{\mathrm{\prime }}$. Since $t$ was arbitrary, $g=g^{\mathrm{\prime }}$ as linearly parametrized segments; hence geodesics between points at distance $<\pi$ are unique up to reparametrization.

Hinge estimate for one common initial point. Fix $p\in X$ and $\mathrm{\ell }<\pi$, let $q,q^{\mathrm{\prime }}\in X$ satisfy $d(p,q),d(p,q^{\mathrm{\prime }})\le \mathrm{\ell }$ and $\mathrm{\Delta }:=d(q,q^{\mathrm{\prime }})<2\pi -2\mathrm{\ell }$, and let $c,c^{\mathrm{\prime }}$ be the linear parametrizations of the unique geodesic segments from $p$ to $q$ and to $q^{\mathrm{\prime }}$. If $d(q,q^{\mathrm{\prime }})$ is small enough that $d(p,q),d(p,q^{\mathrm{\prime }}),\mathrm{\Delta }$ form an admissible triangle with perimeter $<2\pi$, then the triangle with vertices $p,q,q^{\mathrm{\prime }}$ and sides $c,c^{\mathrm{\prime }}$ and a segment from $q^{\mathrm{\prime }}$ to $q$ is a geodesic triangle of perimeter $\le 2\mathrm{\ell }+\mathrm{\Delta }<2\pi$, and the CAT(1) inequality applied to the pair of its points $c(t),c^{\mathrm{\prime }}(t)$ gives $d(c(t),c^{\mathrm{\prime }}(t))\le d_S(\bar{c}(t),{\bar{c}}^{\mathrm{\prime }}(t))$, where $\bar{c}(t),{\bar{c}}^{\mathrm{\prime }}(t)$ lie on the comparison sides at distances $td(p,q)$ and $td(p,q^{\mathrm{\prime }})$ from $\bar{p}$. With $\bar{\gamma }$ the model angle at $\bar{p}$, the cosine rule [F2] gives $\cos d_S(\bar{c}(t),{\bar{c}}^{\mathrm{\prime }}(t))=\cos (tL)\cos (t{L}^{\mathrm{\prime }})+\sin (tL)\sin (t{L}^{\mathrm{\prime }})\cos \bar{\gamma }$ and $\cos \bar{\gamma }=(\cos \mathrm{\Delta }-\cos L\cos L^{\mathrm{\prime }})\mathrm{/}(\sin L\sin L^{\mathrm{\prime }})$, where $L=d(p,q)$, $L^{\mathrm{\prime }}=d(p,q^{\mathrm{\prime }})$; the right-hand side is jointly continuous in $(t,L,L^{\mathrm{\prime }},\mathrm{\Delta })$ on the compact family and equals $1$ when $\mathrm{\Delta }=0$ and $L=L^{\mathrm{\prime }}$, while for $L=0$ or $L^{\mathrm{\prime }}=0$ the bound $d(c(t),c^{\mathrm{\prime }}(t))\le t(L+L^{\mathrm{\prime }})$ is immediate; hence ${\sup }_{t\in [0,1]}d(c(t),c^{\mathrm{\prime }}(t))\to 0$ as $(\mathrm{\Delta },L-L^{\mathrm{\prime }})\to 0$.

Continuous dependence of clause (i). Let $p_k\to p$, $q_k\to q$ with $d(p,q)<\pi$, and let $c_k,c$ be the linear parametrizations of $[p_k,q_k]$, $[p,q]$; for large $k$ all lengths are at most some $\mathrm{\ell }<\pi$. Fix such $k$ and let $c_k^{\mathrm{\prime }}$ be the linear parametrization of the unique segment from $p_k$ to $q$, which exists for large $k$ because $d(p_k,q)\le d(p_k,p)+d(p,q)<\pi$. Then $d(c_k(t),c(t))\le d(c_k(t),c_k^{\mathrm{\prime }}(t))+d(c_k^{\mathrm{\prime }}(t),c(t))$ for all $t$: the first term tends to $0$ uniformly by the hinge estimate at the common initial point $p_k$ with endpoint distance $d(q_k,q)\to 0$, and the second term tends to $0$ uniformly by the hinge estimate applied to the reversed segments, which have common initial point $q$ and endpoint distance $d(p_k,p)\to 0$. Hence the linear parametrizations of $[p_k,q_k]$ converge uniformly to that of $[p,q]$.

A unit-speed local geodesic minimizes on every short interval. Assume the speed is $1$ and fix $s<t$ in $I$ with $t-s\le \pi$. A finite subdivision into local isometry intervals shows that $c{\mathrm{\mid }}_{[s,t]}$ is $1$-Lipschitz and has length $t-s$. Let $S=\{u\in [s,t]:c{\mathrm{\mid }}_{[s,u]}\text{ is a geodesic}\}$. It contains an initial interval by local isometry, and it is closed: the distance equalities on $[s,u]$ pass to the limit as $u$ increases or decreases to an endpoint. Put $u_0=\sup S\in S$. If $u_0<t$, choose $0<\epsilon <u_0-s$ with $u_0+\epsilon <t$, $u_0+\epsilon -s<\pi$, and $c{\mathrm{\mid }}_{[u_0-\epsilon ,u_0+\epsilon ]}$ isometric. This is possible since $u_0-s<t-s\le \pi$. The triangle with vertices $c(s),c(u_0),c(u_0+\epsilon )$ and its two indicated subarcs has perimeter at most $2(u_0+\epsilon -s)<2\pi$. In its model let $\bar{\beta }$ be the angle at $\bar{c}(u_0)$. The local isometry gives $d(c(u_0-\sigma ),c(u_0+\tau ))=\sigma +\tau$ for small positive $\sigma ,\tau <\epsilon$. Comparison and the cosine rule therefore force $\bar{\beta }=\pi$: any smaller angle gives a model cross-distance strictly less than $\sigma +\tau$. Thus the opposite side has length $u_0+\epsilon -s$, and the whole subarc $c{\mathrm{\mid }}_{[s,u_0+\epsilon ]}$ minimizes. Indeed, a strict shortcut between any two of its points, combined with the remaining subarcs, would make its endpoint distance smaller than its length. This contradicts the definition of $u_0$. Hence $u_0=t$ and $d(c(s),c(t))=t-s$.

Clause (ii) with arbitrary speed. If $\lambda =0$, the map is locally constant and therefore constant on the connected interval $I$: the inverse image of each attained value is open and its complement is a union of such open fibers. If $\lambda >0$, set $J=\lambda I$ and $\tilde{c}(u)=c(u\mathrm{/}\lambda )$. This is unit-speed locally; finite partitions show that lengths of corresponding compact restrictions agree. For $s<t$ in $I$, a finite local-isometry subdivision gives $L(c{\mathrm{\mid }}_{[s,t]})=\lambda (t-s)\le L(c)\le \pi$. Applying step 2.2 to $\tilde{c}{\mathrm{\mid }}_{[\lambda s,\lambda t]}$ gives $d(c(s),c(t))=\lambda (t-s)$. Empty and one-point intervals have no unequal pair to test. This proves the distance equality and the stated segment interpretation.

Clause (iii): $\mathrm{\ell }\ge 2\pi$. Suppose a nonconstant closed local geodesic $c\text{\hspace{0.17em}⁣}:S_{\mathrm{\ell }}^1\to X$ had $\mathrm{\ell }<2\pi$, and put $p=c(0)$, $q=c(\mathrm{\ell }\mathrm{/}2)$; the arcs $\alpha (\theta )=c(\theta )$ and $\beta (\theta )=c(\mathrm{\ell }-\theta )$, $\theta \in [0,\mathrm{\ell }\mathrm{/}2]$, are local geodesics of length $\mathrm{\ell }\mathrm{/}2<\pi$, hence geodesic segments from $p$ to $q$ by step 3.1, and $d(p,q)\le \mathrm{\ell }\mathrm{/}2<\pi$. Since $d(p,\alpha (\theta ))=\theta =d(p,\beta (\theta ))$ for all $\theta \in [0,\mathrm{\ell }\mathrm{/}2]$, step 1.1 (uniqueness) gives $c(\theta )=c(\mathrm{\ell }-\theta )$ for every $\theta \in [0,\mathrm{\ell }\mathrm{/}2]$. But $c$ is locally isometric at $p$, so for small $\theta >0$ with $2\theta <\mathrm{\ell }$ one has $d(c(\theta ),c(-\theta ))=2\theta$, where $c(-\theta ):=c(\mathrm{\ell }-\theta )$; this contradicts $c(\theta )=c(\mathrm{\ell }-\theta )$ and $\theta >0$. Hence $\mathrm{\ell }\ge 2\pi$.

Clause (iii): diameter at least $\pi$. Let $c$ be as in clause (iii) with $\mathrm{\ell }\ge 2\pi$ and fix ${\theta }_0$; the restriction of $c$ to $[{\theta }_0,{\theta }_0+\pi ]$ is a local geodesic of length $\pi$ (its length equals the parameter length by the first paragraph of step 2.2), so step 3.1 gives $d(c({\theta }_0),c({\theta }_0+\pi ))=\pi$. Hence the image of $c$ has diameter at least $\pi$, and by definition of diameter a nonconstant closed local geodesic is never contained in a ball of diameter $<\pi$.

Conclusion. Clause (i) is step 2.1; clause (ii) is step 3.1; clause (iii) is steps 4.1 and 4.2. Therefore in a CAT(1) space short geodesics are unique and depend continuously on their endpoints, every constant-speed local geodesic of length at most $\pi$ minimizes between its points, and every nonconstant closed local geodesic has length at least $2\pi$ and diameter at least $\pi$.