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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?

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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.

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    $\begingroup$ Plot of both surfaces: math3d.org/WsnNs862C $\endgroup$
    – Al.G.
    Oct 17, 2023 at 20:32
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    $\begingroup$ Thanks for that, @Al.G. I actually assume that every differential geometry student knows well what the two surfaces look like, but it never hurts :) $\endgroup$ Oct 17, 2023 at 20:34
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    $\begingroup$ Darn, I came to post exactly this is example. In Ivey and Landsberg's "Cartan for Beginners" they give an account of a generalization of this phenomenon called the Bonnet family. In minimal surfaces, every minimal surface comes in a $1$-parameter family by a sort of complex rotation. The part you see as the surface is the real part of an immersion into $\mathbb{C}^3$. The relevant keyword is "Weierstrass-Enneper representation." $\endgroup$ Oct 18, 2023 at 1:08
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    $\begingroup$ @A.ThomasYerger Yes, this is also in Spivak and in most every treatment of minimal surfaces. But I had never before thought of it from the perspective raised in this question. $\endgroup$ Oct 18, 2023 at 1:10
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    $\begingroup$ Oh, @A.ThomasYerger, I missed that aspect of your comment. So you’re no longer integrating a holomorphic (or anti-holomorphic) Gauss map? Interesting. No, I don’t think I’ve thought about this generalization. $\endgroup$ Oct 18, 2023 at 1:18

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