Let $M$ be a class $C^k$-Riemannian manifold and suppose there exists an atlas $\langle U,\psi\rangle$ for $M$ containing only one global chart.

  • Does this imply that the Riemmanian $Log_p\,(:=Exp_p^{-1})$ maps are defined globally?

    Ie: for every point $p\in M$ $Dom(Log_p)=M$?

  • If so can we give a simple and explicit description of these coordinates in terms of the map $\psi$?

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