Let $S$ be a regular connected surface and let $\varphi , \psi : S \to S$ two isometries and we suppose that exists $p \in S$, with $\varphi(p) = \psi(p)$, such that $d {\varphi}_p = d {\psi}_p$. I have to prove that $\varphi = \psi$ in $S$. I have tried to use Egregium Gauss theorem or Minding theorem but I haven't obtain any result. Can you help me? Thank you very much.
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I know very little about this are, but it if I remember correctly, you can use existence and uniqueness of geodesics for regular connected surfaces to prove the result. – Andres Mejia Sep 06 '17 at 18:07
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Define $V:={q\in S: \varphi(q)=\psi(q) \text{ and } d\varphi_q=d\psi_q}$ and show that it is open and closed. Can you finish from there? – Frieder Jäckel Sep 28 '17 at 21:02
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Yes, thank you very much! – joseabp91 Sep 30 '17 at 05:38