Differential geometry

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Differential Geometry ◽  
Vector Fields ◽  
Differential Forms ◽  
Curvilinear Coordinates ◽  
Moving Frames ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Vector Calculus ◽  
Vector Version ◽  
Definition Of

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

scholarly journals CLASSICAL TENSORS AND QUANTUM ENTANGLEMENT I: PURE STATES

2010 ◽  
Vol 07 (03) ◽  
pp. 485-503 ◽  
Author(s):  
P. ANIELLO ◽  
J. CLEMENTE-GALLARDO ◽  
G. MARMO ◽  
G. F. VOLKERT
Keyword(s):  
Hilbert Space ◽  
Symplectic Geometry ◽  
Vector Fields ◽  
Riemannian Metric ◽  
Inner Product ◽  
Tensor Fields ◽  
Metric Tensor ◽  
Separable States ◽  
Pure States ◽  
Complex Projective

The geometrical description of a Hilbert space associated with a quantum system considers a Hermitian tensor to describe the scalar inner product of vectors which are now described by vector fields. The real part of this tensor represents a flat Riemannian metric tensor while the imaginary part represents a symplectic two-form. The immersion of classical manifolds in the complex projective space associated with the Hilbert space allows to pull-back tensor fields related to previous ones, via the immersion map. This makes available, on these selected manifolds of states, methods of usual Riemannian and symplectic geometry. Here, we consider these pulled-back tensor fields when the immersed submanifold contains separable states or entangled states. Geometrical tensors are shown to encode some properties of these states. These results are not unrelated with criteria already available in the literature. We explicitly deal with some of these relations.

scholarly journals CONFORMAL FIELDS: A CLASS OF REPRESENTATIONS OF Vect(N)

1992 ◽  
Vol 07 (26) ◽  
pp. 6493-6508 ◽  
Author(s):  
T.A. LARSSON
Keyword(s):  
Differential Geometry ◽  
Representation Theory ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Tensor Fields ◽  
Normal Ordering ◽  
New Class ◽  
Finite Dimensional ◽  
Conformal Fields

Vect (N), the algebra of vector fields in N dimensions, is studied. Some aspects of local differential geometry are formulated as Vect(N) representation theory. There is a new class of modules, conformal fields, whose restrictions to the subalgebra sl(N+1)⊂ Vect (N) are finite-dimensional sl (N+1) representations. In this regard they are simpler than tensor fields. Fock modules are also constructed. Infinities, which are unremovable even by normal ordering, arise unless bosonic and fermionic degrees of freedom match.

scholarly journals On Some Characterizations of Vector Fields on Manifolds

2012 ◽  
Vol 60 (2) ◽  
pp. 259-263 ◽  
Author(s):  
Khondokar M. Ahmed ◽  
Md. Raknuzzaman ◽  
Md. Showkat Ali
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Vector Fields ◽  
Definition Of ◽  
Tangent Bundles

The basic geometry of vector fields and definition of the notions of tangent bundles are developed in an essential different way than in usual differential geometry. ø-related vector fields are studied and some related properties are developed in our paper. Finally, a theorem 5.04 on our natural injection j of submanifolds which is j-related to vector field X is treated.DOI: http://dx.doi.org/10.3329/dujs.v60i2.11530 Dhaka Univ. J. Sci. 60(2): 259-263, 2012 (July)

Manifolds and tensors

2019 ◽  
pp. 25-36
Author(s):  
Steven Carlip
Keyword(s):  
Differential Geometry ◽  
General Relativity ◽  
Differential Forms ◽  
General Covariance ◽  
Mathematical Basis ◽  
Metric Tensor ◽  
Diffeomorphism Invariance ◽  
Starting Point ◽  
Tangent Vectors

The mathematical basis of general relativity is differential geometry. This chapter establishes the starting point of differential geometry: manifolds, tangent vectors, cotangent vectors, tensors, and differential forms. The metric tensor is introduced, and its symmetries (isometries) are described. The importance of diffeomorphism invariance (or “general covariance”) is stressed.

scholarly journals Classification of Holomorphic Functions as Pólya Vector Fields via Differential Geometry

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1890
Author(s):  
Lucian-Miti Ionescu ◽  
Cristina-Liliana Pripoae ◽  
Gabriel-Teodor Pripoae
Keyword(s):  
Differential Geometry ◽  
Vector Field ◽  
Vector Fields ◽  
Torsion Tensor ◽  
Holomorphic Functions ◽  
Tensor Fields ◽  
Pedagogical Tool ◽  
Complex Integral ◽  
Curvature And Torsion

We review Pólya vector fields associated to holomorphic functions as an important pedagogical tool for making the complex integral understandable to the students, briefly mentioning its use in other dimensions. Techniques of differential geometry are then used to refine the study of holomorphic functions from a metric (Riemannian), affine differential or differential viewpoint. We prove that the only nontrivial holomorphic functions, whose Pólya vector field is torse-forming in the cannonical geometry of the plane, are the special Möbius transformations of the form f(z)=b(z+d)−1. We define and characterize several types of affine connections, related to the parallelism of Pólya vector fields. We suggest a program for the classification of holomorphic functions, via these connections, based on the various indices of nullity of their curvature and torsion tensor fields.

scholarly journals Jet methods in time-dependent Lagrangian biomechanics

Open Physics ◽  
2010 ◽  
Vol 8 (5) ◽  
Author(s):  
Tijana Ivancevic
Keyword(s):  
Human Body ◽  
Degrees Of Freedom ◽  
Vector Fields ◽  
Differential Forms ◽  
Flow Equation ◽  
Body Motion ◽  
Time Dependent ◽  
Metric Tensor ◽  
Geometric Evolution ◽  
Configuration Manifold

AbstractIn this paper we propose the time-dependent generalization of an ‘ordinary’ autonomous human biomechanics, in which total mechanical + biochemical energy is not conserved. We introduce a general framework for time-dependent biomechanics in terms of jet manifolds associated to the extended musculo-skeletal configuration manifold, called the configuration bundle. We start with an ordinary configuration manifold of human body motion, given as a set of its all active degrees of freedom (DOF) for a particular movement. This is a Riemannian manifold with a material metric tensor given by the total mass-inertia matrix of the human body segments. This is the base manifold for standard autonomous biomechanics. To make its time-dependent generalization, we need to extend it with a real time axis. By this extension, using techniques from fibre bundles, we defined the biomechanical configuration bundle. On the biomechanical bundle we define vector-fields, differential forms and affine connections, as well as the associated jet manifolds. Using the formalism of jet manifolds of velocities and accelerations, we develop the time-dependent Lagrangian biomechanics. Its underlying geometric evolution is given by the Ricci flow equation.

Spacetime symmetries

2018 ◽  
Author(s):  
Michael Kachelriess
Keyword(s):  
Riemannian Manifold ◽  
Conservation Laws ◽  
Vector Fields ◽  
Killing Vector ◽  
Geodesic Equation ◽  
Spacetime Symmetries ◽  
Tensor Fields ◽  
Killing Vector Fields ◽  
Covariant Derivatives ◽  
Killing Equation

This chapter introduces tensor fields, covariant derivatives and the geodesic equation on a (pseudo-) Riemannian manifold. It discusses how symmetries of a general space-time can be found from the Killing equation, and how the existence of Killing vector fields is connected to global conservation laws.

Sign in / Sign up

Export Citation Format

Share Document