Homogeneous riemannian manifolds are complete. Trouble understanding proof.

Solution 1:

Homogeneity implies that all metric balls of the same radius are isometric. Therefore if one can extend a geodesic at a point $p$ in each direction by a distance of $\delta$, then one can extend by the same $\delta$ at every point of the manifold.

Solution 2:

Another way to prove this is by Hopf Rinow's Theorem. I'll prove that a homogeneous Riemannian manifold is a complete metric space.

Let $\{x_n\}\subset M$ be a Cauchy sequence and fix $p\in M$. For each $n\in\mathbb{N}$ there is a $g_m\in \operatorname{Iso}(M)$ s.t. $g_m(x_m)=p$. Let $B_\epsilon (p)$ a normal ball around $p$. There is a $N\in\mathbb{N}$ s.t. for every $n\geq N$, $$\epsilon>d(x_N,x_n)=d(g_Np,g_np)=d(p,g_N^{-1}g_np)$$ So $d(p,x_n)=d(p,g_np)\in g_N(B_\epsilon(p))$. Then $\{x_n\}$ admits a convergente subsequence. As $x_n$ is a cauchy sequence, $x_n$ converges to the limit of the subsequence.