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In the celebrated book of Hilbert and Cohn-Vossen, the following sentence appears (p. 230):

Bending is impossible in the case of all closed convex surfaces, such as, for example, the ellipsoids. It is likewise impossible to bend any convex surface having boundary curves, provided each boundary curve has the property that at all of its points the tangent plane is the same. An example of such a surface (with two boundary curves) is the convex part of the torus.

I couldn't find any hint of a proof of the second part of the sentence. I know how to prove that such surfaces are infinitesimally-rigid, but as far as I can tell, Hilbert and Cohn-Vossen refer to global rigidity.

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  • $\begingroup$ Did you check Pogorelov's book? $\endgroup$
    – Moishe Kohan
    Commented Aug 27 at 16:06
  • $\begingroup$ @MoisheKohan I guess you referring to "Extrinsic geometry of convex surfaces"? The only notion of rigidity I found there is infinitesimal (p. 260: a surface is said to be rigid if it admits no nontrivial bending fields). $\endgroup$
    – Raz Kupferman
    Commented Aug 28 at 6:43

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