Udo Hertrich-Jeromin

Curved flats and isothermic surfaces

We show how pairs of isothermic surfaces are given by curved flats in a pseudo Riemannian symmetric space and vice versa. Calapso’s fourth order partial differential equation is derived and, using a solution of this equation, a Mobius invariant frame for an isothermic surface is built.

Möbius geometry of surfaces of constant mean curvature 1 in hyperbolic space

An overview of the various transformations of isothermic surfaces and their interrelations is given using aquaternionic formalism. Applications to the theory of cmc-1 surfaces inhyperbolic space are given and relations between the two theories are discussed. Within this context, we give Möbius geometric characterizations for cmc-1 surfaces in hyperbolic space and theirminimal cousins.

Harmonic maps in unfashionable geometries

Abstract We describe some general constructions on a real smooth projective 4-quadric which provide analogues of the Willmore functional and conformal Gauss map in Lie sphere and projective differential geometry. Extrema of these functionals are characterized by harmonicity of this Gauss map.

The Ribaucour transformation in Lie sphere geometry

Abstract We discuss the Ribaucour transformation of Legendre (contact) maps in its natural context: Lie sphere geometry. We give a simple conceptual proof of Bianchis original Permutability Theorem and its generalisation by Dajczer–Tojeiro as well as a higher dimensional version with the combinatorics of a cube. We also show how these theorems descend to the corresponding results for submanifolds in space forms.

Discrete linear Weingarten surfaces

Discrete linear Weingarten surfaces in space forms are characterized as special discrete Ω-nets, a discrete analogue of Demoulin’s Ω-surfaces. It is shown that the Lie-geometric deformation of Ω-nets descends to a Lawson transformation for discrete linear Weingarten surfaces, which coincides with the well-known Lawson correspondence in the constant mean curvature case.

Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality

We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a concentric quadric.

Conformally flat hypersurfaces with cyclic guichard net

We classify the 3-dimensional conformally flat hypersurfaces with cyclic principal Guichard net. The orthogonal surfaces of the cyclic system turn out to be linear Weingarten surfaces in suitable ambient space forms and we provide explicit parametrizations for the conformally flat hypersurfaces.

CONFORMALLY FLAT HYPERSURFACES WITH BIANCHI-TYPE GUICHARD NET

We obtain a partial classification result for generic

Semi-discrete isothermic surfaces

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On conformally flat hypersurfaces, curved flats and cyclic systems

-dimensional conformally flat hypersurfaces in the conformal