Let $(M^{n},g)$ be a Riemannian manifold with constant sectional $K$. Assume that in the geodesic normal coordinate $ (r,\theta^{1},\cdots, \theta^{n-1}) $, the metric $g$ has the form $g=dr^2+(f(r))^2h_{ij}(\theta)d\theta^{i}d\theta^{j}$, where the (n-1)-dimensional metric $h_{ij}(\theta)d\theta^{i}d\theta^{j}$ has constant sectional curvature 1.
I want to ask how to prove that \begin{align*} f(r)=\begin{cases} \sin(\sqrt{Kr^2})/\sqrt{K},\quad &K>0;\\\ r,\quad &K=0;\\\ \sinh(\sqrt{-Kr^2})/\sqrt{-K},\quad &K<0. \end{cases} \end{align*}