# Asymptotic behaviour of the riemannian metric in polar coordinates

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