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Differential Geometry of Manifolds

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Cheshkova M. A.

Transformation of Bianchi for Minding Top

DOI
10.5922/0321-4796-2020-51-15
Pages
135-142
Gaussian curvature surface of revolution Minding top Bianchi transform

Abstract

The work is devoted to the study of the Bianchi transform for surfac­es of revolution of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, the pseudosphere (Beltrami surface). The study of surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. The connection of the geometric characteristics of pseudospherical surfaces with the theory of networks, with the theory of solitons, with nonlinear differential equations and sin-Gordon equations is established. The sin-Gordon equation plays an important role in modern physics. Bianchi transformations make it possible to obtain new pseudospherical surfaces from a given pseudospherical surface. The Bianchi transform for the Minding top is constructed. Using a mathematical package, Minding's top and its Bianchi transform are constructed.

Reference

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3. Norden, A. P.: On the foundations of geometry. Moscow (1956).