Uniqueness problem for surfaces with given Gauss curvature

Question

We know that the curvature and the torsion of a curve determine uniquely the curve itself (up to position in space). Now what about surfaces. E.g., are there connected surfaces with constant positive Gaussian curvature $ +1 $ which are not (isometric to a subset of) the sphere?

And, more in general, under which additional conditions is a surface uniquely determined?

This is a question that arose in my personal thoughts, and I have no clue on how this problem can be solved; I think it is something that exceeds the status of "exercise".

Thank you in advance.

Answer 1

Except for translations/rotations a surface is uniquely defined in space if the first and second fundamental forms are given. If the first form (FF) alone is given a set of inter-bendable surfaces is defined.They share anything you can define or name from the coefficients of FF, for example Gauss or geodesic curvature, integral curvature, the geodesic lines, Christoffel symbols and any object that can be assembled with them, which are all invariant in isometric mappings.

Answer 2

Liebmann's theorem states that any compact surface which has constant Gaussian curvature is (isometrically) a sphere.

As the other answer states, this is a special case; most surfaces need not so rigid. Probably the two most famous specific examples are:

• Any developable surface (such as a cylinder or a cone) has Gaussian curvature everywhere zero, but need not be planar.
• The helicoid and the catenoid have the same Gaussian curvature as each other, meaning they have the same intrinsic geometry despite being clearly non-isometric as surfaces embedded in $\Bbb{R}^3$.