Let $\left\{ U_{\alpha },\varphi _{\alpha }\right\}$ be a local chart of a regular surface S $\subset \mathbb{R}^{3}$. That is:

$\varphi_{\alpha}:U_{\alpha}\subseteq\mathbb{R}^2\rightarrow \varphi_{\alpha}(U_{\alpha})\subseteq S$ is a $C^\infty$ homeomorphism with injective differential.

We say that a function $f:A\subseteq \mathbb{R} ^{n}\rightarrow \mathbb{R} ^{m} $ is a local diffeomorphism at a point $x\in A$ if there exists a neighborhood $U\subseteq A$ of $x$ such that $f_{|U}:U \rightarrow f(U)$ is bijective, $C^{\infty}$ with $C^{\infty}$ inverse.

Can we conclude that $\varphi_{\alpha}:U_{\alpha}\rightarrow \varphi(U_{\alpha})$ is a local diffeomorphism at every $x\in U_{\alpha}$?

I think it’s true, and I believe that one way to prove it is to apply the inverse function theorem, noting that $d{\varphi_{\alpha}}_{x}:\mathbb{R}^{2}\rightarrow T_{\varphi(x)}S$ is an isomorphism of vector spaces. However, I can’t give a rigorous proof.

  • $\begingroup$ First, a chart is usually defined to go the other way (from an open subset of the surface to an open subset of $\Bbb R^2$). Your definition of diffeomorphism is only for a map on $\Bbb R^n$ to $\Bbb R^n$. Do you know how to define what it means for $f\colon S\to\Bbb R$ to be $C^\infty$ when $S$ is a surface? In fact, once you answer this, $\varphi_\alpha$ is in fact a diffeomorphism, yes [you don't need "local"]. $\endgroup$ – Ted Shifrin Jun 3 at 21:40
  • $\begingroup$ @TedShifrin Now I understand. $f:S\rightarrow\mathbb{R}$ is $C^{\infty}$ at $p\in S$ if for some parameterization $\varphi:U\subseteq\mathbb{R}^{2}\rightarrow S$ such that $p\in\varphi(U)$ we have that $f\circ\varphi:U\rightarrow\mathbb{R}$ is $C^{\infty}$ at $\varphi^{-1}(p)$. I was using wrong definitions. Thank you very much. $\endgroup$ – W.. Jun 3 at 23:51
  • $\begingroup$ @TedShifrin Isn't it common that for books on Differential Geometry of Surface the chart map defined as the OP define them ? As far as i can remember i've had a class where we follow that kind of books. $\endgroup$ – Si Kucing Jun 4 at 2:15
  • $\begingroup$ @Eumenes: Then they're called parametrizations. After all, a chart is like a map in an atlas, so it's representing the surface locally on a piece of paper. Typically, we use parametrizations for submanifolds of Euclidean space and charts for abstract manifolds. (Note the OP even called it a parametrization in the title!) $\endgroup$ – Ted Shifrin Jun 4 at 5:44
  • $\begingroup$ @TedShifrin Ok. I got it now. I've checked on them and you're right ! Thank you. $\endgroup$ – Si Kucing Jun 4 at 5:54

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