I'm studying the section 7 ("Local Geometry in Constant Curvature) of chapter 5 of "Riemannian Geometry" written by Petersen.

At the beginning there is a Lemma which says how behaves the metric $g$ of a Riemannian manifold $M$ around a point $p$ in normal coordinates: $$ g_{ij} = \delta_{ij} + O(r^2). $$ where $r$ is the distance from $p$.

Then the book says in polar coordinates around $p$ any Riemannian metric has the form: $$ g = dr^2 + g_r $$ where $g_r$ is a metric on $S^{n-1}$.

We know also that the Euclidean metric looks like: $$ \delta_{ij} = dr^2 + r^2 ds^2_{n-1}, $$ where $ds^2_{n-1}$ is the canonical metric on $S^{n-1}$. Since these two metrics agrees up to the first order (thanks to the Lemma), we have that: $$ \lim_{r \rightarrow 0}g_r = \lim_{r \rightarrow 0}(r^2 ds^2_{n-1}) = 0 $$ and $$ \lim_{r \rightarrow 0}\Big(\partial_rg_r - \frac{2}{r}g_r\Big) = \lim_{r \rightarrow 0}\Big(\partial_r(r^2 ds^2_{n-1})- \frac{2}{r}(r^2 ds^2_{n-1})\Big) = 0. $$

Everything is quite clear so far. But then the book says that since

$$ \partial_rg_r = 2\text{Hess}r$$

then we get $$ \lim_{r \rightarrow 0}\Big(\text{Hess}r - \frac{1}{r}g_{r}\Big) = 0. $$

I don't understand why $\partial_rg_r = 2\text{Hess}r$. Any help would be really appreciated.

Another question: in the next theorem Petersen talks about space forms $S^n_k$. I know that they should be complete Riemannian manifold of dimnsion $n$ and constant sectional curvature $k$, but do you know where does he define them? What is $sn_k$?


Recall $$ {\rm Hess}\ f (v,w) =(\nabla_v {\rm grad}\ f,w) $$

Hence on a geodesic sphere $S(p,r)$, let $$v\in T_q S(p,r),\ q=\exp_p\ rx \in S(p,r),\ |x|=1,\ x\in T_pM $$ Then we have $$ (d\exp_p)_{rx} v_0=v $$

Define $$ f(s,t)=\exp_p \ sv(t),\ |v|=1,\ v(0)=x,\ v'(0)=v_0 $$

So $ f_t(r,0)=v $ so that $$ {\rm Hess}\ r(v,v)=(\nabla_v {\rm grad}\ r,v)=(\nabla_{f_t} f_s,f_t)=\partial_r \frac{1}{2} g(f_t,f_t) =\partial_r \frac{1}{2}g(v,v) $$


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