Torus 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 circle defined by $x^2 + \paren {y - 2}^2 = 1$ in the open upper half-plane.

Let the smooth local parametrization of $C$ be:

$\map \gamma t = \tuple {\sin t, 2 + \cos t}$

Then the induced metric on $S_C$ is:

$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$

Proof

We have that:

$\map {\gamma'} t = \tuple {\cos t, - \sin t}$

Furthermore:

$\paren {\cos t}^2 + \paren {- \sin t}^2 = 1$

Hence, $\map \gamma t$ is a unit-speed curve.

By the corollary of the induced metric on the surface of revolution:

$g = d t^2 + \paren {2 + \cos t}^2 \, d \theta^2$

$\blacksquare$

Sources

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