I am reading the proof of a proposition in the textbook "Differential Geometry of curves and surface" by do Carmo (2nd edition). It seems to me that the proof presented in the book is incorrect, so I want to check if I am missing something.
Let me first write down some definitions.
- A subset $S$ of $\mathbb{R}^3$ is called a regular surface if it can be covered by parametrizations which will be defined below.
- Let $U$ be an open subset of $\mathbb{R}^2$. A smooth ($C^{\infty}$) map $\mathbf{x} \colon U \to \mathbb{R}^3$ is called a parametrization of $S$ if $\mathbf{x}(U) \subset S$, $\mathbf{x}$ is a homeomorphism to its image, and $d\mathbf{x}_q \colon T_qU \to T_{\mathbf{x}(q)}\mathbb{R}^3$ is injective for all $q \in U$.
- $p \in S$ is called elliptic if the Gaussian curature at $p$ is positive.
(A part of) Proposition 1 in Section 3.3 states that all points on $S$ near an elliptic point $p$ lie on the same side of the tangent plane $T_p S$. In the proof of the proposition, the author chooses a parametrization $\mathbf{x} \colon U \to S$ with $\mathbf{x}(0,0) = p$ near an elliptic point $p$ and then uses the Taylor expansion $$\mathbf{x}(u,v) = \mathbf{x}(0,0) + \mathbf{x}_u u + \mathbf{x}_v v + \frac{1}{2}(\mathbf{x}_{uu}u^2 + 2\mathbf{x}_{uv}uv + \mathbf{x}_{vv}v^2) + \bar{R},$$ where $\bar{R}$ consists of terms of degree $\geq 3$.
However, we can't use the above equation unless we have proved that $\mathbf{x}$ is analytic at $(0,0)$. For instance, if we define $\mathbf{x}(u,v) = \left(u, v, e^{-\frac{1}{u^2+v^2}}\right)$ for $(u,v) \neq (0,0)$ and $\mathbf{x}(0,0) = (0,0,0)$, the map $\mathbf{x}$ is smooth but not analytic at $(0,0)$. The Gaussian curvature at $(0,0,0)$ is zero in this case. So maybe can we say more about elliptic points?
Let me summarize my question.
Is it always possible to choose an analytic parametrization near an elliptic point?