Thesis (Ph. D.)--University of Missouri, 1931. / Vita. "Photo-lithoprint reproduction of author's manuscript." "Presented to the American mathematical society December 31, 1929." Includes bibliographical references (p. 20-21).

Thesis (Ph. D.)--University of Virginia, 1907.

Thesis (Ph. D.)--University of Michigan, 1939. / "Reprinted from the American journal of mathematics, vol. LXII, number 2 [1940]."

Thesis (Ph. D.)--University of Michigan, 1939. / "Reprinted from the American journal of mathematics, vol. LXII, number 2 [1940]."

Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Fulton B. Gonzalez. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 72-73). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

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

Orientadores: Carlos Eduardo Durán Fernandez, Marcos Benevenutto Jardim / Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-16T11:28:16Z (GMT). No. of bitstreams: 1 Vitorio_HenriquedeBarrosCorreia_D.pdf: 1074812 bytes, checksum: e23ca71f5e87d6990c05425cdcb87bee (MD5) Previous issue date: 2010 / Resumo: A presente tese dá continuidade ao recente trabalho de J.C . Álvarez e C.E. Durán acerca dos invariantes geométricos de uma classe genérica de curvas em variedades de Grassmann, ditas "curvas fanning". Mais precisamente, considera-se como tais curvas de planos lagrangeanos comportam-se mediante uma redução simplética, e conclui-se a existência de dois novos invariantes que desempenham um papel fundamental neste contexto, mais notavelmente a maneira pela qual eles generalizam as bem conhecidas fórmulas de O'Neill para submersões isométricas / Abstract: The present thesis gives continuity to the recent work of J.C. Álvarez e C.E. Durán about the geometric invariants of a generic class of curves in the Grassmann manifolds, called "fanning curves". More precisely, we look at how such curves of lagrangean planes behave under a symplectic reduction, and establish the existence of two new invariants which play a fundamental role in that context, more notably the way they generalize the well known O'Neill's formulas for isometric submersions / Doutorado / Matematica / Doutor em Matemática

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common

In a Minkowski three dimensional space, whose metric is based on a strictly convex and centrally symmetric unit ball , we deal with ruled surfaces Φ in the sense of E. Kruppa. This means that we have to look for Minkowski analogues of the classical differential invariants of ruled surfaces in a Euclidean space. Here, at first – after an introduction to concepts of a Minkowski space, like semi-orthogonalities and a semi-inner-product based on the so-called cosine-Minkowski function - we construct an orthogonal 3D moving frame using Birkhoff’s left-orthogonality. This moving frame is canonically connected to ruled surfaces: beginning with the generator direction and the asymptotic plane of this generator g we complete this flag to a frame using the left-orthogonality defined by ; ( is described either by its supporting function or a parameter representation). The plane left-orthogonal to the asymptotic plane through generator g(t) is called Minkowski central plane and touches Φ in the striction point s(t) of g(t). Thus the moving frame defines the Minkowski striction curve S of the considered ruled surface Φ similar to the Euclidean case. The coefficients occurring in the Minkowski analogues to Frenet-Serret formulae of the moving frame of Φ in a Minkowski space are called “M-curvatures” and “M-torsions”. Here we essentially make use of the semi-inner product and the sine-Minkowski and cosine-Minkowski functions. Furthermore we define a covariant differentiation in a Minkowski 3-space using a new vector called “deformation vector” and locally measuring the deviation of the Minkowski space from a Euclidean space. With this covariant differentiation it is possible to declare an “M-geodesicc parallelity” and to show that the vector field of the generators of a skew ruled surface Φ is an M-geodesic parallel field along its Minkowski striction curve s. Finally we also define the Pirondini set of ruled surfaces to a given surface Φ. The surfaces of such a set have the M-striction curve and the strip of M-central planes in common