Pseudospherical surfaces on time scales: a geometric definition and the spectral approach

TL;DR

This paper extends the concept of pseudospherical surfaces to arbitrary time scales, providing new geometric definitions, a unified Gaussian curvature expression, and spectral methods applicable to both discrete and continuous cases.

Contribution

It introduces a generalized geometric framework for pseudospherical surfaces on time scales, including a new curvature formula and spectral tools like the Lax pair and Darboux-Backlund transformation.

Findings

Gaussian curvature formula valid on any time scale

Asymptotic Chebyshev nets have constant negative curvature

Spectral problem and transformations generalized to arbitrary time scales

Abstract

We define and discuss the notion of pseudospherical surfaces in asymptotic coordinates on time scales. Thus we extend well known notions of discrete pseudospherical surfaces and smooth pseudosperical surfaces on more exotic domains (e.g, the Cantor set). In particular, we present a new expression for the discrete Gaussian curvature which turns out to be valid for asymptotic nets on any time scale. We show that asymptotic Chebyshev nets on an arbitrary time scale have constant negative Gaussian curvature. We present also the quaternion-valued spectral problem (the Lax pair) and the Darboux-Backlund transformation for pseudospherical surfaces (in asymptotic coordinates) on arbitrary time scales.