Geometry of Curves and Surfaces — Problem 2.3

Definitions and mathematical statement

Let $M,M'\subset\mathbb R^3$ be smooth connected closed oriented embedded surfaces. Write $N:M\to S^2$ and $N':M'\to S^2$ for their oriented unit normal maps, and $K,K'$ for their Gauss curvatures, the products of the principal curvatures.

Suppose a smooth diffeomorphism $f:M\to M'$ satisfies the two pointwise identities \[ N'\circ f=N,\qquad K'\circ f=K. \] Must the two surfaces be congruent? Precisely, must there exist an orthogonal matrix $Q\in O(3)$ and a vector $a\in\mathbb R^3$ such that $M'=QM+a$?

The correspondence preserves the actual normal vector in the fixed ambient coordinates, not merely the set of directions visited by the Gauss map. It also preserves curvature at corresponding points; matching total curvature is insufficient. No intrinsic isometry, area-preservation assumption, or convexity is imposed in addition to the two displayed equations.

The conclusion concerns congruence of the surfaces as subsets; it does not require the given diffeomorphism itself to be the restriction of that rigid motion. Flat points and regions, where $K=0$ and the Gauss map may fail to be locally invertible, remain within the statement.

Short English statement

Do matching unit normals and matching Gauss curvature under a diffeomorphism force two closed surfaces in space to have the same shape up to a rigid motion?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000012 (AMR-069-0012), 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 2.3, PDF page 10. Original source indexed by AMR-069-0012. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.