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I encounter to the question in reading the following Excise:

Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ 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 Jacobo-field $J$ ($J(0)=0$, $|\dot J(0)|=1$ and $J$ is perpendicular to the base geodesic curve $\gamma(t)$) for 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 $W(t)\bot\dot\gamma(t)=T(t)$.

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.

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This question is firstly post on…, after I read the site meta, I think is more prefer to post it here. – van abel Feb 21 '12 at 0:41
Actually, this is foundational material, from a first course on Riemannian manifolds. MSE is better. – Will Jagy Feb 21 '12 at 0:50
Will, I am not so sure that the question will get a good answer on MSE – Yemon Choi Feb 21 '12 at 1:23
I suggest consulting more than one textbook on Riemannian geometry. There are different ways to derive these formulas, and you might find one approach easier to understand than others. You can do this using moving frames and differential forms; you can also just do everything using local co-ordinates. Just fix a choice of local co-ordinates on the unit sphere and use them along with the radial co-ordinate $r$. – Deane Yang Feb 21 '12 at 11:41
Actually, using Jacobi fields to do this exercise is not such a bad idea. It's actually one of my preferred approaches. But it seems to me that when you are learning this stuff for the first time, you shouldn't worry about doing it the "right" way or the "best" way or the "least messiest" way first. Just do it one way or another, even if it involves pages and pages of awful calculations with Christoffel symbols. Then slowly learn how to do it more elegantly. My advice is to keep struggling with it! – Deane Yang Feb 21 '12 at 21:53

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