Geometry of Curves and Surfaces — Problem 2.1

Definitions and mathematical statement

Fix a closed oriented smooth $n$-manifold $M$, where closed means compact and without boundary. For a $C^1$ immersion $f:M\to\mathbb R^{n+1}$, the derivative has rank $n$ everywhere. The orientations determine a continuous unit normal map $N_f:M\to S^n$, called the Gauss map. Its image describes the normal directions attained by the immersion.

Given a subset $A\subset S^n$, an $A$-directed immersion is one with $N_f(M)\subset A$. Characterize the pairs $(M,A)$ for which such an immersion exists, or equivalently characterize the admissible sets $A$ for each specified $M$.

The inclusion need not be equality: every direction in $A$ need not occur. The immersion may have self-intersections; replacing it by an embedding is a stronger problem. No openness assumption on $A$ is built into the general question, although open sets are an important subcase for existence criteria.

An elementary necessary condition is $A\cup(-A)=S^n$. Indeed, for any direction $u$, the height function $x\mapsto\langle f(x),u\rangle$ attains a maximum on the compact manifold, and its normal there is $u$ or $-u$. The problem seeks a complete criterion, including topological restrictions on $M$, beyond this necessary condition.

Short English statement

Which prescribed sets of normal directions can contain the Gauss image of an immersion of a given closed oriented manifold?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 7000010 (AMR-069-0010), 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 2.1, PDF page 9. Original source indexed by AMR-069-0010. https://people.math.gatech.edu/~ghomi/Papers/op.pdf.