given first fundamental form, what type of surface?

Question

In Pressley book I found problem: Find the first fundamental form and determine what kind of surface is patch $\sigma=\langle u-v,u+v,u^2+v^2\rangle$. It is easy to see that the first fundamental form is $(2+4u^2)du^2+8uv\,du\,dv+(2+4v^2)dv^2$. The answer from the book tells that the surface is paraboloid of revolution. But the surface of revolution of unit-speed curve $\gamma(u)$ around $z$-axis is $\sigma(u,v)=\langle u\cos v, u\sin v,\gamma(u)\rangle$ where $v$ is the angle of rotation.Then the first fundamental form for the surface of rotation is $2du^2+u^2dv^2$, which differs from the result above. Is there an algorithm how to assign unknown surfaces to the typical surfaces relying on the first fundamental form?

Answer 1

A surface $S$ admits many parametrizations/patches $\sigma : D \subset \mathbb{R}^{2}_{(u, v)} \to \mathbb{R}^{3}_{(x,y,z)}$ and in each paramatrization the first fundamental form will potentially take on a different appearance. (Note that this exhibits one of the main features of differential geometry: you are trying to study surfaces independently of the parametrization and must pay attention to how expressions/quantities change when one changes parametrizations (or coordinates).)

In particular, if your surface $S$ is a surface of revolution obtained by revolving a (unit-speed) profile curve $\gamma(u) = (f(u), 0, g(u))$ about the $z$-axis by an angle of $v$ then your parametrization/patch will be of the form $$ \sigma(u, v) = \left\langle f(u)\cos v , f(u) \sin v, g(u)\right\rangle, $$ with corresponding first fundamental form $$ \mathrm{d}s^2 = \mathrm{d}u^2 + f(u)^2\mathrm{d}v^2. $$

But the given parametrization of your surface $S$ is $$ \sigma(u, v) = \left\langle u - v , u + v, u^2 + v^2\right\rangle $$ is not the parametrization obtained by rotating a profile curve about an axis of revolution.

In this instance, the author probably intends that you recognize the surface $S$ as a paraboloid of revolution by noting the following algebraic relation of the component/coordinate functions of the parametrization: \begin{align*} x^2 + y^2 &= x(u, v)^2 + y(u, v)^2\\ &= (u - v)^2 + (u + v)^2\\ &= 2(u^2 + v^2)\\ &= 2z(u, v)\\ &= 2z.\\ \end{align*}

Thus, all points $P(x, y, z)$ on the surface patch satisfy $x^2 + y^2 = 2z$ and lie on the paraboloid of revolution given by the graph of the function $z = F(x, y) = \frac{1}{2}\left(x^2 + y^2\right)$. (Recognizing this would also allow you to know parametrize your surface using Monge patch, in which case the first fundamental form would again change appearance.)