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In classical differential geometry, Liebmann’s theorem states that a compact and connected surface in $\mathbb{R}^{3}$ with constant Gauss curvature is a (standard) sphere; see e.g. Do Carmo's Differential Geometry of Curves and Surfaces (2nd edition, p. 323). It follows that any sphere is rigid.

On the other hand, the sphere is not the only surface in $\mathbb{R}^{3}$ with constant Gauss curvature. For instance, there exist surfaces of revolution with constant Gauss curvature equal to 1 that have no umbilical point. In Kristopher Tapp's Differential Geometry of Curves and Surfaces (p. 222), they are called fake spheres. Here is an image taken from the same book.

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What I do not understand is how the surface corresponding to $a=2$ in the picture fails to be smooth, compact (assuming that it contains its boundary) or connected.

I have always thought that the rigidity of the sphere was referring to the sphere as a closed surface (compact surface without boundary), whereas a single hemisphere could be deformed isometrically in space.

So my question is, why don't we need to assume closedness (instead of compactness) in the statement of Liebmann’s theorem?

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    Liebmann’s theorem definitely talks about closed surfaces (compact and boundless), not only compact (see this wiki page). Otherwise parts of the sphere or pseudo-sphere are all examples. – Zerox Apr 30 '22 at 11:52
  • @Zerox OK, that is what I thought. But some relevant books (Do Carmo, O'Neill, Gray) do not seem to state the theorem under the appropriate assumptions. Am I missing something? –  Apr 30 '22 at 11:59
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    @MK7 Some authors define "surface"/"manifold" etc. to denote objects without boundary unless explicitly allowed. Others will do the opposite. It seems that the books you've cited follow the former convention (which they likely state at some point). – Kajelad Apr 30 '22 at 12:39
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    To complement the existing comments, if being a surface means each point has a neighborhood homeomorphic to an open disk in the plane, then fake spheres are not compact (if they have no boundary), or they are not surfaces (if they contain any of their boundary points). – Andrew D. Hwang Apr 30 '22 at 13:35

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