# Free homotopy class of Riemannian manifold represented by closed geodesic

I'm trying to prove the following result, which is Problem 6-17 in Lee's Introduction to Riemannian Manifolds:

Suppose $$(M,g)$$ is a compact connected Riemannian manifold. Show that every nontrivial free homotopy class in $$M$$ is represented by a closed geodesic that has minimal length among all admissible loops in the given free homotopy class.

This problem comes with a hint, which breaks the solution down into the following steps:

1. Show the given free homotopy class is represented by a geodesic loop;
2. Show the lengths of such loops have positive greatest lower bound;
3. Choose a sequence of geodesic loops whose lengths approach this bound, and show that a subsequence converges uniformly to a geodesic loop whose length is equal to this greatest lower bound;
4. Show the limiting curve is in the given free homotopy class;
5. Apply the first variation formula to show that the limiting curve is in fact a closed geodesic.

I have most of these steps down except 3 and 5. First of all, if I have such a sequence, finding a universally convergent subsequence seems similar in spirit to Arzelà-Ascoli, but finding the right uniformly bounded and equicontinuous sequence of functions so far has eluded me.

Second of all, in proving 5, the first variation formula says that if $$\Gamma_s(t)$$ is a variation (i.e. a homotopy) between the limiting curve $$\gamma(t)$$ and one of the sequence curves $$\gamma_n(t)$$, then $$\frac{d}{ds}\bigg|_{s=0} L_g(\Gamma_s) = -\int_a^b \langle V, D_t\dot\gamma\rangle\:dt - \sum_{i=1}^{k-1} \left\langle V(a_i), \Delta_i \dot\gamma\right\rangle + \langle V(b),\dot\gamma(b)\rangle - \langle V(a), \dot\gamma(b)\rangle$$ where $$V(t) = \frac{d}{ds}\big|_{s=0} \Gamma_s(t)$$ is the variation field along $$\gamma$$, $$(a_0, \ldots, a_k)$$ is an admissible partition for $$V$$, and $$\Delta_i \dot\gamma = \dot\gamma(a_i^+) - \dot\gamma(a_i^-)$$ is the "jump" in the velocity vector field at $$a_i$$. In this case, $$V(a) = V(b)$$, so we need to show $$\dot\gamma(a) = \dot\gamma(b)$$ and $$\Delta_i \dot\gamma = 0$$ for all $$i$$. Since $$\Gamma_0$$ is a critical point for this variation, this forumula becomes $$\int_a^b \langle V, D_t\dot\gamma\rangle\:dt = - \sum_{i=1}^{k-1} \left\langle V(a_i), \Delta_i \dot\gamma\right\rangle + \langle V(b),\dot\gamma(b)\rangle - \langle V(a), \dot\gamma(a)\rangle$$ But why does this show the limiting curve $$\gamma$$ is geodesic? Where in here does it follow that $$D_t\dot\gamma \equiv 0$$?

Edit 1: I think I can use the Arzelà-Ascoli theorem for $$\mathbb R^n$$-valued functions of a compact space in a geodesic ball in normal coordinates around a limit point of an appropriate sequence $$\{x_n\}$$ in $$M$$; say, for example, $$x_n = \gamma_n(0)$$, where $$\{\gamma_n\}$$ is the sequence of geodesic loops.

Edit 2: Using the sequence $$\{x_n\}$$ coming from $$x_n = \gamma_n(0)$$, take a convergent subsequence $$x_n \to p \in M$$, and then let $$w_n = \dot\gamma_n(0) \in T_{x_n}M$$ for each $$n$$. Assume after reparametrization that each $$\gamma_n$$ is unit-speed, so the parallel translations $$v_n \in T_pM$$ along radial geodesics from $$p$$ to $$x$$ are unit tangent vectors. In particular, they too have a convergent subsequence $$v_n \to v$$. I think the resulting geodesic we want is then just $$\gamma(t) = \exp_p(tv)$$.

• The hint says to pick a sequence of geodesic loops. so these should have no jumps but your variation formula contains a list of jumps. I think these should all disappear, so you varation formula simply says critical point if and only if $D_t\dot{\gamma}=0$. – quarague Sep 24 '19 at 8:49
• If the jumps all disappear, then we should get $0 = \langle V(b), \dot\gamma(b) \rangle - \langle V(a), \dot\gamma(a)\rangle$. Why is this equal to $0$? What we really want to show is $\dot\gamma(a) = \dot\gamma(b)$. – D Ford Sep 24 '19 at 15:59
• If it is a geodesic loop there is no jump at the end points either. The end points are only special in your parametrization. – quarague Sep 25 '19 at 7:03

Let $$\gamma_i: [0, 1] \to M$$ be your geodesics, with $$l(\gamma)=l_i$$, and $$l_i \to l$$, the infimum of all distances. We have $$\lVert\gamma_i\rVert=l_i$$. Now, if $$t_1, we can bound the (intrinsic) distance between $$\gamma_i(t_1)$$ and $$\gamma_i(t_2)$$ using $$\gamma_i$$ as a path: $$d(\gamma_i(t_1), \gamma_i(t_2))\leq \int_{t_1}^{t_2} \lVert \gamma_i'(s) \rVert \, ds=(t_2-t_1) l_i.$$ Now, $$\sup l_i<\infty$$, as $$\{l_i\}_{i \in \mathbb{N}}$$ is a convergent sequence. This proves that $$\{\gamma_i\}_{i \in \mathbb{N}}$$ are uniformly Lipschitz functions, and thus equicontinuous.
With that in mind, we can use something like Arzelà-Ascoli: by a diagonal argument and the compactness of $$M$$, we can extract a subsequence $$\{\gamma_{i_j}\}_{j \in \mathbb{N}}$$ such that $$\gamma_{i_j}(t)$$ converges for each rational $$t \in [0, 1]$$. Equicontinuity (and completeness of $$M$$) gives uniform convergence, and now we have a uniform limit $$\gamma$$. In particular, $$\gamma: [0, 1] \to M$$ is a continuous and closed path.