What's the relationship between the Riemannian metric and Jacobi field?

Question

I encounter to the question in reading the following Excise:

Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\dotsc,\theta^{m-1})$ be the (geodesic) polar coordinate. 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 $$ f(r)=\begin{cases} \sin(\sqrt{Kr^2})/\sqrt{K},&\text{if }K>0; \\ r,&\text{if }K=0; \\ \sinh(\sqrt{-Kr^2})/\sqrt{-K},&\text{if }K<0. \end{cases} $$

I already know that the Jacobi-field $J$ ($J(0)=0$, $\lvert\dot J(0)\rvert=1$ and $J$ is perpendicular to the base geodesic curve $\gamma(t)$) for a manifold with constant sectional curvature $K$ is $$ J(t)=f(t)W(t), $$ where $f(t)$ is defined as above and $W(t)$ is a unit parallel vector filed along $\gamma$ with $W(t)\perp\dot\gamma(t)=T(t)$.

So my question is that: can we give a proof that based on the Jacobi-field? If not, what's the relation between the two problem?

In fact, I don't know how the solve the first problem, and try to use Jacobi-field to give a proof without no results.

Answer 1

By Gauss lemma, $$ \newcommand{\rd}{\mathrm d} \rd s^2=(\rd r)^2+\tilde h_{ij}(r)(\theta)\cdot \rd\theta^i\cdot\rd\theta^j, $$ where $\tilde h_{ij}(r)$ is a Riemannian metric on the sphere that depends on the radius $r$.

Note that $\tilde h_{ij}(r)$ is rotationally symmetric. Therefore $h_{ij}(r)=f(r)^2\cdot h_{ij}(\theta)\cdot\rd\theta^i\cdot\rd\theta^j$ for some function $f$. It remains to check that $f''+K\cdot f=0$.

Note that $f(r)=|J(r)|$, where $J=\tfrac{\partial}{\partial \theta_1}$ is a field on the geodesic $t\mapsto (t,0,\dots,0)$. Note that $J$ is a Jacobi field --- it is a field of variation of radial geodesics. It remains to write Jacobi equation and get $f''+K\cdot f=0$ (you know it already).