Surfaces of Constant Curvature
- Minding's theorem (1839) states that all surfaces with the same constant curvature K are locally isometric. A consequence of Minding's theorem is that any surface whose curvature is identically zero can be constructed by bending some plane region. Such surfaces are called developable surfaces. Minding also raised the question whether a closed surface with constant positive curvature is necessarily rigid.
- Liebmann's theorem (1900) answered Minding's question. The only regular (of class C2) closed surfaces in R3 with constant positive Gaussian curvature are spheres.
- Hilbert's theorem (1901) states that there exists no complete analytic (class Cω) regular surface in R3 of constant negative Gaussian curvature. In fact, the conclusion also holds for surfaces of class C2 immersed in R3, but breaks down for C1-surfaces. The pseudosphere has constant negative Gaussian curvature except at its singular cusp.
Read more about this topic: Gaussian Curvature
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