Normal Calculus on Moving Surfaces

2018 ◽  
Author(s):  
Keith C. Afas
Keyword(s):  
Differential Geometry ◽  
Riemannian Geometry ◽  
Orthogonal Vector ◽  
Normal Field ◽  
Inner Products ◽  
Calculus Of Moving Surfaces ◽  
Basis Vectors ◽  
New Framework

This paper presents an extension for principles of Differential Geometry on Surfaces (re-hashed through the budding field of CMS, the Calculus of Moving Surfaces). It analyzes mostly 2D Hypersurfaces with Riemannian Geometry and proposes the construction of a 3D Static Frame combining the Surface Basis Vectors with the Orthogonal Normal Field as a 3D Orthogonal Vector Frame. The paper introduces conventions for manipulating Tensors defined using this 3D Orthogonal Vector Frame as well as Curvature Connections associated with this Vector Frame. It then finally introduces Symbols and Tensors to describe Inner Products and Variance within the 3D Vector Frame and then extends all the above concepts to a surface which is Dynamic utilizing principles from CMS. This formulation has potential to extend identities and concepts from CMS and from Differential Geometry in a compact Tensorial Framework, which agrees with the new Framework proposed by CMS.

A Generalization of Finsler Geometry

1962 ◽  
Vol 14 ◽  
pp. 87-112 ◽  
Author(s):  
J. R. Vanstone
Keyword(s):  
Differential Geometry ◽  
General Theory ◽  
Distance Function ◽  
Riemannian Geometry ◽  
Finsler Geometry ◽  
Riemannian Metric ◽  
Elliptic Partial Differential Equations ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Function

Modern differential geometry may be said to date from Riemann's famous lecture of 1854 (9), in which a distance function of the form F(xi, dxi) = (γij(x)dxidxj½ was proposed. The applications of the consequent geometry were many and varied. Examples are Synge's geometrization of mechanics (15), Riesz’ approach to linear elliptic partial differential equations (10), and the well-known general theory of relativity of Einstein.Meanwhile the results of Caratheodory (4) in the calculus of variations led Finsler in 1918 to introduce a generalization of the Riemannian metric function (6). The geometry which arose was more fully developed by Berwald (2) and Synge (14) about 1925 and later by Cartan (5), Busemann, and Rund. It was then possible to extend the applications of Riemannian geometry.

scholarly journals Arthur Geoffrey Walker. 17 July 1909 — 31 March 2001

2006 ◽  
Vol 52 ◽  
pp. 413-421
Author(s):  
Nigel J. Hitchin
Keyword(s):  
Differential Geometry ◽  
Quantum Theory ◽  
Riemannian Geometry ◽  
Psychiatric Illness ◽  
Grammar School ◽  
Atomic Spectra ◽  
King's College

Arthur Geoffrey Walker was born in Watford, Hertfordshire, on 17 July 1909, and attended Watford Grammar School, from where he won in 1928 an Open Mathematical Scholarship to Balliol College, Oxford. There, his tutor was John William Nicholson FRS, who had been a professor at King's College, London, and was one of the first mathematical physicists to relate quantum theory to atomic spectra. However, in the late 1920s he was suffering from a psychiatric illness and in 1930 was hospitalized, so that Walker had to study on his own a great deal. This perhaps influenced his subsequent method of working on mathematics, which he normally did in the privacy of his room rather than in active consultation with others. He obtained a Second in Moderations, but in 1930 won a Junior Mathematical Exhibition and in 1931 took a First with a Distinction in the special subject of differential geometry, which was to become his life's work. Eisenhart's book Riemannian geometry (Eisenhart 1926) became his bible and he continually referred to it in many of his papers.

scholarly journals A Survey on Differential Geometry of Riemannian Maps Between

2015 ◽  
Vol 0 (0) ◽  
Author(s):  
Bayram Şahin
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifolds ◽  
Riemannian Geometry ◽  
Research Area ◽  
Riemannian Submersions ◽  
Open Problems ◽  
Geometric Objects ◽  
Riemannian Map

AbstractThe main aim of this paper is to state recent results in Riemannian geometry obtained by the existence of a Riemannian map between Riemannian manifolds and to introduce certain geometric objects along such maps which allow one to use the techniques of submanifolds or Riemannian submersions for Riemannian maps. The paper also contains several open problems related to the research area.

scholarly journals Riemannian Geometry and Modern Developments

2019 ◽  
Vol 39 ◽  
pp. 71-85
Author(s):  
AKM Nazimuddin ◽  
Md Showkat Ali
Keyword(s):  
Differential Geometry ◽  
Scalar Curvature ◽  
Constant Curvature ◽  
Riemannian Geometry ◽  
Ricci Tensor ◽  
Software Tool ◽  
3 Dimensional ◽  
Christoffel Symbols ◽  
Implementation Method

In this paper, we compute the Christoffel Symbols of the first kind, Christoffel Symbols of the second kind, Geodesics, Riemann Christoffel tensor, Ricci tensor and Scalar curvature from a metric which plays a fundamental role in the Riemannian geometry and modern differential geometry, where we consider MATLAB as a software tool for this implementation method. Also we have shown that, locally, any Riemannian 3-dimensional metric can be deformed along a directioninto another metricthat is conformal to a metric of constant curvature GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 71-85

scholarly journals Psychology and geometry: I. On the geometry of the human kind

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 545-552
Author(s):  
Leopold Verstraelena
Keyword(s):  
Differential Geometry ◽  
Riemannian Geometry ◽  
Point Of View ◽  
Psychological Point ◽  
History Of ◽  
Present Part ◽  
Human Kind

In this note an attempt is made to describe a personal look at some of the main steps in the history of geometry from a psychological point of view, hereby basing on and sometimes merely formulating again parts of some previous papers, like [1-11]. For general references on elementary differential geometry, pseudo Riemannian geometry and geometry of submanifolds, see e.g. [12-22]. In reference [23], part II of some of the author?s reflections on psychology and geometry, an attempt is made to describe relativistic spacetimes in a way as kind of a supplement to the contents of the present part I.

scholarly journals Surfaces and Curves Induced by Nonlinear Schrödinger-Type Equations and Their Spin Systems

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1827
Author(s):  
Akbota Myrzakul ◽  
Gulgassyl Nugmanova ◽  
Nurzhan Serikbayev ◽  
Ratbay Myrzakulov
Keyword(s):  
Differential Geometry ◽  
Spin Systems ◽  
Heisenberg Ferromagnet ◽  
Integrable Equations ◽  
Nonlinear Schrödinger ◽  
Space Curves ◽  
The Third ◽  
Integrable Spin ◽  
Frenet Equation ◽  
Basis Vectors

In recent years, symmetry in abstract partial differential equations has found wide application in the field of nonlinear integrable equations. The symmetries of the corresponding transformation groups for such equations make it possible to significantly simplify the procedure for establishing equivalence between nonlinear integrable equations from different areas of physics, which in turn open up opportunities to easily find their solutions. In this paper, we study the symmetry between differential geometry of surfaces/curves and some integrable generalized spin systems. In particular, we investigate the gauge and geometrical equivalence between the local/nonlocal nonlinear Schrödinger type equations (NLSE) and the extended continuous Heisenberg ferromagnet equation (HFE) to investigate how nonlocality properties of one system are inherited by the other. First, we consider the space curves induced by the nonlinear Schrödinger-type equations and its equivalent spin systems. Such space curves are governed by the Serret–Frenet equation (SFE) for three basis vectors. We also show that the equation for the third of the basis vectors coincides with the well-known integrable HFE and its generalization. Two other equations for the remaining two vectors give new integrable spin systems. Finally, we investigated the relation between the differential geometry of surfaces and integrable spin systems for the three basis vectors.

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