Geometry of Curves and Surfaces — Problem 8.2

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $f:M\to\mathbb R^3$ be a smooth regular immersion of a surface. Locally choose a unit normal $n$; the shape operator is $A=-dn$, and its eigenvalues $k_1,k_2$ are the principal curvatures. An umbilic is a point where $k_1=k_2$. Away from umbilics the two eigendirections of $A$ give the principal curvature line fields. Let $p$ be an isolated umbilic. On a sufficiently small positively oriented loop around $p$, express either principal line direction in a continuous oriented local tangent frame by an angle $\theta$, taken modulo $\pi$. A continuous lift of $\theta$ around the loop defines the line-field index \[\operatorname{ind}_p=\frac{\theta(2\pi)-\theta(0)}{2\pi}\in\tfrac12\mathbb Z.\] The two perpendicular principal fields have the same index. The immersion is smooth at the umbilic itself; a point where it is only $C^1$ and not $C^2$ is outside this question.

Statement recorded in the supplied source.

Does every isolated umbilic of every smooth immersed surface in $\mathbb R^3$ have principal-line-field index at most $1$? [S2]

Short English statement

Can the principal curvature directions wind more than once around an isolated umbilic of a smooth surface?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 7000029 (AMR-069-0029). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Naoya Ando, Toshifumi Fujiyama and Masaaki Umehara, $C^1$-umbilics with arbitrarily high indices, arXiv:1508.03685v1 (2015), Introduction, pp. 1--2. https://arxiv.org/pdf/1508.03685v1.
• S3 Mohammad Ghomi and Ralph Howard, Normal curvatures of asymptotically constant graphs and Carathéodory's conjecture, arXiv:1101.3031v2 (2011), Introduction, p. 3. https://arxiv.org/pdf/1101.3031v2.