Possible correction for Theorem 6.13 (Hopf-Rinow) Lee's book Riemannian Manifold 2nd Ed?

Question

I think i found a notational error for this theorem from Lee's book Riemannian manifold p.108.

I think the expression at the last sentence it should be $\tilde{\gamma}(t) = \gamma(t_j+t) $ instead of $\tilde{\gamma}(t) = \sigma(t_j+t)$. Or probably more clear $\sigma(t) = \tilde{\gamma}(t) = \gamma(t_j+t)$. Did anyone ever following this theorem and find this confusing ? I already seen the errata for this book (https://sites.math.washington.edu/~lee/Books/Riemannian/errata.pdf) but not found any correction for it. Is this true ? Or its just my ignorance.

Answer 1

You're right that the formula is incorrect. But it should have been $$ \tilde\gamma(t) = \sigma(t-t_j) \text{ for $t\in (t_j-\delta,t_j+\delta)$}. $$ (This is equivalent to the formula proposed by John Ma, but perhaps a little clearer.) I had noticed this and fixed it in my draft second edition, but neglected to add the fix to my correction list for the first edition. It's now added. Thanks for pointing it out.

Answer 2

We want $\tilde \gamma : [0, t_j+ \delta) \to M$ so that $\tilde\gamma$ is an extension of $\gamma$ and $\tilde \gamma$ is a geodesic. We define $\tilde \gamma (t) = \gamma(t)$ on $[0,t_j)$. For $[t_j, t_j + \delta)$, define

$$\tilde \gamma (t_j+s) = \sigma (s).$$

where $\sigma : [0, \delta)\to M$ is the geodesic with $\sigma (0) = \gamma(t_j)$ and $\sigma'(0) = \gamma'(t_j)$. Note that $\tilde \gamma(t) = \gamma(t)$ on $[t_j, b)$: Since they both satisfy

$$\gamma(t_j) = \tilde\gamma (t_j), \ \ \gamma'(t_j) = \tilde\gamma'(t_j) \ \ (= \sigma'(0)).$$