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 $C \times_f \Bbb S^1$.