Unit Cylinder as Surface of Revolution
Jump to navigation
Jump to search
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:
- $g = d t^2 + d \theta^2$
$\blacksquare$
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Methods for Constructing Riemannian Metrics