COSET MODELS AND DIFFERENTIAL GEOMETRY

Ruled surfaces, also known as scrolls, are the surfaces built up by moving a straight line along a certain course in the 3D Euclidean space (E³). Many geometers have studied ruled surfaces for centuries to gain rational knowledge about how the surfaces in E³ are generated, and their findings are now essential in all monographs on differential geometry. In the area of optical scanning, scan fields produced by mirror scanning devices of different configurations are surfaces with a light ray as its generating element. This implies, within the accuracy of geometrical optics, that all scan fields are ruled surfaces by nature, and the theory of ruled surfaces in differential geometry provides a land with precious deposits for structural analysis of the scan fields.

As a general reference to the topological material covered in this chapter, see Seifert-Threlfall: Lehrbuch der Topologie.

This chapter briefly reviews the fundamental concepts and theorems in algebraic topology [1], surface differential geometry [6], and surface Ricci flow [4, 7]. Detailed discussion on Ricci flow on general Riemannian manifolds can be found in [5]. Advanced topics on differential geometry related to Yamabe equations can be found in [9].

The following notation will be used: There will be a “parameter plane”, E2, with Cartesian coordinates (u,v) or (u1,u2) and a Euclidean space E3 with Cartesian coordinates (x,y,z) or (x1,x2,x3).KeywordsTangent VectorFundamental FormTangent PlanePlane CurveGeodesic LineThese 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.

This text for advanced undergraduates emphasizes the logical connections of the subject. The derivations of formulas from the axioms do not make use of models of the hyperbolic plane until the axioms are shown to be categorical; the differential geometry of surfaces is developed far enough to establish its connections to the hyperbolic plane; and the axioms and proofs use the properties of the real number system to avoid the tedium of a completely synthetic approach. The development includes properties of the isometry group of the hyperbolic plane, tilings, and applications to special relativity. Elementary techniques from complex analysis, matrix theory, and group theory are used, and some mathematical sophistication on the part of students is thus required, but a formal course in these topics is not a prerequisite.

Differential Geometry of Surfaces in Three-Dimensional Euclidean Space

A method for accurate registration on point clouds is presented in the paper. Manual alignment or the use of landmarks is avoided in the process of multi-view point clouds. Firstly, the differential geometric information is extracted from the point clouds. The extended Gaussian sphere and combination features are used to define the corresponding points of crude alignment. Secondly, the optimal algorithm，the point-to-point Iterated Closest Point, is applied to the accurate registration on point clouds. Thus, the complete point cloud can be obtained in the method.

Gauss as a geometer

Analysis of human back shape using surface curvatures

Level-Rank Duality of WZW Theories and Isomorphisms of N = 2 Coset Models

The Gauss map is a fundamental concept in surface differential geometry. Although the algorithm of the Gauss map for free-form surface is current more matures, it often can not be used when encountered with nested case. In this paper, a calculation algorithm for the Gauss map is designed in a nested case through which the global Gauss map can be drawn when mapping the Gauss map for free-form surface to the parameters domain. We discuss its decomposition, and inverse mapping back to the surface and get all of its smallest closed area boundary. Finally, we improve the algorithm and validate it with a numerical example.

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.

Conformal field theories corresponding to two-dimensional electrically charged black holes and to two-dimensional anti-de Sitter space with a covariantly constant electric field are simply constructed as SL(2,R)/Z Wess-Zumino-Witten coset models. Four-dimensional spacetime solutions are obtained by tensoring these two-dimensional theories with $\frac{\mathrm{SU}(2)}{Z(m)}$ coset models. These describe a family of dyonic black holes and the Bertotti-Robinson universe.

K-theoretical boundary rings in [formula omitted] coset models

We analyze the relation between a topological coset model based on super SL(2, R)/U(1) coset and non-critical string theory by using free field realization. We show that the twisted N=2 algebra of the coset model can be naturally transformed into that of non-critical string. The screening operators of the coset models can be identified either with those of the minimal matters or with the cosmological constant operator. We also find that another screening operator, which is intrinsic in our approach, becomes the BRST non-trivial state of ghost number 0 (generator of the ground ring for c=1 gravity). The relation between non-critical strings and topological field theories is the subject of current interest. It has long been suggested that the latter theories describe the unbroken phase of gravity,1 but their precise relation has not been clear. It has been known that the twisting of N=2 superconformal field theory gives rise to topological theory.1,2 This suggests that any non-critical string theories may have hidden N=2 superconformal symmetry. Indeed, several authors have observed that the BRST current and the antighost field b(z) generate an algebra that is quite similar but apparently not identical to the N=2 superconformal algebra.3 It turns out that the BRST current can be modified by total derivative terms so that the antighost and the physical BRST current exactly generate a topologically twisted N=2 superconformal algebra.4,5 This does not identify, however, the structure of the models with N=2 symmetry. Recently, rather non-trivial correspondence between super SL(2, R)k/U(1) coset model6 and c=1 string has been analyzed through twisted N=2 structure. Mukhi and Vafa7 have revealed an amazing correspondence between these two models for k=3. In this letter, we discuss the relation of these models and the generalization of the correspondence to the minimal models coupled to gravity by means of the free field realization. We find that there is another interesting correspondence for k=1. Super SL(2, R)k/U(1) model is described by the bosonic coset model of SL(2, R)k×U(1)/U(1).8 For a representation of SL(2, R)k, we use the following