Geometry of Curves and Surfaces — Problem 1.1

Definitions and mathematical statement

Let $M$ be a compact connected smooth two-manifold without boundary, and let $f_0:M\to\mathbb R^3$ be a $C^2$ embedding. Its first fundamental form is the metric $g_u(v,w)=\langle df_0(v),df_0(w)\rangle$ on tangent vectors at $u$. A $C^2$ isometric flex is a continuous family of $C^2$ embeddings $f_t:M\to\mathbb R^3$, $0\le t\le1$, with \[ \langle df_t(v),df_t(w)\rangle=g_u(v,w) \quad\text{for every }u,v,w,t. \] Continuity is taken in the $C^2$ topology. Thus intrinsic lengths are unchanged at every stage, not just at the endpoints.

Does there exist such a family for which at least one image $f_t(M)$ is not congruent to $f_0(M)$ by a Euclidean rigid motion? The initial surface may be nonconvex and of any allowable genus. Every intermediate surface must remain embedded and $C^2$.

Two isolated isometric but noncongruent surfaces do not provide a flex: a continuous path joining them is required. Likewise, flexible polyhedra or $C^1$ constructions do not meet the $C^2$ regularity requirement. "Closed" means compact with no boundary, so bending a sheet with a free edge is outside the question.

Short English statement

Can a closed twice-continuously-differentiable surface in ordinary three-dimensional space bend continuously into a different shape without stretching or self-intersection?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000001 (AMR-069-0001), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Ghomi - Open Problems in Geometry of Curves and Surfaces (2019), Problem 1.1, PDF page 5. Original source indexed by AMR-069-0001. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.