I'm trying to teach myself the classical differential geometry of 2D surfaces in 3D Euclidean space but I'm struggling to understand exactly how much information the Gaussian and mean curvature provide.
If I've understood the Bonnet theorem correctly, the Gauss–Codazzi equations are both necessary and sufficient conditions on the first and second fundamental forms to determine a surface (up to rigid motions). As elaborated on in this question Gaussian curvature and mean curvature sufficient to characterize a surface?, this implies that the Gaussian curvature and mean curvature alone should generally be insufficient to determine a surface.
Can someone give me an explicit example of two different surfaces, parameterised over the same patch, which have the same Gaussian and mean curvature everywhere?
Best Answer
First, you need simple connectivity to get a global result for sufficiency. The equality of Gaussian and mean curvatures tells you that the shape operators $\text{I}^{-1}\text{II}$ have the same eigenvalues. This is not telling you that the first and second fundamental forms of the two surfaces are equal.
Here's the easiest example I can think of. Take the catenoid $$x(u,v)=(u\cos v,u\sin v,v)$$ and the helicoid $$y(u,v)=\left(\sqrt{1+u^2}\cos v,\sqrt{1+u^2}\sin v, \sinh^{-1}u\right).$$ In fact, these two surfaces are locally isometric (and the first fundamental forms agree with these parametrizations), so they have the same Gaussian curvature. They are both minimal, so the mean curvatures are both $0$. But the surfaces are not even locally congruent: One is ruled and the other is not.
P.S. You may be more advanced at this point, but you might want to check out my undergraduate differential geometry text, (freely) linked in my profile.