Преобразование Бианки катушки Миндинга

The work is devoted to the study of the Bianchi transformation for surfaces of constant negative Gaussian curvature. The surfaces of rotation of constant negative Gaussian curvature are the Minding top, the Minding coil, and the pseudosphere (Beltrami surface). Surfaces of constant negative Gaussian curvature also include Kuen’s surface and the Dini’s surface. Studying the surfaces of constant negative Gaussian curvature (pseudospherical surfaces) is of great importance for the interpretation of Lobachevsky planimetry. Geometric characteristics of pseudospherical surfaces are found to be related to the theory of networks, the theory of solitons, nonlinear differential equations, and sin-Gordon equations. 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 transformation for the Kuen’s surface is constructed using a mathematical software package.

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.

The work is devoted to the study of the Bianchi transform for surfa­ces 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). Surfaces of constant negative Gaus­sian curvature also include Kuens surface and the Dinis 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 pseudosphere is constructed. Using a mathematical package, the pseu­dos­phere and its Bianchi transform are constructed.

This article is an application of the author’s paper (Kobayashi, Nonlinear d’Alembert formula for discrete pseudospherical surfaces, 2015, [9]) about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d’Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition in Lemma 3.1. As an application, we draw figures of discrete constant negative Gaussian curvature surfaces given by this method (Figs. 1 and 2).

Wave maps (or Lorentzian-harmonic maps) from a $1+1$-dimensional Lorentz space into the $2$-sphere are associated to constant negative Gaussian curvature surfaces in Euclidean 3-space via the Gauss map, which is harmonic with respect to the metric induced by the second fundamental form. We give a method for constructing germs of Lorentzian-harmonic maps from their $k$-jets and use this construction to study the singularities of such maps. We also show how to construct pseudospherical surfaces with prescribed singularities using loop groups. We study the singularities of pseudospherical surfaces and obtain their bifurcations in generic 1-parameter families of such surfaces.

In this paper, we have considered surfaces with constant negative Gaussian curvature in the simply isotropic 3-Space by defined Sauer and Strubeckerr. Firstly, we have studied the isotropic II-flat, isotropic minimal and isotropic II-minimal, the constant second Gaussian curvature, and the constant mean curvature of surfaces with constant negative curvature (SCNC) in the simply isotropic 3-space. Surfaces with symmetry are obtained when the mean curvatures are equal. Further, we have investigated the constant Casorati, the tangential and the amalgamatic curvatures of SCNC.

On Differential Systems Describing Surfaces of Constant Curvature

In this note, we study an overdetermined system of partial differential equations whose solutions determine the minimal surfaces in of constant Gaussian curvature. If the Gaussian curvature is positive, the solution to the global problem was found by [Calabi], while the solution to the local problem was found by [Wallach]. The case of nonpositive Gaussian curvature is more subtle and has remained open. We prove that there are no minimal surfaces in Sn of constant negative Gaussian curvature (even locally). We also find all of the flat minimal surfaces in and give necessary and sufficient conditions that a given two-torus may be immersed minimally, conformally, and flatly into S. 0. Introduction. In this paper, we classify the connected minimal surfaces of constant Gaussian curvature in the unit n-sphere Sn C Enll for all n. It is a classical fact (and follows easily from the structure equations) that the only examples up to rigid motion in S3 are the open subsets of the geodesic spheres (with K 1) and the Clifford torus (with K 0). In an early paper Boruvka constructed a linearly full immersion S2 C S2m for each m > 0, where the induced metric had K 2/m(m + 1). Later [Calabi] showed that, up to rigid motion, these Boruvka spheres were the only compact minimal surfaces with K Ko > 0 in S' for any n. [Wallach] proved that any connected piece of minimal surface with K a positive constant in Sn was a subset of a Boruvka sphere. [Kenmotsu, 1976] found all of the flat minimal surfaces (K 0) in S' and quite recently, [Kenmotsu, 1983] showed that K Ko < 0 is impossible for minimal surfaces in S4. The techniques used in the above proofs range from harmonic analysis to rather involved calculations with the moving frame. On the other hand, in this paper, we take a somewhat different point of view. If (M2, ds2) is a surface of constant Gaussian curvature K, then the minimal surfaces in S n C E n ? 1 of constant Gaussian curvature K are given locally by smooth maps f: M En' + lwhich satisfy three conditions: first, Kf, f) 1; second, Af =2f (A is the Laplace-Beltrami operator of ds2); and third, that f should be an isometry, Kdf, df) = ds2. We study the more general class of maps f: M En + 1 which only satisfy the first two conditions. Note that this set of equations is already overdetermined (by one equation). Received by the editors August 14, 1984. 1980 Mathematics Subject Classification. Primary 53A10; Secondary 53B25 53C40.

Line congruences as surfaces in the space of lines

A surface in E3 is called parallel to the surface M if it consists of the ends of constant length segments, laid on the normals to the surfaces M at points of this surface. The tangent planes at the corresponding points will be parallel. For surfaces in E3 the theorem of Bonnet holds: for any surface M that has constant positive Gaussian curvature, there exists a surface parallel to it with a constant mean curvature. Using Bonnet's theorem for a surfaces of revolution of constant positive Gaussian curvature, surfaces of constant mean curvature are constructed. It is proved that they are also surfaces of revolution. A family of plane curvature lines (meridians) is described by means of elliptic integrals. The surfaces of constant Gaussian curvature are also described by means of elliptic integrals. Using the mathematical software package, the surfaces under consideration are constructed.

The geometric Cauchy problem for a class of surfaces in a pseudo-Riemannian manifold of dimension 3 is to find the surface which contains a given curve with a prescribed tangent bundle along the curve. We consider this problem for constant negative Gauss curvature surfaces (pseudospherical surfaces) in Euclidean 3-space, and for timelike constant non-zero mean curvature (CMC) surfaces in the Lorentz-Minkowski 3- space. We prove that there is a unique solution if the prescribed curve is non-characteristic, and for characteristic initial curves (asymptotic curves for pseudospherical surfaces and null curves for timelike CMC) it is nec- essary and sufficient for similar data to be prescribed along an additional characteristic curve that intersects the first. The proofs also give a means of constructing all solutions using loop group techniques. The method used is the infinite dimensional d'Alembert type representation for surfaces as- sociated with Lorentzian harmonic maps (1-1 wave maps) into symmetric spaces, developed since the 1990's. Explicit formulae for the potentials in terms of the prescribed data are given, and some applications are consid- ered.

We classify minimal surfaces with constant Gaussian curvature in a 4 4 -dimensional space form without any global assumption. As a corollary of the main theorem, we show there is no isometric minimal immersion of a surface with constant negative Gaussian curvature into the unit 4 4 -sphere even locally. This gives a partial answer to a problem proposed by S. T. Yau.

Soliton equations and pseudospherical surfaces

The explicit study of surfaces of constant negative total curvature goes back to the work of Minding in 1838. Thus, in that year, Minding's theorem established the important result that all such surfaces are isometric, that is, they can be placed in one-to-one correspondence in such a way that the metric is preserved. Beltrami subsequently gave the term pseudospherical to these surfaces and made important connections with Lobachevski's non-Euclidean geometry.

The (scalar) sine-Gordon equation appears in many interesting contexts, such as the description of surfaces of constant negative Gaussian curvature (Bour, 1862; Eisenhart, 1960), magnetic flux propagation in Josephson junctions (McLaughlin & Scott, 1978; Mineev & Shmidt, 1980) and propagation of deformations along the DNA double helix (Gaeta et al., 1994; Lennholm & Hornquist, 2003; Salerno, 1991; Yakushevich, 2004). Because of the importance of the applications in which the sine-Gordon equation arises (see Aktosun et al., 2010 for details), it is not surprising that different approaches, such as the IST (Ablowitz et al., 1973), and Darboux and Backlund transformations (Gu et al., 2005; Rogers & Schief, 2002), were developed to get exact solutions. These exact solutions are written in terms of elementary functions and, in Lamb (1980) and Poppe (1983), the reader can find many of the solutions coming from the methods cited above. In Aktosun et al. (2010), we have presented a family of explicit solutions to the sine-Gordon equation uxt = sin(u) (1)