Metric tensor in Radial normal coordinates

Question

Consider an $n-$ dimensional complete Riemannian manifold $M$. For $n= 2 \mbox{ and } 3$, metric tensor $ds^{2}$ on $M$ can be written as $dr^{2} + f(r,u) du^{2},$ where $f(r,u) du^{2}$ is the component tangential to geodesic sphere. My question is that can we write $f(r,u)$ as some function of Jacobi fields on $M$.

Answer 1

It can be shown that if $(M,g)$ is a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ are geodesic polar coordinates, then one can prove that,

If $M$ is a space form, i.e., with the constant (sectional) curvature $K$, then $g$ has the following expression (by Gauss's Lemma): $$ \newcommand{\rd}{\mathrm d} \rd s^2=(\rd r)^2+(f(r))^2h_{ij}(\theta)\rd\theta^i\rd\theta^j, $$ where the $m-1$-dimensional metric $(\rd\sigma)^2=h_{ij}(\theta)\rd\theta^i\rd\theta^j$ has constant sectional curvature $1$, and $$ f(r)=\begin{cases} \sin(\sqrt{Kr^2})/\sqrt{K},&K>0\\ r,&K=0\\ \sinh(\sqrt{-Kr^2})/\sqrt{-K},&K<0. \end{cases} $$