I am reading the 4.1 Theorem of do Carmo's Riemannian Geometry (as pictures below). The well-definedness of red line is not obvious for me.
In fact, I can accept that $\exp_p^{-1}$ map $S^n-\{q\}$ to $B(0,\pi)\subset T_pM$, where $B(0,\pi)$ is open ball of $T_pM$. Besides, $i$ is ok. But $\exp_{\tilde p}$ is not clear. So, I want to know why the red line is well-defined.
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1$M$ is complete, so $\tilde{M}$ is too and hence $\exp_{\tilde{p}}$ is defined on the whole of $T_{\tilde{p}}\tilde{M}$. – user10354138 Sep 08 '23 at 03:18
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@user10354138 Thanks. – Enhao Lan Sep 08 '23 at 12:16