Basis Vector Fields and the Metric Tensor

2007 ◽  
pp. 63-88
Keyword(s):  
Vector Fields ◽  
Basis Vector ◽  
Metric Tensor

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 A gravitational action with stringy Q and R fluxes via deformed differential graded Poisson algebras

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Eugenia Boffo ◽  
Peter Schupp
Keyword(s):  
Vector Fields ◽  
Poisson Algebra ◽  
Gravitational Theory ◽  
Point Particle ◽  
Linear Functions ◽  
Metric Tensor ◽  
Higher Spin ◽  
Spin Fields ◽  
Generalized Differential ◽  
Differential Graded Poisson Algebras

Abstract We study a deformation of a 2-graded Poisson algebra where the functions of the phase space variables are complemented by linear functions of parity odd velocities. The deformation is carried by a 2-form B-field and a bivector Π, that we consider as gauge fields of the geometric and non-geometric fluxes H, f, Q and R arising in the context of string theory compactification. The technique used to deform the Poisson brackets is widely known for the point particle interacting with a U(1) gauge field, but not in the case of non-abelian or higher spin fields. The construction is closely related to Generalized Geometry: with an element of the algebra that squares to zero, the graded symplectic picture is equivalent to an exact Courant algebroid over the generalized tangent bundle E ≅ TM ⊕ T∗M, and to its higher gauge theory. A particular idempotent graded canonical transformation is equivalent to the generalized metric. Focusing on the generalized differential geometry side we construct an action functional with the Ricci tensor of a connection on covectors, encoding the dynamics of a gravitational theory for a contravariant metric tensor and Q and R fluxes. We also extract a connection on vector fields and determine a non-symmetric metric gravity theory involving a metric and H-flux.

scholarly journals On the Orbits of One Non-Solvable 5-Dimensional Lie Algebra

2019 ◽  
pp. 5-20
Author(s):  
Artem Atanov ◽  
Alexander Loboda
Keyword(s):  
Lie Algebra ◽  
Lie Algebras ◽  
Fourth Order ◽  
Algebraic Surface ◽  
Vector Fields ◽  
Symmetry Algebra ◽  
Basis Vector ◽  
Invariant Polynomial ◽  
Real Hypersurfaces ◽  
Symmetry Algebras

This paper studies holomorphic homogeneous real hypersurfaces in C3 associated with the unique non-solvable indecomposable 5-dimensional Lie algebra 𝑔5 (in accordance with Mubarakzyanov’s notation). Unlike many other 5-dimensional Lie algebras with “highly symmetric” orbits, non-degenerate orbits of 𝑔5 are “simply homogeneous”, i.e. their symmetry algebras are exactly 5-dimensional. All those orbits are equivalent (up to holomorphic equivalence) to the specific indefinite algebraic surface of the fourth order. The proofs of those statements involve the method of holomorphic realizations of abstract Lie algebras. We use the approach proposed by Beloshapka and Kossovskiy, which is based on the simultaneous simplification of several basis vector fields. Three auxiliary lemmas formulated in the text let us straighten two basis vector fields of 𝑔5 and significantly simplify the third field. There is a very important assumption which is used in our considerations: we suppose that all orbits of 𝑔5 are Levi non-degenerate. Using the method of holomorphic realizations, it is easy to show that one need only consider two sets of holomorphic vector fields associated with 𝑔5. We prove that only one of these sets leads to Levi non-degenerate orbits. Considering the commutation relations of 𝑔5, we obtain a simplified basis of vector fields and a corresponding integrable system of partial differential equations. Finally, we get the equation of the orbit (unique up to holomorphic transformations) (𝑣 − 𝑥2𝑦1)2 + 𝑦2 1𝑦2 2 = 𝑦1, which is the equation of the algebraic surface of the fourth order with the indefinite Levi form. Then we analyze the obtained equation using the method of Moser normal forms. Considering the holomorphic invariant polynomial of the fourth order corresponding to our equation, we can prove (using a number of results obtained by A.V. Loboda) that the upper bound of the dimension of maximal symmetry algebra associated with the obtained orbit is equal to 6. The holomorphic invariant polynomial mentioned above differs from the known invariant polynomials of Cartan’s and Winkelmann’s types corresponding to other hypersurfaces with 6- dimensional symmetry algebras.

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 Non-orthogonalisable vector fields on spheres

1984 ◽  
Vol 27 (3) ◽  
pp. 275-281
Author(s):  
Martin Raussen
Keyword(s):  
Vector Fields ◽  
Fibre Bundle ◽  
Basis Vector ◽  
Image Position

A (k – l)-field on Sn-1 may be given as a section ϕ of the fibre bundlewith fibre Vn-1, k-1 or, equivalently, as a semi-orthogonal map, i.e., a mapwhich is isometric in the second variable and such that for the basis vector e1∈Rk and every x∈Rn

scholarly journals On equitorsion concircular tensors of generalized Riemannian spaces

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 463-471 ◽  
Author(s):  
Milan Zlatanovic ◽  
Irena Hinterleitner ◽  
Marija Najdanovic
Keyword(s):  
Vector Fields ◽  
Metric Tensor ◽  
The Subject ◽  
Curvature Tensors ◽  
Riemannian Spaces

In this paper we consider concircular vector fields of manifolds with non-symmetric metric tensor. The subject of our paper is an equitorsion concircular mapping. A mapping f : GRN?GRN? is an equitorsion if the torsion tensors of the spaces GRN and GRN? are equal. For an equitorsion concircular mapping of two generalized Riemannian spaces GRN and GRN, we obtain some invariant curvature tensors of this mapping Z?,? = 1,2,... 5, given by equations (3.14, 3.21, 3.28, 3.31, 3.38). These quantities are generalizations of the concircular tensor Z given by equation (2.5).

Field Approach for General Relativity

2021 ◽  
Vol 5 (1) ◽  
Author(s):  
Ali Rıza ŞAHİN ◽  
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Vector Fields ◽  
Einstein Equations ◽  
Light Deflection ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Metric Tensor ◽  
Perihelion Precession ◽  
Field Approach

The general theory of relativity is based on expressing gravity by means of the metric tensor and its elements instead of some fields as in electrodynamics. This work starts with by defining some vector fields for metric or metric tensor. After metric and metric tensor are expressed in terms of these fields, the geodesic equations and Einstein equations are derived for these fields. Finally, perihelion precession and light deflection are recalculated, as two different applications of the introduced fields.

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