Gauss Lemma: metric and coordinates

In my studies of Riemannian Geometry I ran into the following formulation of the Gauss Lemma:

Gauss Lemma: Around any point $p$ in a Riemannian Manifold $M^n$ there exists a system of coordinates $(r, \theta^1, \dots, \theta^{n-1}) $, such that $r$ vanishes at $p$ and $\theta=(\theta^1, \dots, \theta^{n-1})$ denotes the standard coordinate system on the sphere $S^{n-1}$, in which the metric takes the form: $$ds^2=dr^2+f(r, \theta)^2d\theta^2_{n-1}$$ Where $d\theta^2_{n-1}$ denotes the standard metric on $S^{n-1}$ in the coordinate system $(\theta^1, \dots, \theta^{n-1})$.

My questions is: how can we deduce this formulation of the Gauss Lemma from the usual formulation of the lemma, for exam from P.Petersen's Riemannian Geometry which says:

Gauss Lemma: On a normal neighborhood $U$ of $p$, the distance function $r$ satisfies $\nabla r=\partial_r$, where $\partial_r=D\,exp(\partial_r)$.

Thank you in advance!

Solution 1:

The "usual" formulation of Gauss Lemma is one of:

Gauss Lemma I: For all $p\in M$, $v\in T_pM$, $w\in T_vT_pM\cong T_pM$, we have $g_{\exp_p(v)}(\mathrm{d}\exp_p(v),\mathrm{d}\exp_p(w))=g_p(v,w)$ (as long as $\exp_p(v)$ exists).

Gauss Lemma II: The geodesic spheres $\Sigma_p(r)$ are perpendicular to their radii $\operatorname{d}\exp_p(\partial_r)$ for $0<r<\operatorname{inj}_p M$.

and you can deduce the metric in geodesic polar coordinates:

Corollary: The metric $g$ in geodesic polar coordinates has local expression $g=\mathrm{d}r^2 + g\vert_{\Sigma_r(p)}$.

Petersen's formulation of Gauss Lemma is not the quite usual one, as he was trying to deduce as much as possible from a distance-like function alone. However, you can see how you can get Gauss Lemma II from his formulation and vice versa, since $\nabla r$ is orthogonal to $r=\operatorname{const}$, i.e., geodesic spheres and hence deduce the corollary.