The Stieltjes integral

Call number: LD2668 .R4 1965 H58

On Ruled Surfaces in three-dimensional Minkowski Space

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 commo

Isometric immersions of complete surfaces with non-positive curvature.

by Fan Xuqian.Thesis (M.Phil.)--Chinese University of Hong Kong, 2000.Includes bibliographical references (leaves 99-100).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.5Chapter 2 --- The Theorem of Efimov --- p.7Chapter 2.1 --- The Idea of the Proof of the Efimov's Theorem --- p.8Chapter 2.2 --- Proof of the Efimov's Main Lemma --- p.12Chapter 2.3 --- Proof of Lemma 2.3 --- p.48Chapter 2.4 --- Proof of Lemma 2.4 --- p.52Chapter 3 --- Isometric Immersion into R3 of Complete Surfaces with Negative Curvature --- p.62Chapter 3.1 --- The Sketch of the Proof of Theorem 3.1 --- p.66Chapter 3.2 --- Proof of Lemma 3.4 --- p.75Chapter 3.3 --- Proof of Lemma 3.5 --- p.76Chapter 3.4 --- Proof of Lemma 3.6 --- p.86Chapter 3.5 --- Proof of Lemma 3.7 --- p.89Chapter 3.6 --- The Geometric Properties of the Immersion --- p.9

Matemaatika- ja mehhaanika-alaseid töid. Struktuurid muutkondadel = Труды по математике и механике. Структуры на многообразиях

• В. Абрамов. Об общенные BEST - суперсимметрии • Summary • Т. Вировере. Полярная плоскость картанова подмногообразия Мм с плоской нормальной связностью В Е2м+1 • Summary • Т. Вировере. Геометрия некоторых специальных картановых подмногообразий Мм с плоской нормальной связностью B Е2м+1 • Summary • H. Вяльяс. Подмногообразие Дюпена-Мангейма и конусы Клиффорда в Еп+m • Summary • X. Кильп. Структурные уравнения и флаговые структуры квазилинейней системы типа S1м2[m] • Summary • Ü. Lumiste. Decomposition of semi-symmetrlc submanlfolds • Резюме • Ü. Lumiste. Classlflcatlon of two-codimenslonal semi-symmetric submanlfolds • Резюме • К. Рийвес. О двух классах полусимметрических подмногообразий • Summary • Е. Ферапонтов. Слабо-нелинейные полугамильтоновы системы трех дифференциальных уравнений с точки зрения теории тканей • Summary • Л. филоненко. Квадратичная гиперполоса и нормальные связности подмногообразия конформного пространства • Summary • Фляйшер. Алгебры инвариантных псевдоримановых связностей на однородных пространствах • Summaryhttp://tartu.ester.ee/record=b1265025~S1*es

Thread-wire surfaces

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.Includes bibliographical references (p. 183-190) and Index.This thesis studies surfaces which minimize area, subject to a fixed boundary and to a free boundary with length constraint. Based on physical experiments, I make two conjectures. First, I conjecture that minimizers supported on generic wires have finitely many surface components. I approach this conjecture by proving that surface components of near-wire minimizers are Lipschitz graphs in wire Frenet coordinates, and appear near maxima of wire curvature. Second, I conjecture and prove that surface components of near-wire minimizers are C1 at corners where the thread touches the wire interior. Moreover, the limit of the surface normal field is the Frenet binormal of the wire at the corner point. This shows local wire geometry dominates global wire geometry in influencing the surface corner. Third, I show that these two conjectures are related: assuming additional regularity up to the corner, the finiteness conjecture follows.by Benjamin K. Stephens.Ph.D