Theorem: If a Riemannian $n$-manifold $(M, g)$ has constant sectional curvature $k=1$, then every point in $M$ has a neighborhood that is isometric to an open subset of the space form $S^n$. (cf. Page136, GTM171, Riemannian Geometry, by Peter Petersen).
I have some difficulties in details of the proof, although I know the sketch of the proof of this theorem. First we introduce a metric of constant curvature $k=1$ on a polar neighborhood $U$: $\tilde{g} = dr^2 + sin^2(r)ds_{n-1}^2 =dr^2 + \tilde{g_r}$
Second, we know the origin metric on $U$ is given by $g = dr^2 + g_r$.
Now, the book states that both $g_r$ and $\tilde g_r$ solve the equation $$\partial_rg_r = 2\frac{\cos(r)}{\sin(r)}g_r, \ \ \lim_{r\to 0}g_r = 0$$and indicates this shows they are equal (Here $\partial_rg_r$, in my opinion, should mean the Lie derivative $L_{\partial_r}g_r$). However, I cannot figure out the details of this step. More importantly, why does the fact that both $g_r$ and $\tilde g_r$ solve this equation imply $g_r = \tilde g_r$? The equation is not the usual ODE or PDE, and it involves the metrics.
The second question is a consequence of a general result in differential equations: two solutions of a (partial) differential equation with initial conditions are equal.
https://en.wikipedia.org/wiki/Cauchy%E2%80%93Kowalevski_theorem