Meromorphic open-string vertex algebras and modules over two-dimensional orientable space forms

Geodesics, in particular minimal geodesics, are of focal geodetic interest. In the paper we apply the Maupertuis variational principle of least action in the Newton mechanics to transform the geodesic flow on the twodimensional sphere S 2R with the radius R and on the biaxial ellipsoid E 2A,B with the semi-major axis A and semi-minor axis B into the Newton form. A geodesic flow on a twodimensional Riemann manifold takes the form of the Newton law if two assumptions are met: 1. The twodimensional Riemann manifold is represented by conformal coordinates (isometric coordinates, isothermal coordinates), 2. The arc length s as the curve parameter of a geodesic flow is replaced by the dynamic time t according to the Maupertuis gauge $$ds = {\lambda ^2}\left( {{q^1},{q^2}} \right)dt$$ where A2 is the factor of conformality and q1,q2 are the conformal coordinates which form a local chart of the twodimensional Riemann manifold. KeywordsNewton MechanicGeodesic FlowLocal ChartUniversal Transverse MercatorMinimal GeodesicThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Lax representations for the Euler ideal hydrodynamics equation in vorticity form on a two-dimensional Riemannian manifold

1.1. Let (M2, ds 2) be a two-dimensional Riemannian manifold M 2 with metric ds 2 and let p ~ M be a point where the Gaussian curvature K of ds 2 is nonpositive. The Ricci theorem [2, pg. 41 l] states that a necessary and sufficient condition for some neighborhood of p in M to be isometrically and minimally immersed in the euclidean space R 3 is that the metric ~, K ds 2 be flat around p. This condition, usually called the Ricci condition, can be shown to be equivalent to the fact that the metric (- K) ds 2 has constant Gaussian curvature equal to one. The purpose of the present paper is to obtain a generalization of the latter version of the Ricci condition for hypersurfaces of a space M"+l(c) of constant curvature c, c any real number. More precisely, we prove Theor. (1.2) below, for the statement of which we need some notation. We will denote by RicM the quadratic form in a Riemannian manifold (M, ds 2) defined by the Ricci tensor of ds 2. (,) and V will denote the inner product and the Levi-Civitta connection, respectively, of ds 2. We will be considering the case where (n-~ 1)(-RicM+cds 2) is positive definite: then (n- I)(-RicM + cds 2 ) defines a further Riemannian metric on M, and we will denote by ((,)) and ~' the inner product and the Riemannian connect'ion, respectively, of such a metric. Finally, S(M) will denote the bundle over M whose fiber at q ~ M in the space of symmetric linear maps of T~(M).

(3+3+2) warped-like product manifolds with [formula omitted] holonomy

We consider the Laplace–Beltrami operator in tubular neighborhoods of curves on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and time preserving boundary conditions. We are interested in the interplay between the geometry and spectrum. After introducing a suitable Hilbert space framework in the general situation, which enables us to realize the Laplace–Beltrami operator as an m-sectorial operator, we focus on solvable models defined on manifolds of constant curvature. In some situations, notably for non-Hermitian Robin-type boundary conditions, we are able to prove either the reality of the spectrum or the existence of complex conjugate pairs of eigenvalues, and establish similarity of the non-Hermitian m-sectorial operators to normal or self-adjoint operators. The study is illustrated by numerical computations.

Let M be a two-dimensional Riemannian manifold with nowhere zero curvature and let D1, D2 be two smooth one-dimensional distributions on M orthogonal to each other at any point. We present a method of seeing whether M may be locally and isometrically immersed in R3 so that D1 and D2 give principal distributions, and in the case where M may be immersed in such a manner, we specify the values that may become principal curvatures at each point of M. In addition, we study relations among the first fundamental form, the principal distributions and the principal curvature functions on each of a parallel curved surface and a surface with constant mean curvature in R3.

Mappings from a left two-dimensional Riemann manifold to a right two-dimensional Riemann manifold, simultaneous diagonalization of two matrices, mappings (isoparametric, conformal, equiareal, isometric, equidistant), measures of deformation (Cauchy–Green deformation tensor, Euler–Lagrange deformation tensor, stretch, angular shear, areal distortion), decompositions (polar, singular value), equivalence theorems of conformal and equiareal mappings (conformeomorphism, areomorphism), Korn–Lichtenstein equations, optimal map projections.

The isometric embedding problem is a fundamental problem in differential geometry. A longstanding problem is considered in this paper to characterize intrinsic metrics on a two-dimensional Riemannian manifold which can be realized as isometric immersions into the three-dimensional Euclidean space. A remarkable connection between gas dynamics and differential geometry is discussed. It is shown how the fluid dynamics can be used to formulate a geometry problem. The equations of gas dynamics are first reviewed. Then the formulation using the fluid dynamic variables in conservation laws of gas dynamics is presented for the isometric embedding problem in differential geometry.

We consider geodesic flows of Riemannian metrics on T 2 and S 2 integrable by an integral depending on the momenta linearly or quadratically (in what follows such integrals are referred to as linear and quadratic integrals) and show that there exists an effective criterion for topological equivalence of two arbitrary flows. The criterion is based on comparing the "codes" of the corresponding Riemannian metrics (see below). A. T. Fomenko [1, 2] found a new topological invariant I(H, Q) of integrable Hamiltonian systems, making it possible to classify integrable Hamiltonians up to the coarse topological equivalence. In this case the labeled topological invariant I*(H, Q) discovered by A. T. Fomenko and H. Wsishang [3] (also called the labeled molecule [4]) classifies integrable Hamiltonians up to a finer topological equivalence. Let M 4 be a smooth symplectic manifold, and let v = sgrad H be an integrable Hamiltonian system with smooth Hamiltonian H . Following [1] we say that an integral F on an isoenergetic surface Q3 = {H = const} is a Bott integral if all its critical points form nondegenerate critical submanifolds. Definition (see [4]). A pair (P1, K1), where P1 is a compact orientable surface and K1 a nonempty finite connected graph on P1, is called a letter-atom if the following conditions hold: 1) the degree of each vertex of 1(1 is equal to 0, 2, or 4; 2) each connected component of the difference P1 \ K1 is homeomorphic to the annulus S 1 x (0, 1]; 3) the set of the annuli forming P1 \ :K~ can be divided into two parts (positive and negative annuli) so that exactly one positive and one negative annulus adjoin each edge of K1. A molecule W, which is a union of atoms, is a pair ( P , K ) , where K is a graph embedded in a two-dimensional surface p2. A labeled molecule W* is obtained by supplementing W with numerical (rational or integer) labels. To each unlabeled molecule W we assign its complexity (m, n) , where m is the number of vertices in K (coinciding with the total number of critical circles of the Bott integral) and n is the number of connected components of P \ K (or the number of cylinders S 1 x S 1 x (0, 1) into which the isoenergetic manifold Q3 splits On deleting all singular leaves of the corresponding Liouville foliation). Fomenko posed the problem of describing all labeled molecules W* for the above type of integrable geodesic flows on T 2 and S 2 and of finding the corresponding (m, n)-domain in the molecular complexity table. For the torus T 2 this problem has been completely investigated by Selivanova whose results, as it turned out later, can be applied effectively to the topological classification of geodesic flows on the sphere S 2 with nontrivial quadratic integral. Here we also consider geodesic flows on S 2 with linear integral and thus complete the investigation of integrable geodesic flows on two-dimensional Riemannian manifolds (with additional integral of degree no higher than 2). As is known [6, 9], the geodesic flows on two-dimensional surfaces of genus > 1 have no additional integrals analytic with respect to the momenta. In §2 we give the basic facts concerning the above type of geodesic flows that are used in the subsequent presentation and are based on the results of Darboux [10], Levi-Civita [11], Birkhoff [5], and Kolokol'tsov

The definition of the cut locus was introduced by Poincare ([ 68 ]), and he first investigated the structure of the cut locus of a point on a complete, simply connected and real analytic Riemannian 2-manifold. Myers ([ 60, 61 ]) determined the structure of the cut locus of a point in a 2-sphere and Whitehead [ 107 ] proved that the cut locus of a point on a complete two-dimensional Riemannian manifold carries the structure of a local tree. In this chapter we will determine the structure of the cut locus and distance circles of a Jordan curve in a complete Riemannian 2-manifold and will prove the absolute continuity of the distance function of the cut locus. Preliminaries Throughout this chapter ( M , g ) always denotes a complete connected smooth two-dimensional Riemannian manifold without boundary. Let γ : [0, ∞) → M be a unit-speed geodesic. Note that any geodesic segment γ : [0, a ) → M is extensible to [0, ∞) according to Theorem 1.7.3. If, for some positive number b , γ | [0, b ] is not a minimizing geodesic joining its endpoints, let t 0 be the largest positive number t such that γ | [0, t ] is minimizing. Note that there always exists such a positive number t 0 , by Lemma 1.2.2. The point γ ( t 0 ) is called a cut point of γ (0) along the geodesic γ. For each point p on M , let C ( p ) denote the set of all cut points along the geodesics emanating from p .

where the equality holds if and only if C is a circle. By using different methods many authors have given different proofs for this inequality (1.1), and extended the inequality to surfaces in ordinary euclidean space E3 under different conditions on the gaussian and mean curvatures of the surfaces (for this see [3], [1], [5], [10], [9], [11]). The purpose of this paper is to further extend the inequality to the most general two-dimensional riemannian manifolds with boundary, which can be imbedded in a euclidean space EN+2 of dimension N + 2 for any N > 0. The results are contained in Theorems 4.1 and 4.2.

We study open discrete maps of two-dimensional Riemannian manifolds from the Sobolev class. For these mappings, we establish the lower estimates of distortions of the moduli of families of the curves. As a consequence, we establish the equicontinuity of Sobolev classes at the interior points of the domain.

We consider two natural problems arising in geometry which are equivalent to the local solvability of specific equations of Monge-Ampère type. These two problems are: the local isometric embedding problem for two-dimensional Riemannian manifolds, and the problem of locally prescribed Gaussian curvature for surfaces in $\mathbb{R}^3$. We prove a general local existence result for a large class of Monge-Ampère equations in the plane, and obtain as corollaries the existence of regular solutions to both problems, in the case that the Gaussian curvature vanishes to arbitrary finite order on a single smooth curve.

The elastic flow, which is the $L^2$-gradient flow of the elastic energy, has several applications in geometry and elasticity theory. We present stable discretizations for the elastic flow in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk, the elliptic plane as well as any conformal parameterization of a two-dimensional manifold in ${\mathbb R}^d$, $d\geq 3$. Numerical results show the robustness of the method, as well as quadratic convergence with respect to the space discretization.

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e. conformally equivalent to the Euclidean plane. Examples include the hyperbolic plane, the hyperbolic disc and the elliptic plane, as well as any conformal parameterization of a two-dimensional manifold in ${{\mathbb{R}}}^d$, $d\geqslant 3$. In these spaces we introduce stable numerical schemes for curvature flow and curve diffusion, and we also formulate schemes for elastic flow. Variants of the schemes can also be applied to geometric evolution equations for axisymmetric hypersurfaces in ${{\mathbb{R}}}^d$. Some of the schemes have very good properties with respect to the distribution of mesh points, which is demonstrated with the help of several numerical computations.