Surface of Revolution as Warped Product Manifold

Theorem

Let $H$ be the open upper half-plane.

Let $S_C \subseteq \R^3$ be the surface of revolution, where $C$ is its generating curve.

Endow $C$ with the Riemannian metric induced from the Euclidean metric on $H$.

Let $\Bbb S^1$ be the $1$-dimensional sphere, that is, a circle endowed with its standard metric.

Let $f : C \to \R$ be the distance to the axis of rotation, say the $x$-axis:

$\map f {x, y} = y$

Then $S_C$ is isometric to the warped product manifold $C \times_f \Bbb S^1$.

Proof

This theorem requires a proof. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Sources

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