6 papers · 1 filter
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…
Isotropic submanifolds of and their focal sets
Nikos Georgiou, Brendan Guilfoyle, Morgan Robson
Families of oriented lines in are studied via their identification with submanifolds of . In particular, families of oriented lines which are orth…
The three obdurate conjectures of differential geometry
Brendan Guilfoyle, Wilhelm Klingenberg
We explore the role of symmetry in three obdurate conjectures of differential geometry: the Carathéodory, the Willmore and the Lawson Conjectures. All three Conjectures concern su…
Minimal surfaces in the Riemannian product of surfaces
Nikos Georgiou, Brendan Guilfoyle
Minimal surfaces in the Riemannian product of surfaces of constant curvature have been considered recently, particularly as these products arise as spaces of oriented geodesics of…
Properties and Transformations of Weingarten Surfaces
Brendan Guilfoyle, Morgan Robson
The Weingarten relations satisfied by rotationally symmetric surfaces in Euclidean 3-space E3 are considered from three points of view: restrictions on the slope of the relation at…
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…