Geometry

Supportive Intersection (Without Analyticity)

Authors: Bin Wang
Comments: 22 Pages.

Part I. On a manifold, we apply the analysis in Part II below to define an intersection called supportive intersection for singular cycles. It has a topological descend to the cup-product. The result is motivated by a problem in cohomology theory. The tool is the notion of currents. A current which is a functional was first introduced by de Rham in 1955. Ever since then, currents played a central role in geometry. However, the part about the support has not been in focus. For instance, the cup-product has been extensively studied in the past. Yet, there is no adequate control on the support of cohomological classes. So, we would like to introduce the supportive intersection that will catch this property. The purpose of this paper is to build the foundation for exploring further.In the end, we'll give an application in this direction.

Part II. This is the technical foundation for the geometry above, but it may have an independent interest. It consists of a functional analysis on a very specific type of convergence of currents. In terms of classical analysis, it is an extension of mollifiers. Classically, mollifier is mostly applied as a smoother for a distribution which is usually viewed as a current of degree $0$. We extend the mollifier to currents where the degrees are positive.
Category: Geometry

On Curves in N-Dimensional Euclidean Spaces

Authors: Emanuels Grinbergs
Comments: 20 Pages. Translated (from Latvian) and submitted by Dainis Zeps

This work examines curves in n-dimensional spaces, as well as varieties contained in such spaces, with the main focus on curves and osculating linear and spherical varieties. Absolute dierential calculus a method almost exclusively used in n-dimensional dierential geometry in recent times is convenient in systematic terms because it enables the determination of all dierential invariantsusing classical techniques. However, it is cumbersome and inconvenient. Since I have sought to examine purely geometric properties and their relationships, I have consistently used vector analysis both independently and in conjunction with direct geometric reasoning, as wellas with Cartesian coordinates.
Category: Geometry

Introduction to Differential Geometry of Space Curves and Surfaces

Authors: Taha Sochi
Comments: 252 Pages.

This book is about differential geometry of space curves and surfaces. The formulation and presentation are largely based on a tensor calculus approach. It can be used as part of a course on tensor calculus as well as a textbook or a reference for an intermediate-level course on differential geometry of curves and surfaces. The book is furnished with an index, extensive sets of exercises and many cross references, which are hyperlinked for the ebook users, to facilitate linking related concepts and sections. The book also contains a considerable number of 2D and 3D graphic illustrations to help the readers and users to visualize the ideas and understand the abstract concepts. We also provided an introductory chapter where the main concepts and techniques needed to understand the offered materials of differential geometry are outlined to make the book fairly self-contained and reduce the need for external references.
Category: Geometry