By Andrey Popov, Andrei Iacob

ISBN-10: 3319056689

ISBN-13: 9783319056685

ISBN-10: 3319056697

ISBN-13: 9783319056692

This monograph offers the elemental techniques of hyperbolic Lobachevsky geometry and their attainable purposes to fashionable nonlinear utilized difficulties in arithmetic and physics, summarizing the findings of approximately the final hundred years. The critical sections conceal the classical development blocks of hyperbolic Lobachevsky geometry, pseudo round surfaces conception, web geometrical investigative thoughts of nonlinear differential equations in partial derivatives, and their purposes to the research of the actual types. because the sine-Gordon equation appears to be like to have profound “geometrical roots” and diverse purposes to trendy nonlinear difficulties, it really is handled as a common “object” of research, connecting the various difficulties mentioned.

The objective of this publication is to shape a basic geometrical view at the diversified difficulties of recent arithmetic, physics and common technology quite often within the context of non-Euclidean hyperbolic geometry.

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**Additional info for Lobachevsky Geometry and Modern Nonlinear Problems**

**Example text**

24) of the Lobachevsky 3 Straight lines can also be represented by diameters. 2. Model interpretations of Lobachevsky’s planimetry 39 plane (in the same variables), allows one to apply in their study the tools and methods of Riemannian geometry, of the theory of curves and surfaces, and so on. Based on these, in Chapter 2 we will obtain important geometric characteristics of various geometric elements of Lobachevsky planimetry. 5) conformally onto the upper half-plane Π = {w = u + iv ∈ W, Im w > 0}.

To this aim, as before, we consider in the Lobachevsky plane some straight line m and a point M not on it, as well as two straight lines b and c that pass through M parallel to m. 3). 3). In other words, drop from the point M the perpendicular M H to m and denote it by h ≡ M H. Consider the angles that arise in this way. 1). Note also that in the models of the Lobachevsky plane used (in particular, in the Cayley-Klein model), the “model Euclidean” angles do not necessarily coincide with the corresponding angles of the Lobachevsky planimetry.

3) in which the following notations for partial derivatives are used: p= ∂z , ∂x q= ∂z , ∂y r= ∂2z , ∂x2 s= ∂2z , ∂x∂y t= ∂2z . 3) with respect to the function z = z(x, y) means to describe, in the Euclidean space E3 , all surfaces with a priori given curvature K according to their shape and position in space. 3) cannot be integrated exactly. Nevertheless, when the curvature of the surface is constant, important particular typical cases can be studied exhaustively. F. Minding carefully studied surfaces of constant positive curvature, as well as surfaces of constant negative curvature.

### Lobachevsky Geometry and Modern Nonlinear Problems by Andrey Popov, Andrei Iacob

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