Definition of $\partial_r$ and Gauss Lemma (Petersen's Riemannian Geometry)

Question

From page 43 of Petersen's Riemannian Geometry:

Let $U$ be an open subset of a Riemannian manifold $(M,g)$ and $r:U\to \mathbb R$ a distance function, then $\partial_r=\nabla r$ is the gradient.

So far, everything seems clear, but then I am confused on page $133$:

The Gauss Lemma: On $(U, g)$ the function $r$ satisfies $\nabla r = \partial_r$, where $\partial_r = D\exp_p(\partial r)$.

Here $U$ is a suitable neighborhood of $p\in M$ and the equation $\nabla r = \partial_r$ is clearly not considered to be a definition. Furthermore, I am confused by the equation $\partial_r = D\exp_p(\partial_r)$. Is this an abuse of notation or is this equation telling us that $D\exp_p$ maps $\partial_r$ onto itself?

Answer 1

I think that Petersen is mixing up two different, yet equivalent, definitions of $r$ here.

1. The first is as the distance function $r = d(\cdot,p)$. In that case, its gradient $\nabla r$ (in a punctured neighbourhood) is well defined as a local vector field on $M$.
2. The second is as a component in some specific coordinates, namely, the polar exponential coordinates $(r,\theta) \in (0,\varepsilon)\times \Bbb S \mapsto \exp_p(r\theta) \in M$, where $\Bbb S$ is the unit sphere of $T_pM$. On $(0,\varepsilon)\times \Bbb S$, there is a canonical vector field $\frac{d}{d r}$ which induces a local vector field $\frac{\partial}{\partial r} = (\exp_p)_*(\frac{d}{dr})$ on $M$.

It then follows from Gauss Lemma that these two notions give the same vector field, i.e. that $\nabla r = \frac{\partial}{\partial r}$. It is usually denoted $\partial_r$ and called the radial vector field.