Unit Sphere as Surface of Revolution

Theorem

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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:

$g = \d \phi^2 + \sin^2 \phi \rd \theta^2$

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:

\(\ds g\)

\(=\)

\(\ds \paren {\paren {\cos' \phi}^2 + \paren {\sin' \phi}^2} \rd \phi^2 + \sin^2 \phi \rd \theta^2\)

\(\ds \)

\(=\)

\(\ds \paren {\sin^2 \phi + \cos^2 \phi} \rd \phi^2 + \sin^2 \phi \rd \theta^2\)

\(\ds \)

\(=\)

\(\ds \rd \phi^2 + \sin^2 \phi \rd \theta^2\)

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

$\blacksquare$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics