1970 (Publication #5)
Non-rigidity theorems for tight polyhedral tori
Archiv der Mathematik, 21 (1970), 416-423.
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Given a surface M in E^3, M is said to be rigid in a given class of surfaces if the intrinsic properties of M completely determine (within that class) its extrinsic properties. Cauchy  proved the first rigity theorems, which applied to convex polyhedra, Cohn-Vossen  has proven rigity theorems for convex real analytics surfaces, and Herglotz  proved rigity theorems for convex surfaces which are several times differentiable. A. D. Alexandroff  generalized the idea of convexity to apply to a particular set of real analytic surfaces which he named T-surfaces and proved to be rigid. Alexandroff's condition, although stated in terms of minimal total absolute curvature, is equivalen to the notion of a tight surface. (A surface is M^2 in E^3 is tight if the intersection of M^2 and H^3 is connected for every half space H^3 in E^3.) Nirenberg, in 1963,
proved a rigidity theorem for several times differentiable tight surfaces under of M^2 and H^3 is connected for every half space H^3 in E^3.) Nirenberg, in 1963, proved a rigidity theorem for several times differentiable tight surfaces under additional assumptions.
The paper primarily seeks to prove that the corresponding statement regarding rigidity of tight polyhedra is false, even for "square tori." Examples of isometric but non-congruent tight tori are given to demonstrate this point.
Stoker  has recently provided rigidity theorems for some classes of non-convex polyhedra with vertices of prescribed types. A discussion of Stoker's restrictions on square tori is given, and it is proven that any square torus satisfying these conditions is necessarily rigid in the class of square tori.