Some General Remarks on Closed Surfaces in 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.

String propagation on a curved background defines an embedding problem of surfaces in differential geometry. Using this, we show that in a wide class of backgrounds the classical dynamics of the physical degrees of freedom of the string involves 2-dim sigma-models corresponding to coset conformal field theories.

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

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.

A high efficient surface-based method for predicting part distortions in machining and shot peening

We review the connections between condensed matter physics, symmetry, and topology. Physics goes back to at least the time of Galileo, but condensed matter physics, or solid-state physics, is a much newer, emerging only as a separate subject in the 1940s. The subject of symmetry, which is the mathematics of groups and representations, only came to the fore with the advent of quantum mechanics. Early applications to crystalline solids include Bloch’s theorem, the symmetry of electronic and phononic energy bands, and selection rules. Topology, on the other hand, did not exist as a mathematical subject before the twentieth century, but has had a profound influence on physics in general, and on condensed matter physics in particular. The quantum Hall effect is recognized as the first solid-state topological phenomenon and, along with the Berry phase, led to the development of topological materials. This, in turn, led to the topological description of energy bands and to the development of topological quantum chemistry and the energy band representation. Topology has also led to the description of martensitic transformations and the shape memory effect in terms of topological transformations. Apart from a concise statement of martensitic transformations, topology provides a fast-screening method for the discovery of new shape-memory materials. We review these phenomena, providing background material in topology and differential geometry to enable the reader to understand applications to topological materials and to materials physics.