[proofplan] Use compactness of $M$ to fix a uniform injectivity radius $\varepsilon > 0$ (so $\exp_p|_{B(0, \varepsilon)}$ is a diffeomorphism for every $p$). The infimum $\ell$ of lengths in the homotopy class $[\alpha]$ is strictly positive — any loop shorter than $\varepsilon$ lies in a normal ball, hence is null-homotopic, contradicting non-triviality of $[\alpha]$. Take a minimising sequence $\gamma_k \in [\alpha]$ with $\ell(\gamma_k) \to \ell$, replace each by a piecewise-geodesic loop on a fixed mesh of size $\varepsilon/2$ which has the same length-up-to-$\varepsilon$ and the same homotopy class, and use compactness of $M$ to extract a subsequence whose vertices converge. The limit is a piecewise-geodesic closed loop $\gamma$ in $[\alpha]$ of length exactly $\ell$. Local length-minimality on each segment plus minimality of the global length forces the broken vertices to be smooth: any corner could be smoothed via a geodesic shortcut, decreasing the length and producing a competitor in $[\alpha]$ — contradicting the minimality of $\gamma$. Hence $\gamma$ is a smooth closed geodesic and a length-minimiser in $[\alpha]$. [/proofplan] [step:Fix a uniform injectivity radius via compactness of $M$] For each $p \in M$, the [Exponential Map as a Local Diffeomorphism](/theorems/2712) provides an open neighbourhood $U_p \ni p$ and $\varepsilon_p > 0$ such that $\exp_q$ restricted to $\{v \in T_q M : |v|_g < \varepsilon_p\}$ is a diffeomorphism onto its image, for every $q \in U_p$. The cover $\{U_p\}_{p \in M}$ of the compact manifold $M$ admits a finite subcover $\{U_{p_1}, \dots, U_{p_K}\}$. Set \begin{align*} \varepsilon := \tfrac{1}{2} \min(\varepsilon_{p_1}, \dots, \varepsilon_{p_K}) > 0. \end{align*} Then for every $p \in M$, $\exp_p|_{B(0, \varepsilon)}$ is a diffeomorphism onto a normal neighbourhood of $p$, which we denote $V_p := \exp_p(B(0, \varepsilon))$. In particular, by the local minimisation property in [Geodesics Minimize Length Locally](/theorems/2720), if $p, q \in M$ with $d(p, q) < \varepsilon$, there is a unique minimal geodesic from $p$ to $q$ of length $d(p, q)$, lying entirely inside $V_p$. [/step] [step:Establish $\ell := \inf_{[\alpha]} \ell > 0$] Set $\ell := \inf_{\gamma \in [\alpha]} \ell(\gamma)$, where the infimum is taken over piecewise $C^1$ closed loops $\gamma$ in the free homotopy class $[\alpha]$. Suppose, for contradiction, $\ell < \varepsilon$. Then there exists $\gamma \in [\alpha]$ with $\ell(\gamma) < \varepsilon$. Pick any $p \in \gamma([0, T])$. The image $\gamma([0, T])$ has diameter at most $\ell(\gamma) < \varepsilon$ (any two points $\gamma(t_1), \gamma(t_2)$ satisfy $d(\gamma(t_1), \gamma(t_2)) \le \ell(\gamma|_{[t_1, t_2]}) \le \ell(\gamma) < \varepsilon$), so $\gamma([0, T]) \subseteq V_p$. Since $V_p$ is the diffeomorphic image of a Euclidean ball under $\exp_p$, $V_p$ is contractible. The free homotopy class of any loop in a contractible set is the zero class (every loop is null-homotopic, by composing the loop with the contraction). Hence $[\alpha] = 0$, contradicting non-triviality of $[\alpha]$. Therefore $\ell \ge \varepsilon > 0$. [/step] [step:Replace a minimising sequence by piecewise-geodesic loops on a fixed mesh] Let $(\gamma_k)$ be a minimising sequence: $\gamma_k \in [\alpha]$, parametrised by arc length on $[0, L_k]$ with $L_k = \ell(\gamma_k)$, and $L_k \to \ell$ as $k \to \infty$. We may assume $L_k \le \ell + 1$ for all $k$. For each $k$, choose an integer $N_k \in \mathbb{N}$ with $L_k / N_k < \varepsilon/4$, and set $\Delta_k := L_k / N_k$. Note $N_k \le \lceil 4 L_k / \varepsilon \rceil \le \lceil 4(\ell + 1)/\varepsilon \rceil =: N^*$ — so the mesh size $N_k$ is bounded uniformly in $k$. By passing to a subsequence we may assume $N_k = N$ is independent of $k$, with $\Delta_k = L_k / N < \varepsilon/4$. Define mesh points $t_{k, j} := j L_k / N$ for $j = 0, 1, \dots, N$, and put $p_{k, j} := \gamma_k(t_{k, j}) \in M$. Note $p_{k, 0} = p_{k, N}$ since $\gamma_k$ is closed. For each $j$, the points $p_{k, j}, p_{k, j+1}$ satisfy $d(p_{k, j}, p_{k, j+1}) \le \ell(\gamma_k|_{[t_{k, j}, t_{k, j+1}]}) = \Delta_k < \varepsilon/4$. By Step 1, there is a unique minimal geodesic segment $\sigma_{k, j} : [0, 1] \to M$ from $p_{k, j}$ to $p_{k, j+1}$ of length $d(p_{k, j}, p_{k, j+1})$. Define the piecewise-geodesic loop $\tilde\gamma_k : [0, L_k] \to M$ by concatenating these segments: on $[t_{k, j}, t_{k, j+1}]$, $\tilde\gamma_k$ traverses $\sigma_{k, j}$ at constant speed $\Delta_k / d(p_{k, j}, p_{k, j+1})$ if $p_{k, j} \ne p_{k, j+1}$, and is constant otherwise (in which case the segment contributes zero length). The length of $\tilde\gamma_k$ is \begin{align*} \ell(\tilde\gamma_k) = \sum_{j=0}^{N-1} d(p_{k, j}, p_{k, j+1}) \le \sum_{j=0}^{N-1} \ell(\gamma_k|_{[t_{k, j}, t_{k, j+1}]}) = \ell(\gamma_k) = L_k. \end{align*} [claim:$\tilde\gamma_k$ is freely homotopic to $\gamma_k$] [proof] On each subinterval $[t_{k, j}, t_{k, j+1}]$ both $\gamma_k$ and $\sigma_{k, j}$ are paths from $p_{k, j}$ to $p_{k, j+1}$, both lying in the normal neighbourhood $V_{p_{k, j}}$ of $p_{k, j}$ (which is a contractible set, being diffeomorphic to a Euclidean ball via $\exp_{p_{k, j}}$): for $\gamma_k|_{[t_{k, j}, t_{k, j+1}]}$, the image has diameter at most $\Delta_k < \varepsilon/4$ from any reference point in it, hence sits in a ball of radius $\varepsilon/2 < \varepsilon$ around $p_{k, j}$, hence lies in $V_{p_{k, j}}$. Inside $V_{p_{k, j}}$, any two paths with the same endpoints are homotopic rel endpoints (because $V_{p_{k, j}}$ is diffeomorphic to a Euclidean ball, hence simply connected and indeed contractible). Concatenating these segment-by-segment homotopies (rel endpoints) and adjusting via reparametrisation, we obtain a free homotopy from $\gamma_k$ to $\tilde\gamma_k$. Therefore $\tilde\gamma_k \in [\alpha]$. [/proof] [/claim] The minimality property gives $\ell(\tilde\gamma_k) \ge \ell$, hence \begin{align*} \ell \le \ell(\tilde\gamma_k) \le L_k \to \ell, \end{align*} so $\ell(\tilde\gamma_k) \to \ell$ as well: $(\tilde\gamma_k)$ is a piecewise-geodesic minimising sequence in $[\alpha]$. [/step] [step:Extract a convergent subsequence of vertex sequences and pass to the limit] For each fixed $j \in \{0, 1, \dots, N\}$, the sequence $(p_{k, j})_{k \in \mathbb{N}}$ lies in the compact manifold $M$. By compactness and a diagonal extraction over $j \in \{0, \dots, N\}$ (a finite index set), there is a subsequence — relabelled $(\gamma_k)$ — and points $p_j^* \in M$ with \begin{align*} p_{k, j} \to p_j^* \quad \text{as } k \to \infty, \text{ for each } j = 0, 1, \dots, N. \end{align*} Since $p_{k, 0} = p_{k, N}$ for every $k$, $p_0^* = p_N^*$. For each $j$, the lengths $L_k / N \to \ell/N$ and $d(p_{k, j}, p_{k, j+1}) \to d(p_j^*, p_{j+1}^*)$ by continuity of $d$. From $d(p_{k, j}, p_{k, j+1}) \le L_k/N < \varepsilon/4$ and the limit $L_k \to \ell$, we get $d(p_j^*, p_{j+1}^*) \le \ell/N \le \varepsilon/4$. Define the limit piecewise-geodesic loop $\gamma : [0, \ell] \to M$ by concatenating the unique minimal geodesic segments from $p_j^*$ to $p_{j+1}^*$ for $j = 0, \dots, N - 1$, parametrised so that on $[j\ell/N, (j+1)\ell/N]$ the curve is the constant-speed minimal geodesic from $p_j^*$ to $p_{j+1}^*$. The total length is \begin{align*} \ell(\gamma) = \sum_{j=0}^{N-1} d(p_j^*, p_{j+1}^*) = \lim_{k \to \infty} \sum_{j=0}^{N-1} d(p_{k, j}, p_{k, j+1}) = \lim_{k \to \infty} \ell(\tilde\gamma_k) = \ell. \end{align*} [claim:$\gamma$ is freely homotopic to $\alpha$] [proof] For each fixed $j$ and large $k$, the points $p_{k, j}$ and $p_j^*$ both lie in the normal neighbourhood $V_{p_j^*}$ — choose $k$ large enough that $d(p_{k, j}, p_j^*) < \varepsilon/4$ for all $j$ (possible by simultaneous convergence on the finite index set). The minimal geodesic segments from $p_{k, j}$ to $p_{k, j+1}$ and from $p_j^*$ to $p_{j+1}^*$ both lie in a normal neighbourhood (since both pairs of endpoints are close together and close to $p_j^*$); within a normal neighbourhood, the two segments are homotopic rel endpoints. A segment-by-segment free homotopy then connects $\tilde\gamma_k$ to $\gamma$. Combined with the homotopy $\tilde\gamma_k \sim \gamma_k$ from Step 3, we obtain $\gamma \sim \gamma_k$ in $[\alpha]$. [/proof] [/claim] So $\gamma \in [\alpha]$, with $\ell(\gamma) = \ell = \inf_{[\alpha]} \ell$. [/step] [step:Show that the limit $\gamma$ has no corners — it is a smooth closed geodesic] The limit $\gamma$ is a piecewise-geodesic loop achieving the infimum length in $[\alpha]$. We now show it is smooth at every vertex $p_j^*$ for $j = 0, 1, \dots, N - 1$, hence a smooth closed geodesic. Fix an interior vertex index $j$ (the cyclic structure makes every vertex interior). Set \begin{align*} v^- &:= \dot\gamma(t_j^-) \in T_{p_j^*} M, \\ v^+ &:= \dot\gamma(t_j^+) \in T_{p_j^*} M, \end{align*} where $t_j = j \ell/N$. Both are constant-norm vectors with $|v^\pm|_g = \ell/N \cdot \frac{1}{d(p_{j-1}^*, p_j^*)}$ on the relevant side after reparametrisation, but the cleanest argument works with arc-length reparametrisation directly: parametrise $\gamma$ by arc length, so $|v^-|_g = |v^+|_g = 1$ and the joining condition is $v^- = v^+$ if and only if $\gamma$ is smooth at $p_j^*$. Suppose $v^- \ne v^+$. We construct a curve in $[\alpha]$ of length strictly less than $\ell(\gamma) = \ell$, contradicting the minimisation property. The construction is the geodesic shortcut. To compare the broken path's length with the geodesic distance between its endpoints, we need both endpoints in a *common* totally normal (convex) neighbourhood — not merely each in $V_{p_j^*}$. By Whitehead's theorem on totally normal neighbourhoods (a standard consequence of the local-diffeomorphism property of $\exp$ combined with continuity of the metric), every point $p \in M$ admits a *totally normal* (or *convex*) neighbourhood $W_p$: an open neighbourhood with the property that any two points $q_1, q_2 \in W_p$ are joined by a unique minimising geodesic in $M$, and that geodesic lies entirely inside $W_p$. Apply this at each vertex $p_j^*$ to obtain $W_{p_j^*}$. Pick $\delta > 0$ small enough that simultaneously $\delta < \varepsilon/4$ and $\gamma(t_j \pm \delta) \in W_{p_j^*}$ for every vertex index $j$ (possible: by arc-length parametrisation $d(\gamma(t_j \pm \delta), p_j^*) = \delta \to 0$, and $W_{p_j^*}$ is open with $p_j^* \in W_{p_j^*}$). Take the minimum over the finitely many vertex thresholds. By Whitehead's property in $W_{p_j^*}$, the unique minimising geodesic $\sigma$ from $\gamma(t_j - \delta)$ to $\gamma(t_j + \delta)$ lies in $W_{p_j^*}$ and has length $\ell(\sigma) = d(\gamma(t_j - \delta), \gamma(t_j + \delta))$. The broken path $\gamma|_{[t_j - \delta, t_j + \delta]}$ has length $2\delta$, joins the same endpoints, but has a corner at $t_j$ (since $v^- \ne v^+$) — a true geodesic, being a smooth solution of $\nabla_{\dot\gamma}\dot\gamma = 0$ with continuous $\dot\gamma$, cannot have a corner. So the broken path is not the unique minimising geodesic, and its length strictly exceeds the geodesic distance: \begin{align*} d(\gamma(t_j - \delta), \gamma(t_j + \delta)) < 2\delta. \end{align*} Replace the broken segment $\gamma|_{[t_j - \delta, t_j + \delta]}$ by the unique minimal geodesic $\sigma$ from $\gamma(t_j - \delta)$ to $\gamma(t_j + \delta)$ in $V_{p_j^*}$. The resulting curve $\gamma'$ is a piecewise-geodesic loop with length \begin{align*} \ell(\gamma') = \ell(\gamma) - 2\delta + d(\gamma(t_j - \delta), \gamma(t_j + \delta)) < \ell(\gamma) = \ell. \end{align*} [claim:$\gamma' \in [\alpha]$] [proof] The replaced segment $\gamma|_{[t_j - \delta, t_j + \delta]}$ and the new geodesic segment $\sigma$ both lie in the totally normal neighbourhood $W_{p_j^*}$: the broken segment because $d(\gamma(t), p_j^*) \le \delta$ for $|t - t_j| \le \delta$ (after possibly shrinking $\delta$ so this $\delta$-ball sits in $W_{p_j^*}$), and the geodesic $\sigma$ because Whitehead's theorem guarantees the unique minimising geodesic between two points of $W_{p_j^*}$ lies in $W_{p_j^*}$. The neighbourhood $W_{p_j^*}$ is contractible (an image under $\exp_{p_j^*}$ of a star-shaped subset of $T_{p_j^*}M$ on which $\exp_{p_j^*}$ is a diffeomorphism), so two paths with the same endpoints in $W_{p_j^*}$ are homotopic rel endpoints. Concatenating with the unchanged remainder of $\gamma$, we obtain a free homotopy from $\gamma$ to $\gamma'$. So $\gamma' \in [\alpha]$. [/proof] [/claim] But then $\ell(\gamma') < \ell = \inf_{[\alpha]} \ell$, contradicting the definition of $\ell$. Therefore $v^- = v^+$, i.e., $\gamma$ has no corner at $p_j^*$. This holds for every vertex $j$, so $\gamma$ is smooth at every vertex. Combined with being geodesic on each segment, $\gamma$ is a smooth closed geodesic on $[0, \ell]$, with $\gamma \in [\alpha]$ and $\ell(\gamma) = \inf_{[\alpha]} \ell$. [guided] We have arrived at the analytic heart of the proof: showing the piecewise-geodesic minimiser $\gamma$ has no corners. The strategy is *proof by contradiction via a geodesic shortcut*. Suppose a vertex were truly a corner. Then we could splice in a tiny minimising geodesic across the corner, producing a curve in $[\alpha]$ that is strictly shorter than $\gamma$ — but $\gamma$ already attains the infimum, which is impossible. To set up the shortcut, parametrise $\gamma$ by arc length on $[0, \ell]$ and fix an interior vertex index $j$ (every vertex is interior since $\gamma$ is closed). Define the one-sided velocities at $p_j^*$: \begin{align*} v^- &:= \dot\gamma(t_j^-) \in T_{p_j^*} M, \\ v^+ &:= \dot\gamma(t_j^+) \in T_{p_j^*} M, \end{align*} where $t_j = j \ell/N$. Arc-length parametrisation gives $|v^-|_g = |v^+|_g = 1$, so smoothness of $\gamma$ at $p_j^*$ is equivalent to $v^- = v^+$. Suppose for contradiction $v^- \ne v^+$ — that is, $\gamma$ has a genuine corner at $p_j^*$. Now a subtle point: to compare a broken path's length with the geodesic distance between its endpoints, both endpoints must lie in a *common* totally normal (convex) neighbourhood — having each endpoint inside its own $V_p$ is not enough, because the local minimisation property of [Geodesics Minimize Length Locally](/theorems/2720) only applies between two points that share a normal neighbourhood. This is exactly what Whitehead's theorem on totally normal neighbourhoods gives us: every point $p \in M$ admits a *totally normal* neighbourhood $W_p$ such that any two points $q_1, q_2 \in W_p$ are joined by a unique minimising geodesic in $M$, and that geodesic lies entirely inside $W_p$. Apply this at each vertex $p_j^*$ to obtain $W_{p_j^*}$. We now choose a small parameter $\delta$ to slice out the corner. Pick $\delta > 0$ small enough that simultaneously $\delta < \varepsilon/4$ and $\gamma(t_j \pm \delta) \in W_{p_j^*}$ for every vertex index $j$. Why is this possible? By arc-length parametrisation, $d(\gamma(t_j \pm \delta), p_j^*) = \delta$, and $W_{p_j^*}$ is open with $p_j^* \in W_{p_j^*}$, so a small enough $\delta$-ball lies in $W_{p_j^*}$. We take the minimum of these thresholds across the finitely many vertices. Why must the broken path be strictly longer than the straight geodesic between its endpoints? Because of a key rigidity fact: a true geodesic, being a smooth solution of $\nabla_{\dot\gamma}\dot\gamma = 0$ with continuous velocity, *cannot have a corner*. Whitehead guarantees a unique minimising geodesic $\sigma$ from $\gamma(t_j - \delta)$ to $\gamma(t_j + \delta)$ inside $W_{p_j^*}$, with length equal to the geodesic distance. The broken segment $\gamma|_{[t_j - \delta, t_j + \delta]}$ has length $2\delta$ (by arc-length parametrisation) and joins the same endpoints, but it has a corner at $t_j$ — hence it cannot be the unique minimiser. Therefore its length strictly exceeds the geodesic distance: \begin{align*} d(\gamma(t_j - \delta), \gamma(t_j + \delta)) < 2\delta. \end{align*} This is the strict inequality that drives the contradiction. We now perform the surgery: replace the broken segment $\gamma|_{[t_j - \delta, t_j + \delta]}$ by the unique minimal geodesic $\sigma$ from $\gamma(t_j - \delta)$ to $\gamma(t_j + \delta)$. The resulting curve $\gamma'$ is a piecewise-geodesic loop, and the length bookkeeping is direct: we removed length $2\delta$ from $\gamma$ and inserted $\sigma$, which has length $d(\gamma(t_j - \delta), \gamma(t_j + \delta))$. Hence \begin{align*} \ell(\gamma') = \ell(\gamma) - 2\delta + d(\gamma(t_j - \delta), \gamma(t_j + \delta)) < \ell(\gamma) = \ell. \end{align*} It remains to verify that the surgery preserves the homotopy class. The replaced segment and the new geodesic $\sigma$ both sit inside the totally normal neighbourhood $W_{p_j^*}$: the broken segment because $d(\gamma(t), p_j^*) \le \delta$ for $|t - t_j| \le \delta$ (after possibly shrinking $\delta$), and $\sigma$ because Whitehead's theorem confines the unique minimising geodesic to $W_{p_j^*}$. Since $W_{p_j^*}$ is contractible (it is the diffeomorphic image under $\exp_{p_j^*}$ of a star-shaped subset of $T_{p_j^*}M$), any two paths with the same endpoints inside $W_{p_j^*}$ are homotopic rel endpoints. Concatenating this local homotopy with the unchanged remainder of $\gamma$ produces a free homotopy from $\gamma$ to $\gamma'$, so $\gamma' \in [\alpha]$. But now we have produced a competitor: $\gamma' \in [\alpha]$ with $\ell(\gamma') < \ell = \inf_{[\alpha]} \ell$, contradicting the definition of the infimum. The only way to escape the contradiction is to deny the assumption $v^- \ne v^+$. Hence $v^- = v^+$, and $\gamma$ has no corner at $p_j^*$. This argument applies at every vertex $j \in \{0, 1, \dots, N - 1\}$. Combined with the fact that $\gamma$ is geodesic on each segment $[t_j, t_{j+1}]$ by construction, $\gamma$ is a smooth closed geodesic on $[0, \ell]$, with $\gamma \in [\alpha]$ and $\ell(\gamma) = \inf_{[\alpha]} \ell$. The two essential hypotheses earned their keep: compactness of $M$ supplied both the uniform injectivity radius and the sequential compactness of the vertex extraction, while non-triviality of $[\alpha]$ kept $\ell$ strictly positive (Step 2) so that the infimum was a genuine length and not zero. [/guided] [/step]