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<p>The article contributes to the theory of infinitesimal bendings of smoothsurfaces in Euclidean 3-space.We derive a first-order linear differential equation, which previouslydid not appear in&#xa0;the literature and which is satisfied by any Darbouxrotation field of a smooth surface.We&#xa0;show that, for some surfaces, this additional equation is functionallyindependent of the three standard equations that the Darboux rotationfield satisfies (and by which it is determined).As&#xa0;a&#xa0;consequence of this additional equation, we prove the maximumprinciple for the components of the Darboux rotation field for a classof disk-homeomorphic surfaces containing not only surfaces of positive Gaussian curvature.</p>

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

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • V. A. Alexandrov

摘要

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