Additional First-Order Equation for Infinitesimal Bendings of Smooth Surfaces in the Isothermal Coordinates

The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a first-order linear differential equation, 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.