scholarly journals ENGLERT-TYPE SOLUTIONS OF d = 11 SUPERGRAVITY

2013 ◽  
Vol 10 (04) ◽  
pp. 1320005
Author(s):  
E. K. LOGINOV
Keyword(s):  
Riemannian Geometry ◽  
Equations Of Motion ◽  
Connected Space ◽  
Geodesic Loop ◽  
Structure Constants ◽  
Affinely Connected Space

A family of geometries on S7 arise as solutions of the classical equations of motion in 11 dimensions. In addition to the conventional Riemannian geometry and the two exceptional Cartan–Schouten compact flat geometries with torsion, one can also obtain non-flat geometries with torsion. This torsion is given locally by the structure constants of a non-associative geodesic loop in the affinely connected space.

scholarly journals HIGHER EQUATIONS OF MOTION IN LIOUVILLE FIELD THEORY

2004 ◽  
Vol 19 (supp02) ◽  
pp. 510-523 ◽  
Author(s):  
AL. ZAMOLODCHIKOV
Keyword(s):  
Field Theory ◽  
2D Gravity ◽  
Equations Of Motion ◽  
Equation Of Motion ◽  
Operator Product ◽  
Conformal Field ◽  
Structure Constants ◽  
Positive Integers ◽  
Infinite Set ◽  
Liouville Field Theory

An infinite set of the operator-valued relations in the Liouville field theory is established. These relations are enumerated by a pair of positive integers (m, n), the first (1,1) representative being the usual Liouville equation of motion. The relations are proven in the framework of conformal field theory on a basis of the exact structure constants in the Liouville operator product expansions. Possible applications in 2D gravity are discussed.

scholarly journals Some cosmological solutions of a nonlocal modified gravity

Filomat ◽  
2015 ◽  
Vol 29 (3) ◽  
pp. 619-628 ◽  
Author(s):  
Ivan Dimitrijevic ◽  
Branko Dragovich ◽  
Jelena Grujic ◽  
Zoran Rakic
Keyword(s):  
Gravitational Field ◽  
Modified Gravity ◽  
Riemannian Geometry ◽  
Equations Of Motion ◽  
Nonlocal Term ◽  
Action Functional ◽  
Einstein Theory ◽  
Metric Tensor ◽  
Differentiable Functions ◽  
Cosmological Solutions

We consider nonlocal modification of the Einstein theory of gravity in framework of the pseudo- Riemannian geometry. The nonlocal term has the form H(R)F(?)G(R), where H and G are differentiable functions of the scalar curvature R, and F(?) = ??n=0 fn?n is an analytic function of the d?Alambert operator ?. Using calculus of variations of the action functional, we derived the corresponding equations of motion. The variation of action is induced by variation of the gravitational field, which is the metric tensor g?v. Cosmological solutions are found for the case when the Ricci scalar R is constant.

scholarly journals Recurrence Decompositions in Finsler Space

2020 ◽  
Vol 1 (1) ◽  
pp. 77-86
Author(s):  
Adel M. A. Al-Qashbari
Keyword(s):  
Differential Geometry ◽  
Covariant Derivative ◽  
Finsler Space ◽  
Applied Mathematics ◽  
Torsion Tensor ◽  
Finsler Geometry ◽  
Second Order ◽  
Connected Space ◽  
Projective Curvature Tensor ◽  
Affinely Connected Space

Finsler geometry is a kind of differential geometry originated by P. Finsler. Indeed, Finsler geometry has several uses in a wide variety and it is playing an important role in differential geometry and applied mathematics of problems in physics relative, manual footprint. It is usually considered as a generalization of Riemannian geometry. In the present paper, we introduced some types of generalized $W^{h}$ -birecurrent Finsler space, generalized $W^{h}$ -birecurrent affinely connected space and we defined a Finsler space $F_{n}$ for Weyl's projective curvature tensor $W_{jkh}^{i}$ satisfies the generalized-birecurrence condition with respect to Cartan's connection parameters $\Gamma ^{\ast i}_{kh}$, such that given by the condition (\ref{2.1}), where $\left\vert m\right. \left\vert n\right. $ is\ h-covariant derivative of second order (Cartan's second kind covariant differential operator) with respect to $x^{m}$ \ and $x^{n}$ ,\ successively, $\lambda _{mn}$ and $\mu _{mn~}$ are\ non-null covariant vectors field and such space is called as a generalized $W^{h}$ -birecurrent\ space and denoted briefly by $GW^{h}$ - $BRF_{n}$ . We have obtained some theorems of generalized $W^{h}$ -birecurrent affinely connected space for the h-covariant derivative of the second order for Wely's projective torsion tensor $~W_{kh}^{i}$ , Wely's projective deviation tensor $~W_{h}^{i}$ in our space. We have obtained the necessary and sufficient condition forsome tensors in our space.

scholarly journals Riemannian Geometry Framed as a Generalized Heisenberg Lie Algebra

2019 ◽  
Author(s):  
Joseph E. Johnson
Keyword(s):  
Quantum Theory ◽  
Lie Algebra ◽  
Riemannian Geometry ◽  
Fourier Transforms ◽  
Heisenberg Algebra ◽  
Space Time ◽  
Einstein Equations ◽  
Curved Space ◽  
Mathematical Generalization ◽  
Structure Constants

The Heisenberg Lie algebra (HA) plays an important role in mathematics with Fourier transforms, as well as for the foundations of quantum theory where it expresses the operators of space-time, X, and their commutation rules with the momentum operators, D, that execute infinitesimal translations in X. Yet it is known that space-time is curved and thus the D operators must interfere thus giving “structure constants” that vary with location which suggests a mathematical generalization of the concept of a Lie algebra to allow for “structure constants” that are functions of X. We here investigate the mathematics of such a “generalized Heisenberg algebra” (GHA) which has “structure constants” that are functions of X and thus are in the enveloping algebra rather than constants. As expected, the Jacobi identity no longer holds globally but only in small regions of space-time where the [D, X] commutator can be considered locally constant and thus where one has a true Lie algebra. We show that one is able to reframe Riemannian geometry in this GHA. As an example, it is then shown that one can express the Einstein equations of general relativity as commutation rules. If one requires that the GHA commutator reduces to the HA of quantum theory in the limit of no curvature, then there are observable effects for quantum theory in this curved space time.

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