Proof that every Riemannian geodesic is locally minimizing

I'am reading John M.Lee's Riemannian Manifolds-An Introduction to curvature, p.107

Here, 'radial geodesic' means a geodesic starting at $p$ and lying in a normal neighborhood of $p$. (his book p.78)

And an open set $W \subset M$ is called 'uniformly normal' if there exists some $\delta > 0$ such that $W$ is contained in a geodesic ball of radius $\delta$ around each of its points(his book p.78)

I'm trying to understand the underlined statement. Any two geodesic lying in the same geodesic ball are same? Or perhaps, is there any relation between them so we can deduce that "$\gamma$ must itself be this minimizing geodesic"?

Can anyone helps for proving this statement rigorously?

Hmmm ... you're right -- I didn't explain this well at all. It doesn't follow directly from Proposition 6.10; instead there are two steps to the argument: first note that the proposition on properties of normal coordinates (Prop. 5.11 in the first edition, 5.24 in the second) shows that every geodesic segment lying in a geodesic ball and starting at the center of the ball is part of a radial geodesic, and then Prop. 6.10 in the first edition (6.11 in the second) shows that each radial geodesic segment is minimizing.

Embarassingly, this same omission was carried over to the second edition of my book, where I thought I had eradicated all of these logical leaps. Oh well.

I've added corrections for both editions here. Thanks for pointing this out.

EDIT: Here are some answers to the questions you asked comments.

First, since geodesic ball is a normal neighborhood, by the definition of radial geodesic( your book p.78)every geodesic segment lying in a geodesic ball is automatically radial geodesic?

No, every geodesic lying in a geodesic ball and starting at the center of that ball is automatically a radial geodesic.

So, if Prop.6.10 means that every radial geodesic segment is minimizing, then in particular $\gamma_{[t_1,t_2]}$ is minimizing. Is this argument correct?

Yes. (A better notation is $\gamma|_{[t_1,t_2]}$ to denote the restriction of $\gamma$ to the interval $[t_1,t_2]$.)

Second, but..you wrote in prop.6.10 that " 'the' radial geodesic from $p$ to $q$ is the unique minimizing curve from $p$ to $q$ in $M$. " Is there a special reason for attaching 'the' to radial geodesic? What is 'the' radial geodesic?

The definition of "radial geodesic" (which appears just after Proposition 5.11) is any geodesic starting at $p$ and lying in a normal neighborhood of $p$. Proposition 5.11 shows that they all are given by $\gamma_V(t) = (tV^1,\dots,tV^n)$ for various choices of $(V^1,\dots,V^n)$. You can check that for any point $q$ in that neighborhood, up to reparametrization, there is exactly one such geodesic from $p$ to $q$. (One such parametrization is obtained by taking $(V^1,\dots,V^n)$ to be the coordinates of $q$, and taking the parameter interval to be $[0,1]$.)

And.. as your answers, for a given geodesic segment (in particular, $\gamma_{[t_1,t_2]}$) in a geodesic ball, can we provide an explicit radical geodesic that has the geodesic(γ[t1,t2]) as a part? How?

As I noted above, the radial geodesic from $p$ to $q$ is $\gamma(t) = (tV^1,\dots,tV^n)$, where $(V^1,\dots,V^n)$ are the coordinates of $q$ and $t$ ranges from $0$ to $1$.

(And what is the rigorous definition of "a part of radial geodesic"?

Just the restriction of the radial geodesic to a subinterval of its domain.

And in our argument, you suggested to use Prop 5.11 (first edition). And it just states that we can represent the $\gamma_V$ in normal coordinate as simple form. How to apply Prop.5.11 to our argument. Perhaps, we maybe use the Prop. 5.7-(b)

Actually, the very definition of "radial geodesic" is "a geodesic starting at $p$ and lying in a normal neighborhood of $p$." What I used Proposition 5.11 for is the claim that there's only one such geodesic from $p$ to $q$ (up to reparametrization, anyway), which follows from the explicit formula.

Consider a set $S$ which is a geodesic ball of radius $r$ around $p$ and also a normal neighborhood of $p$. Since $S$ is a geodesic ball, it contains all geodesics of length at most $r$ starting at $p$. Since $S$ is a normal neighborhood of $p$, all geodesics within this neighborhood starting at $p$ end at different points. Hence, for any point $q$ within $S$, the geodesic within $S$ from $p$ to $q$ is shorter than any geodesic from $p$ to $q$ not within $S$.