📂Geometry

Two Rotational Surfaces with Positive Curvature are Locally Isometric

Theorem¹

Let $M_{1}$, $M_{2}$ be the unit speed curves $\boldsymbol{\alpha}_{1}$, $\boldsymbol{\alpha}_{2}$ of the surfaces of revolution respectively. If $M_{1}$, $M_{2}$ have a constant curvature $a^{2} \gt 0$, then $M_{1}$, $M_{2}$ are locally isometric.

Proof

Lemma

The following two propositions are equivalent:

• The two surfaces $M$ and $N$ are locally isometric.

• For all $p \in M$, there exist two coordinate patch mappings $\mathbf{x} : U \to M$, $\mathbf{y} : U \to N$ $(p \in \mathbf{x}(U))$ such that open set $U \subset \mathbb{R}^{2}$ and the coefficients of the first fundamental form $g_{ij}$ are the same.

According to the lemma above, it’s sufficient to prove that all surfaces of revolution with a curvature of $K = a^{2}$ can have the same metric matrix. Therefore, the following claim is to be proved.

Claim: For all surfaces of revolution with a curvature of $K = a^{2}$, there exists a coordinate patch mapping that has $\left[ g_{ij} \right] = \begin{bmatrix} 1 & 0 \\ 0 & \dfrac{1}{a^{2}}\cos^{2}as \end{bmatrix}$ as its metric coefficient matrix.

The coordinate patch mapping of the surface of revolution generated by the unit speed curve $\boldsymbol{\alpha}(s) = (r(s), z(s))$ is $\mathbf{x}(s, \theta) = \left( r(s)\cos\theta, r(s)\sin\theta, z(s) \right)$. The metric of the coordinate system $(s, \theta)$ for a surface of revolution with a curvature of $K = a^{2} > 0$ is as follows.

$$\begin{bmatrix} g_{ij} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & A^{2} \cos^{2}(as) \end{bmatrix}$$

Let’s consider a new coordinate system $(s, \phi), \phi = a A \theta$. If $f : (s, \phi) \mapsto (s, \theta)$ is defined, then the Jacobian of $f$ is as follows.

$$J = \begin{bmatrix} \dfrac{\partial s}{\partial s} & \dfrac{\partial s}{\partial \phi} \\[1em] \dfrac{\partial \theta}{\partial s} & \dfrac{\partial \theta}{\partial \phi} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & \dfrac{1}{aA} \end{bmatrix}$$

Hence, if the metric matrix of the new coordinate system is denoted as $\begin{bmatrix}\overline{g}_{\alpha \beta}\end{bmatrix}$, according to the relationship between coordinate transformation and metric,

$$\begin{align*} \begin{bmatrix}\overline{g}_{\alpha \beta}\end{bmatrix} &= J^{t} \begin{bmatrix} g_{ij} \end{bmatrix} J \\ &= \begin{bmatrix} 1 & 0 \\ 0 & \dfrac{1}{aA} \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & A^{2} \cos^{2}(as) \end{bmatrix} \begin{bmatrix} 1 & 0 \\ 0 & \dfrac{1}{aA} \end{bmatrix} \\ &= \begin{bmatrix} 1 & 0 \\ 0 & \dfrac{1}{a^{2}}\cos^{2}(as) \end{bmatrix} \end{align*}$$

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