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math.DG2026★ 4 cited
An index bound for smooth umbilic points
Brendan Guilfoyle, Wilhelm Klingenberg
We prove that the half-integer valued local index of an isolated umbilic point on a -smooth convex surface in Euclidean 3-space is less than two. The approach is to study…
math.DG2024
Proof of the Caratheodory Conjecture
Brendan Guilfoyle, Wilhelm Klingenberg
A well-known conjecture of Caratheodory states that the number of umbilic points on a closed convex surface in must be greater than one. In this paper we prove this…