📂Geometry

Positive Gaussian Curvature Surfaces of Revolution

Overview¹

The surface of revolution created by the unit speed curve $\boldsymbol{\alpha}(s) = \left( r(s), z(s) \right)$ is called $M$.

$$ M = \left\{ \left( r(s)\cos\theta, r(s)\sin\theta, z(s) \right) : 0 \le \theta \le 2\pi, s \in (s_{0}, s_{1}) \right\} $$

The coordinate patch mapping $\mathbf{x}$ of $M$ is as follows.

$$ \mathbf{x}(s, \theta) = \left( r(s)\cos\theta, r(s)\sin\theta, z(s) \right) $$

At this time, the Gaussian curvature of this surface of revolution is as follows.

$$ K = -\dfrac{r^{\prime \prime}}{r} $$

Describes the surface of revolution with the Gaussian curvature being $K = -\dfrac{r^{\prime \prime}}{r} = a^{2}$ positive.

Description

Based on the above assumptions, we obtain $r^{\prime \prime} + a^{2}r = 0$. Since this is a second-order ordinary differential equation, the solution is as follows.

$$ r(s) = a_{1}\cos(as) + a_{2}\sin(as) = A\cos(as + b) $$

Let’s denote this as $b=0$. Then, assuming that there was $r>0$, $A > 0$, and therefore, $\left| s \right| \lt \pi/2a$. Also, since $z^{\prime} = \pm\sqrt{1 - (r^{\prime})^{2}}$, the following theorem is obtained.

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Theorem

Let’s call $M$ a surface of revolution created by the unit speed curve $\boldsymbol{\alpha}(s)$, and a constant $K = a^{2}$ with positive Gaussian curvature. Then, $\boldsymbol{\alpha}(s) = \left( r(s), z(s) \right)$ is given as follows.

$$ \begin{align} r(s) &= A\cos (as),\quad \left| s \right| \lt \pi/2a \nonumber \\ z(s) &= \pm \int_{0}^{s} \sqrt{1 - a^{2}A^{2} \sin^{2}(at)}dt + C \end{align} $$

Here, $A > 0, C$ is a constant.

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If $A \ne \dfrac{1}{a}$, the given integral $(1)$ does not appear as an elementary integral, which is called an elliptic integral.

If $A = \dfrac{1}{a}$, it’s simply $z(s) = \pm {\displaystyle \int}_{0}^{s} \cos(at)dt + C = \pm\frac{1}{a}\sin (as) + C$. Therefore, since $r^{2} + z^{2} = \frac{1}{a^{2}}$, the revolution surface $M$ is a sphere.

The shape of the surface of revolution depending on the value of $A$ is as follows:

Also, since $r = A\cos(as)$, the matrix of metric coefficients is:

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