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Anastasios Mallios - One of the best experts on this subject based on the ideXlab platform.
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Fundamentals for symplectic $$ \mathcal{A} $$ -modules. Affine Darboux theorem
Rendiconti del Circolo Matematico di Palermo, 2009Co-Authors: Anastasios Mallios, Patrice P. NtumbaAbstract:In his [9–11], the first author shows that the sheaf-theoreti-cally based Abstract Differential Geometry incorporates and generalizes Classical Differential Geometry . Here, we undertake to explore the implications of Abstract Differential Geometry to Classical symplectic Geometry . The full investigation will be presented elsewhere.
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Fundamentals for symplectic \mathcal{A}-modules. Affine Darboux theorem
Rendiconti del Circolo Matematico di Palermo, 2009Co-Authors: Anastasios Mallios, Patrice P. NtumbaAbstract:In his [9–11], the first author shows that the sheaf-theoreti-cally based Abstract Differential Geometry incorporates and generalizes Classical Differential Geometry. Here, we undertake to explore the implications of Abstract Differential Geometry to Classical symplectic Geometry. The full investigation will be presented elsewhere.
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Fundamentals for Symplectic $\mathcal{A}$-modules
arXiv: Symplectic Geometry, 2007Co-Authors: Anastasios Mallios, Patrice P. NtumbaAbstract:Sheaf theoretically based Abstract Differential Geometry incorporates and generalizes all the Classical Differential Geometry. Here, we undertake to partially explore the implications of Abstract Differential Geometry to Classical symplectic Geometry. The full investigation will be presented elsewhere.
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Geometry and Physics Today
International Journal of Theoretical Physics, 2006Co-Authors: Anastasios MalliosAbstract:“Geometry,” in the sense of the Classical Differential Geometry of smooth manifolds (CDG), is put under scrutiny from the point of view of Abstract Differential Geometry (ADG). We explore potential physical implications of viewing things under the light of ADG, especially matters concerning the “gauge theories” of modern physics, when the latter are viewed (as they are actually regarded currently) as “physical theories of a geometrical character.” Thence, “physical Geometry,” in connection with physical laws and the associated with them, within the background spacetime manifoldless context of ADG, “Differential” equations, are also being discussed.
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Quantum gravity and "singularities"
2006Co-Authors: Anastasios MalliosAbstract:It is our aim in the present article to point out that the phenomenal disagreement of the Classical Differential Geometry (CDG, viz. Differential Geometry of smooth manifolds), when applied in the quantum domain (quantum field theory, for instance) is just due to the entanglement in that procedure of the underlying (:smooth)-manifold, hence, the appearance of the so-called "singularities", while the inherent, quite algebraic, in character, Differential-geometric mechanism of the same Classical theory still remains intact, without having to resort to any carrier space at all (!), supporting that mechanism, as it was the case, so far. Therefore, a potential application of the same, even in the quantum regime.
Kazimieras Navickis - One of the best experts on this subject based on the ideXlab platform.
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Osculating surfaces of curves on surfaces
Lietuvos matematikos rinkinys, 2020Co-Authors: Kazimieras NavickisAbstract:Oosculating sphere have been studied in Classical Differential Geometry [1]. In this article the osculating surfaces of higher order of space curves on surfaces in Euclidean space is considered. We study the intrinsic Differential Geometry of curves on surfaces by analyzing their contact with surfaces of higher order.
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Osculating curves and surfaces
Lietuvos matematikos rinkinys, 2012Co-Authors: Kazimieras NavickisAbstract:Osculating circle and osculating sphere have been studied in Classical Differential Geometry [1]. In this article the osculating curves and surfaces of higher order of plane and space curves in Euclideann-space (n = 2, 3) is considered. We study the intrinsic Differential Geometry of curves by analyzing their contact with curves and surfaces of higher order.
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Osculating hypersurfaces of higher order
Lietuvos matematikos rinkinys, 2011Co-Authors: Kazimieras NavickisAbstract:Oscurating surfaces of second order have been studied in Classical Differential Geometry [1]. In this article we generalize this notion to osculating hyper-surfaces of higher order of hyper-surfaces inEuclidean n-space. Various related results are obtained using the derivatives of higher order.
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Families of osculating paraboloids
Lietuvos matematikos rinkinys, 2010Co-Authors: Kazimieras NavickisAbstract:In this article we generalize some notions in Classical Differential Geometry to families of osculating paraboloids.
Huang Bao-jun - One of the best experts on this subject based on the ideXlab platform.
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The Geodesics on the Several Special Surfaces
College Mathematics, 2008Co-Authors: Huang Bao-junAbstract:We will depict concretely the geometric characters of the geodesics on the cylinders,cones and surface of revolution.Our results not only correct erroneous proposition for the geodesics on the cones in some literature but also extend the description for the properties of the geodesics on the surfaces of revolution in Classical Differential Geometry.
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Geometric Character of the Geodesics on the Special Surfaces
Journal of Huaibei Coal Industry Teachers College, 2007Co-Authors: Huang Bao-junAbstract:We will depict concretely the geometric characters of the geodesics on the cylinders,cones and surfaces of revolution.Our results not only correct erroneous proposition for the geodesics on the cones in some literature but also extend the description for the properties of the geodesics on the surfaces of revolution in Classical Differential Geometry.
J P Miles - One of the best experts on this subject based on the ideXlab platform.
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2 Classical Differential Geometry of space curves
Basic Structured Grid Generation#R##N#with an introduction to unstructured grid generation, 2003Co-Authors: M Farrashkhalvat, J P MilesAbstract:This chapter focuses on the Classical Differential Geometry of space-curves. It focuses on the smooth curves in E 3 specified in terms of rectangular cartesian coordinates x, y, z (or y 1 , y 2 , y 3 ). Such curves are generated by three smooth functions of a single real parameter so that the position vector r of points on the curve relative to some origin O is given by r = r(t) = x( t )i + y( t )j + z( t )k. While discussing the serret-frenet equations, the chapter explains that given a unit tangent vector t and a unit principal normal n at a point on a curve in E 3 , one can define a third unit vector b , called the unit binormal vector, orthogonal to both of them, such that b = t x n . . It may be instructive to derive the Serret-Frenet formulas. In the process, the chapter introduces the concept of intrinsic differentiation. In the context of grid generation, space-curves appear as boundaries of surfaces and as edges of three-dimensional blocks, and it is convenient to map a given finite length of space-curve onto an interval of the ξ-axis. A uniformly spaced set of points in the ξ -interval will then map to a set of points along the curve.
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Basic Structured Grid Generation - 2 – Classical Differential Geometry of space-curves
Basic Structured Grid Generation, 2003Co-Authors: M Farrashkhalvat, J P MilesAbstract:This chapter focuses on the Classical Differential Geometry of space-curves. It focuses on the smooth curves in E 3 specified in terms of rectangular cartesian coordinates x, y, z (or y 1 , y 2 , y 3 ). Such curves are generated by three smooth functions of a single real parameter so that the position vector r of points on the curve relative to some origin O is given by r = r(t) = x( t )i + y( t )j + z( t )k. While discussing the serret-frenet equations, the chapter explains that given a unit tangent vector t and a unit principal normal n at a point on a curve in E 3 , one can define a third unit vector b , called the unit binormal vector, orthogonal to both of them, such that b = t x n . . It may be instructive to derive the Serret-Frenet formulas. In the process, the chapter introduces the concept of intrinsic differentiation. In the context of grid generation, space-curves appear as boundaries of surfaces and as edges of three-dimensional blocks, and it is convenient to map a given finite length of space-curve onto an interval of the ξ-axis. A uniformly spaced set of points in the ξ -interval will then map to a set of points along the curve.
Yuri B. Suris - One of the best experts on this subject based on the ideXlab platform.
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discrete Differential Geometry integrable structure
2008Co-Authors: Alexander I. Bobenko, Yuri B. SurisAbstract:Classical Differential Geometry Discretization principles. Multidimensional nets Discretization principles. Nets in quadrics Special classes of discrete surfaces Approximation Consistency as integrability Discrete complex analysis. Linear theory Discrete complex analysis. Integrable circle patterns Foundations Solutions of selected exercises Bibliography Notations Index.
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On organizing principles of discrete Differential Geometry. Geometry of spheres
Russian Mathematical Surveys, 2007Co-Authors: Alexander I. Bobenko, Yuri B. SurisAbstract:Discrete Differential Geometry aims to develop discrete equivalents of the geometric notions and methods of Classical Differential Geometry. This survey contains a discussion of the following two fundamental discretization principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem treated here is discretization of curvature-line parametrized surfaces in Lie Geometry. Systematic use of the discretization principles leads to a discretization of curvature-line parametrization which unifies circular and conical nets.
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Discrete Differential Geometry. Consistency as integrability
arXiv: Differential Geometry, 2005Co-Authors: Alexander I. Bobenko, Yuri B. SurisAbstract:A new field of discrete Differential Geometry is presently emerging on the border between Differential and discrete Geometry. Whereas Classical Differential Geometry investigates smooth geometric shapes (such as surfaces), and discrete Geometry studies geometric shapes with finite number of elements (such as polyhedra), the discrete Differential Geometry aims at the development of discrete equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. Recent progress in discrete Differential Geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the Classical Differential Geometry and of the theory of integrable systems. The goal of this book is to give a systematic presentation of current achievements in this field.
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Discrete and smooth orthogonal systems: $C^\infty$-approximation
arXiv: Differential Geometry, 2003Co-Authors: Alexander I. Bobenko, Daniel Matthes, Yuri B. SurisAbstract:Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of Classical Differential Geometry. We develop techniques that allow us to extend this known qualitative analogy to rigorous convergence results. In particular, we prove the $C^\infty$-convergence of discrete conjugate/orthogonal coordinate systems to smooth ones. We also show how to construct the approximating discrete nets. Coordinate systems and their transformations are treated on an equal footing, and the approximation results hold for transformations as well.