# A duality for Guichard nets

## TL;DR

This paper explores the relationship between Guichard nets and G-surfaces, introducing duality and transformation techniques that deepen understanding of these geometric structures.

## Contribution

It establishes a duality for Guichard nets via Combescure transformations and develops a Bäcklund-type transformation with a permutability theorem.

## Key findings

- Coordinate surfaces of Guichard nets are G-surfaces.
- A duality for Guichard nets is constructed using Combescure transformations.
- A Bäcklund-type transformation with a permutability theorem is introduced.

## Abstract

In this paper we study G-surfaces, a rather unknown surface class originally defined by Calapso, and show that the coordinate surfaces of a Guichard net are G-surfaces. Based on this observation, we present distinguished Combescure transformations that provide a duality for Guichard nets. Another class of special Combescure transformations is then used to construct a B\"acklund-type transformation for Guichard nets. In this realm a permutability theorem for the dual systems is proven.