An umbilic point, also called simply an umbilic, is a point on a surface at which the curvature is the same in any direction.
Carathéodory's conjecture predicts at least two umbilic points on every closed strictly convex smooth surface in 
. Loewner's
conjecture gives the related local index bound. Ando and Umehara (2026) reported
a smooth non-real-analytic strictly convex closed surface with exactly one umbilic
point of index 2. Independent verification of the full construction has not been
established. Ando and Umehara (2026) report AI assistance with the mathematical construction.
