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When I read the 4.4 Proposition of Chapter 8 of do Carmo's Riemannian Geometry, I can't understand the red line in picture below. I think the $e$ is identity mapping. But, obviously, $\gamma$ has a eigenvalue equal to 1 do not means that $\gamma$ is identity mapping. How to understand it ?

enter image description here

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Let me recall that $\Gamma$ acts in a totally discontinuous manner on $S^n$, for every $x \in S^n$ there exists a neighborhood $U$ of $x$ such that $g(U) \cap U =\emptyset$, for all $g \in \Gamma \setminus \{e\}$. Now, let $\gamma$ as in the proof that you are studying. So far, we have that there exists $p \in S^n$ such that $\gamma(p)=p$. Let $U$ be any neighborhood of $p$. Then $\gamma(U) \cap U \supset \{p\} \neq \emptyset$. Since $\Gamma$ acts in a totaly discontinuous manner on $S^n$, $\gamma$ have to be the identity in order to no obtain a contradiction.

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