Constant Negative Gaussian Curvature Research Articles

Discrete surfaces with constant negative Gaussian curvature and the Hirota equation

Discrete surfaces with constant negative Gaussian curvature and the Hirota equation

SOME STUDIES ON ARITHMETICAL CHAOS IN CLASSICAL AND QUANTUM MECHANICS

Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. The latter consists of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetical features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of closed geodesics (periodic orbits). The most significant one is an exponential growth of degeneracies in these geodesic length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of self-adjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical systems have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor how the spectral statistics in arithmetical quantum chaos can be understood by the properties of the corresponding classical geodesic length spectra. A decisive role is played by the exponentially increasing multiplicities of lengths. The model developed for the level spacings distribution and for the number variance is compared to the corresponding quantities obtained from quantum energies for a specific arithmetical system. Finally, the convergence properties of a representation for the Selberg zeta function as a Dirichlet series are studied. It turns out that the exceptional classical and quantum mechanical properties shared by the arithmetical systems prohibit a convergence of this important function in the physically interesting domain.

Vitreous B[Formula: see text]O[Formula: see text] : a geometrical study

Vitreous boron trioxide (B 2 O 3 ) is an intrinsic noncrystalline material with planar equilateral triangles (BO 3 triangles) as structural units, presenting an overall two-dimensional character, a solid-like membrane structure. The local structural similarities between that glass and the negatively curved Bethe lattice motivated us to build an ideal model for vitreous B 2 O 3 , propagating its local order on a surface of constant negative Gaussian curvature (the hyperbolic plane H 2 ) and using non-Euclidean hierarchical lattices as structural substrates. Based on the metric and symmetry properties of such lattices, we make an analytical investigation of the structure of the ideal glass model. This way, we obtain the peaks of the geometrical radial distribution function for the ideal glass structure, which are in good agreement with experimental data and theoretical studies for vitreous B 2 O 3 . Those facts suggest the evidence of non-Euclidean local order for that amorphous solid, indicating the arrangement of planar triangular units on a negatively curved surface, forming few or no six-membered (boroxol) rings

QUANTIZATION RULES FOR STRONGLY CHAOTIC SYSTEMS

We discuss the quantization of strongly chaotic systems and apply several quantization rules to a model system given by the unconstrained motion of a particle on a compact surface of constant negative Gaussian curvature. We study the periodic-orbit theory for distinct symmetry classes corresponding to a parity operation which is always present when such a surface has genus two. Recently, several quantization rules based on periodic orbit theory have been introduced. We compare quantizations using the dynamical zeta function Z(s) with the quantization condition [Formula: see text] where a periodic-orbit expression for the spectral staircase N(E) is used. A general discussion of the efficiency of periodic-orbit quantization then allows us to compare the different methods. The system dependence of the efficiency, which is determined by the topological entropy τ and the mean level density [Formula: see text], is emphasized.

Flux quantization and quantum mechanics on Riemann surfaces in an external magnetic field

The authors investigate the possibility of applying an external constant magnetic field to a quantum mechanical system consisting of a particle moving on a compact or non-compact two-dimensional manifold of constant negative Gaussian curvature and of finite volume. For the motion on compact Riemann surfaces they find that a consistent formulation is only possible if the magnetic flux is quantized, as it is proportional to the (integrated) first Chern class of a certain complex line bundle over the manifold. In the case of non-compact surfaces of finite volume they obtain the striking result that the magnetic flux has to vanish identically due to the theorem that any holomorphic line bundle over a non-compact Riemann surface is holomorphically trivial.

Minimal surfaces of constant curvature in $S\sp n$

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.

Minimal surfaces with constant curvature in 4-dimensional space forms

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

All the soliton equations in 1 + 1 dimensions that can be solved by the AKNS 2 × 2 inverse scattering method (for example, the sine-Gordon, KdV or modified KdV equations) are shown to describe pseudospherical surfaces, i.e., surfaces of constant negative Gaussian curvature. This result provides a unified picture of all these soliton equations. Geometrical interpretations of characteristic properties like infinite numbers of conservation laws and Bäcklund transformations and of the soliton solutions themselves are presented. The important role of scale transformations as generating one parameter families of pseudospherical surfaces is pointed out.