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This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound "geometrical roots" and numerous applications to modern nonlinear problems, it is treated as a universal "object" of investigation, connecting many of the problems discussed.
The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry.
List of contents
Introduction.- 1 Foundations of Lobachevsky geometry: axiomatics, models, images in Euclidean space.- 2 The problem of realizing the Lobachevsky geometry in Euclidean space.- 3 The sine-Gordon equation: its geometry and applications of current interest.- 4 Lobachevsky geometry and nonlinear equations of mathematical physics.- 5 Non-Euclidean phase spaces. Discrete nets on the Lobachevsky plane and numerical integration algorithms for 2-equations.- Bibliography.- Index.
Summary
This monograph presents the basic concepts of hyperbolic Lobachevsky geometry and their possible applications to modern nonlinear applied problems in mathematics and physics, summarizing the findings of roughly the last hundred years. The central sections cover the classical building blocks of hyperbolic Lobachevsky geometry, pseudo spherical surfaces theory, net geometrical investigative techniques of nonlinear differential equations in partial derivatives, and their applications to the analysis of the physical models. As the sine-Gordon equation appears to have profound “geometrical roots” and numerous applications to modern nonlinear problems, it is treated as a universal “object” of investigation, connecting many of the problems discussed.
The aim of this book is to form a general geometrical view on the different problems of modern mathematics, physics and natural science in general in the context of non-Euclidean hyperbolic geometry.
Additional text
“The main aim of this book is to look at the potential of the geometry developed by Lobachevskii in the context of its emergence in various branches of current interest in contemporary mathematics and science, especially in nonlinear problems of mathematical physics. … the book is well written, very readable, and nicely illustrated throughout with many graphs and figures, especially figures of surfaces. … This unique book makes this difficult subject interesting and within reach.” (Paul F. Bracken, Mathematical Reviews, August, 2015)
“The book is original in its form and content. It covers a wide spectrum of geometry and analysis and it displays the Lobachevsky plane as a central object in the study of the classical equations of mathematical physics. The style is expository and clear. This book is a valuable addition to the geometric literature.” (Athanase Papadopoulos, zbMATH 1311.51002, 2015)
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"The main aim of this book is to look at the potential of the geometry developed by Lobachevskii in the context of its emergence in various branches of current interest in contemporary mathematics and science, especially in nonlinear problems of mathematical physics. ... the book is well written, very readable, and nicely illustrated throughout with many graphs and figures, especially figures of surfaces. ... This unique book makes this difficult subject interesting and within reach." (Paul F. Bracken, Mathematical Reviews, August, 2015)
"The book is original in its form and content. It covers a wide spectrum of geometry and analysis and it displays the Lobachevsky plane as a central object in the study of the classical equations of mathematical physics. The style is expository and clear. This book is a valuable addition to the geometric literature." (Athanase Papadopoulos, zbMATH 1311.51002, 2015)
