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Suppose we have a Riemannian manifold $M$ with an open subset $U$ and a smooth map $\theta: U \to M$. If there is point $q\in U$ such that $\theta(q)=q$ prove that $d\theta_q=Id$ as a map $d\theta_q:T_qM\to T_qM$.

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Is this a homework question? Why do you ask a question in the imperative mode? It sounds demeaning. FYI, your question is impossible -- any such proof would be wrong. –  Ryan Budney Oct 9 '12 at 22:21
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This is not true like this. Think about a rotation in Euclidean plane, for example.. –  Berci Oct 9 '12 at 22:23
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