Besides the sphere, is there any other surface with constant and positive Gaussian Curvature $K$?
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Yes. There's lots. – Ryan Budney May 24 '14 at 22:20
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2Wikipedia: Surfaces of constant Gaussian curvature "The only regular (of class $C^2$) closed surfaces in $\mathbb R^3$ with constant positive Gaussian curvature are spheres." (emphasis mine) – May 24 '14 at 22:22
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2It's also interesting to note that the counterpart answer to this question is the pesudosphere. – IAmNoOne May 24 '14 at 22:32
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4Any open subset of the sphere! – Qiaochu Yuan May 24 '14 at 23:06
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And projective plane. – Moishe Kohan May 24 '14 at 23:53
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@ Rahul: This is not regular or closed? http://virtualmathmuseum.org/Surface/k1-sor/k1-sor.html Regards – Narasimham Sep 11 '14 at 20:27
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@Narasimham: certainly not regular. – Willie Wong Jul 13 '16 at 18:30
4 Answers
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The word "surface" is too vague. If it means a smooth subset of $\mathbb{R}^3$ that can be parametrized by two varibles it will "look" like the sphere of radius $1/\sqrt{K}$ near any point (will be locally isometric to a sphere in technical language). That could be any open piece of the sphere, or a disjoint collection of them.
If we additionally assume that this subset must be closed in $\mathbb{R}^3$ then the entire sphere is the only possibility, up to moving it around. If we still want it closed, but allow it to be a subset of $\mathbb{R}^n$ with large enough $n$, then there is a second possibility, the real projective plane. It still "looks" like the sphere near any point, but has a different overall shape.
The most abstract interpretation is to treat "surface" as not sitting inside any space (two-dimensional manifold), but still having some residual structure (called Riemannian metric), that makes the notion of curvature meaningful. However, even this does not lead to anything other than disjoint open pieces of spheres and real projective planes.
Conifold
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Suppose that $M$ is a connected complete Riemannian 2-dimensional manifold of constant curvature $1$. Then $M$ is isometric either to unit sphere or its quotient, projective plane.
Moishe Kohan
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... and if its curvature is constant $K>1$, then it is isometric to a dilated sphere or dilated projected plane. :) – Neal May 25 '14 at 00:25
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sieverts surface is a second example besides spherical surface. I guess that you want this answer.
Hao Wu
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