We develop a general computational model for a elastic rod which allows for extension and shear. / Physics

An intrinsic geometric flow is an evolution of a Riemannian metric by a two-tensor. An extrinsic geometric flow is an evolution of an immersion of a manifold into Euclidean space. An extrinsic flow induces an evolution of a metric because any immersed manifold inherits a Riemannian metric from Euclidean space. In this paper we discuss the inverse problem of specifying an evolution of a metric and then seeking an extrinsic geometric flow which induces the given metric evolution. We limit our discussion to the case of manifolds that are rotationally symmetric and embeddable with codimension one. In this case, we reduce an intrinsic geometric flow to a plane curve evolution. In the specific cases we study, we are able to further simplify the evolution to an evolution of a function of one variable. We provide soliton equations and give proofs that some soliton metrics exist.

Semi-Riemannian manifolds is a subject popular in physics, with applications particularly to modern gravitational theory and electrodynamics. Semi-Riemannian geometry is a branch of differential geometry, similar to Riemannian geometry. In fact, Riemannian geometry is a special case of semi-Riemannian geometry where the scalar product of nonzero vectors is only allowed to be positive. This essay approaches the subject from a mathematical perspective, proving some of the main theorems of semi-Riemannian geometry such as the existence and uniqueness of the covariant derivative of Levi-Civita connection, and some properties of the curvature tensor. Finally, this essay aims to deal with the physical applications of semi-Riemannian geometry. In it, two key theorems are proven - the equivalenceof the Einstein field equations, the foundation of modern gravitational physics, and the Schwarzschild solution to the Einstein field equations. Examples of applications of these theorems are presented.

Purpose: traditional optical design is a numerical process based on ray tracing theory. The traditional method has the limitation of the application of the spectacle lens because of the necessity of initial configurations and the evaluations of the aberrations of the lens. This study is an initial attempt to investigate an analytic lens design method, Lagrange, which has a potential application in modern spectacle lens for eliminating the limitation of the traditional method. Methods: the Lagrange method can derive the differential equations of an optical system in term of its output and input. The generalized Snell???s law in three-dimensional space and the normal of a refracting surface in fundamental differential geometry are applied to complete the derivation. Based on the Lagrange method, the solution of a refracting surface to perfectly image a point at infinity is obtained. Results: a Plano-convex lens and a Bi-convex lens from this solution were designed. In spherical coordinates, the differential equations of the single surface system and its solution were obtained. The optical design software, ZEMAX, was used to simulate the lenses and evaluate their image qualities. The results illustrated that both of the two lenses were aberration free. Conclusions: the Lagrange solves unknown lens surface based on definable inputs and outputs according to customer requirements. The method has the potential applicants of the modern customized lens design. Moreover, the definable outputs make the simultaneous elimination of several aberrations possible.

Thesis (Ph. D.)--Technische Universität Berlin, 1998. / Includes bibliographical references (p. 52-54).

Thesis (Ph.D.)--Mathematics, Georgia Institute of Technology, 2008. / Committee Chair: Bellissard, Jean; Committee Member: Baker, Matt; Committee Member: Bakhtin, Yuri; Committee Member: Garoufalidis, Stavros; Committee Member: Putnam, Ian

Thesis (Ph. D.)--Duke University, 2007. / Includes bibliographical references.

This thesis studies Ricci solitons of cohomogeneity one and uniform Poincaré inequalities for differentials on Riemann surfaces. In the two summands case, which assumes that the isotropy representation of the principal orbit consists of two inequivalent Ad-invariant irreducible summands, complete steady and expanding Ricci solitons have been detected numerically by Buzano-Dancer-Gallaugher-Wang. This work provides a rigorous construction thereof. A Lyapunov function is introduced to prove that the Ricci soliton metrics lie in a bounded region of an associated phase space. This also gives an alternative construction of non-compact Einstein metrics of non-positive scalar curvature due to Böhm. It is explained how the asymptotics of the Ricci flat trajectories induce Böhm's Einstein metrics on spheres and other low dimensional spaces. A numerical study suggests that all other Einstein metrics of positive scalar curvature which are induced by the generalised Hopf fibrations occur in an entirely non-linear regime of the Einstein equations. Extending the theory of cohomogeneity one steady and expanding Ricci solitons, an estimate which allows to prescribe the growth rate of the soliton potential at any given time is shown. As an application, continuous families of Ricci solitons on complex line bundles over products of Fano Kähler Einstein manifolds are constructed. This generalises work of Appleton and Stolarski. The method also applies to the Lü-Page-Pope set-up and allows to cover an optimal parameter range in the two summands case. The Ricci soliton equation on manifolds foliated by torus bundles over products of Fano Kähler Einstein manifolds is discussed. A rigidity theorem is obtained and a preserved curvature condition is discovered. The cohomogeneity one initial value problem is solved for m-quasi-Einstein metrics and complete metrics are described. L^p-Poincaré inequalities for k-differentials on closed Riemann surfaces are shown. The estimates are uniform in the sense that the Poincaré constant only depends on p ≥1, k ≥ 2 and the genus γ ≥ 2 of the surface but not on its complex structure. Examples show that the analogous estimate for 1-differentials cannot be uniform. This part is based on joint work with Melanie Rupflin.

Orientador: Francesco Mercuri / Dissertação (mestrado) - Universidade Estadual de Campinas, Intituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-05T13:33:06Z (GMT). No. of bitstreams: 1 Min_LienKuan_M.pdf: 1157875 bytes, checksum: 65f63453a02a7c1365c0a9b3524a1602 (MD5) Previous issue date: 2006 / Resumo: O teorema de Bernstein é um marco importante na teoria das superfícies mínimas. Nesta dissertação apresentaremos três demonstrações deste teorema, cada uma levando a generalizações em diferentes direções / Abstract: The Bernstein's theorem is an important landmark in the theory of the minimal surfaces. In this dissertation we will present three demonstrations of this theorem, each one leading to generalizations in different directions / Mestrado / Geometria Diferencial / Mestre em Matemática

Orientador: Carlos Eduardo Duran Fernandez / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-13T06:35:44Z (GMT). No. of bitstreams: 1 Moreira_AnaClaudiadaSilva_M.pdf: 1048893 bytes, checksum: 74079b9ae1dc0f7d9eee36e181cf4377 (MD5) Previous issue date: 2009 / Resumo: O presente trabalho tem por objetivo estudar o Método do Referencial Móvel de Cartan aplicado a curvas, através de diversos exemplos, desde problemas simples, passando por publicações dos anos 60 e 70 até artigos recentes. Embora existam teorias gerais para encontrar referenciais de Cartan, optamos por estudar uma forma um pouco mais "artesanal" de construção dos referenciais móveis; a ênfase está na absorção das variadas técnicas e intuições que se adaptam a cada geometria / Abstract: The aim of this work is to present the Cartan's Moving Frame Method applied to curves, through several examples, starting with simple problems, going through publications of the 60's, 70's, and up to recent results. Although there are general theories for finding Cartan's moving frames, we chose to study a slightly more "handcraft" way of building the required moving frame; the emphasis being on the absorption of the different techniques and intuitive understanding adapted to each geometry / Mestrado / Mestre em Matemática