Additional first order equation for infinitesimal bendings of smooth surfaces in the isothermal coordinates

TL;DR

This paper introduces a new first-order linear differential equation for infinitesimal bendings of smooth surfaces, revealing additional constraints on Darboux rotation fields and extending understanding beyond classical equations.

Contribution

It derives a novel first-order differential equation for Darboux rotation fields, expanding the theoretical framework of infinitesimal bendings in differential geometry.

Findings

New first-order differential equation for Darboux rotation fields

Functional independence of the new equation for certain surfaces

Maximum principle established for components of the Darboux rotation field

Abstract

The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied by any Darboux rotation field of a smooth surface. We show that, for some surfaces, this additional equation is functionally independent of the three standard equations that the Darboux rotation field satisfies (and by which it is determined). As a consequence of this additional equation, we prove the maximum principle for the components of the Darboux rotation field for a class of disk-homeomorphic surfaces containing not only surfaces of positive Gaussian curvature.