I am reading the John Lee's Introduction to Riemannian manifold, proof of the Proposition 11.3 ( first paragraph ) and stuck at some statement.
Let $(M,g)$ be a Riemannian manifold, $U\subseteq M$ is a normal neighborhood of $p\in M$. Given any normal coordinates $(x^i)$ on $U$ at $p$, define the radial distance function $r: U \to \mathbb{R}$ by $$r(x):= \sqrt{(x^1)^2 + \cdots +(x^n)^2},$$
and the radial vector field on $U-\{p\}$, denoted by $\partial_r$, by
$$ \partial_r=\frac{x^i}{r(x)}\frac{\partial}{\partial x^i}.$$
Let $q \in U-\{p\}$, and let $\gamma : [0,b] \to U$ be the unit speed radial geodesic from $p$ to $q$ (so $b=r(q)$ ).
Then $ \gamma'(t) = \partial_r|_{\gamma(t)}$ ? If so, why is it true?
I think that we may use the John Lee's book, Proposition 5.24-(c) :
Proposition 5.24 ( Properties of Normal coordinates). Let $(M,g)$ be a Riemannian or pseudo-Riemannian $n$-manifold, and let $(U, (x^i))$ be any normal coordiane chart centered at $p\in M$.
(c) For every $v= v^i \partial_i|_p \in T_p M$, the geodesic $\gamma_v$ starting at $p$ with initial velocity $v$ is represented in normal coordinates by the line $$ \gamma_v (t) = (tv^1, \dots tv^n),$$ as long as $t$ is in some interval $I$ containing $0$ such that $\gamma_v(I) \subseteq U$.
Can anyone help?
Best Answer
Let $V:= \operatorname{exp}_p^{-1}(q)$. Now consider $\gamma(t) := \operatorname{exp}_p(t |V|^{-1}V)$
This $\gamma$ is the unit-speed radial geodesic from $p$ to $q$ (?) ( $\because $ By the chain rule this now was constant unit speed (first evaluate $\dot\gamma(0)$ then use that $\lambda \mapsto \exp_p(\lambda v)$ has constant speed) ). Now, in normal coordinates, this is given by the curve $$ \tilde{\gamma}(t) = |V|^{-1}(tV^1, \ldots, tV^n) $$ and therefore its tangent vector is $\dot{\tilde{\gamma}}(t) = |V|^{-1}(V^1, \ldots, V^n)$. But, as $\partial/\partial x^i$ is just the $i$th standard basis vector in coordinates, we have $$ \frac{\partial}{\partial r} = \frac{x^i}{r}\frac{\partial}{\partial x^i} = \frac{1}{r}x, $$ where $x = (x^1, \ldots, x^n)$ is the coordinates of a point on the manifold. Thus for instance when $x = \tilde{\gamma}(t) = t|V|^{-1}V$, we have $\frac{\partial}{\partial r} = r^{-1}x = |t|^{-1}t|V|^{-1}V = |V|^{-1}V = \dot{\tilde{\gamma}}(t)$. In this last line I have abused notation, writing $V$ for the coordinate representation $(V^1, \ldots, V^n)$.
Correct ?