Unit Cylinder 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 straight line in the open upper half-plane.

Let the smooth local parametrization of $C$ be:


 * $\map \gamma t = \tuple {t, 1}$

Then the induced metric on $S_C$ is:


 * $g = d t^2 + d \theta^2$

Proof
We have that:


 * $\map {\gamma'} t = \tuple {1, 0}$

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

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


 * $g = d t^2 + d \theta^2$