Induced Metric on Surface of Revolution/Corollary

Theorem

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

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

Let the smooth local parametrization of $C$ be:

$\map \gamma t = \tuple {\map x t, \map y t}$

Let $\gamma$ be a unit-speed curve.

Then the induced metric on $S_C$ is:

$g = d t^2 + \map y t^2 d \theta^2$

Proof

By definition of the unit-speed curve:

$\size {\map {\gamma'} t}_g = 1$

In our case we are working with the Euclidean space.

Hence:

$\sqrt {\map {x'^2} t + \map {y'^2} t} = 1$

or:

$ \map {x'^2} t + \map {y'^2} t = 1$

Substitution of this into the induced metric of $S_C$ yields the desired result.

$\blacksquare$

Sources

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