Unit Sphere as Surface of Revolution

Theorem
Let $\struct {\R^3, d}$ be the Euclidean space.

Let $S_C \subseteq \R^3$ be the surface of revolution.

Let $C$ be a semi-circle defined by $x^2 + y^2 = 1$ in the open upper half-plane.

Let the smooth local parametrization of $C$ be:


 * $\map \gamma \phi = \tuple {\cos \phi, \sin \phi}$

where $\phi \in \openint 0 \pi$.

Then the induced metric on $S_C$ is:


 * $g = d \phi^2 + \sin^2 \phi \, d \theta^2$

Proof
By Smooth Local Parametrization of Surface of Revolution, the smooth local parametrization of $S_C$ can be written as:


 * $\map X {\phi, \theta} = \tuple {\sin \phi \cos \theta, \sin \phi \sin \theta, \cos \phi}$

By Induced Metric on Surface of Revolution:

This is a metric of a unit sphere with the points on the $x$-axis removed.