If $X$ is a smooth manifold and $I : X \rightarrow X$ is the identity map on $X$ (with the same smooth structure on both sides) then I can show that $dI_x : T_x X \rightarrow T_x X$ is the identity map on the tangent space $T_x X$ of $X$ at some point $x \in X$. Here $dI_x$ is the differential of $I$ at the point x.
Is the converse of this statement also true? That is if I have a map $f : X \rightarrow X$ such that $f(x)=x $ for some $x \in X$ and $df_x : T_x X \rightarrow T_x X$ is the identity map on $T_x X$ then $f$ is the identity map?
For example, if I have a map $f : X \rightarrow X$ that is identity restricted to some open subset $U$ of $X$ then $T_x X = T_x U$ for $x\in U$ so $df_x$ is the identity map, but $f$ need not be. Is that right?
Thanks!
Consider the smooth map $f:\mathbb{R}\to\mathbb{R},\;x\mapsto\sin x$. We have $f(0)=0$, and $df_0=id$. However, $f$ is not the identity, not even on some neighborhood of $0$.