Geometrical Theory of Embeddings of Vector Bundles

Miron, Radu; Anastasiei, Mihai

doi:10.1007/978-94-011-0788-4_4

Cited by 9 publications

(53 citation statements)

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“…On the other hand, the vector y a is attached as an internal variable to each point x µ , as previously stated. In this sense, y a may be regarded as the spacetime fluctuations (micro internal freedom) associated to the point x µ [19], [20]. Accordingly, the vertical field equations, we conjecture, may express some kind of micro (or quantum) phenomena.…”

Section: Discussion and Concluding Remarksmentioning

confidence: 89%

“…This could be partially achieved if an appropriate physical interpretation of the directional argument y is given. One possible interpretation is the following: The vector y a is attached as an internal variable to each point x. Consequently, the y-dependence may be combined with the notion of anisotropy or non-locality [20].…”

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…The material covered in the present section may be found in [19] and [20]. Some related topics may be found in [2].…”

Section: Basic Prelimenariesmentioning

confidence: 99%

“…For example, Almost Finslerian Lagrange (AFL-) spaces [20] were used as a model for a theory of electromagnetism in this wider framework [9], [10]. Moreover, R. Miron developed a Lagrangian theory of relativity, which is a more generalized version of the Finslerian theory of relativity.…”

Section: Motivation and Introductionmentioning

confidence: 99%

“…Moreover, R. Miron developed a Lagrangian theory of relativity, which is a more generalized version of the Finslerian theory of relativity. In his theory, four kinds of new Einstein-like equations and two kinds of conservation-like laws are constructed [20].…”

Section: Motivation and Introductionmentioning

confidence: 99%

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Teleparallel Lagrange geometry and a unified field theory

Wanas

Youssef

Sid-Ahmed

2010

Class. Quantum Grav.

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Abstract. In this paper, we construct a field theory unifying gravity and electromagnetism in the context of Extended Absolute Parallelism (EAP-) geometry. This geometry combines, within its structure, the geometric richness of the tangent bundle and the mathematical simplicity of Absolute Parallelism (AP-) geometry. The constructed field theory is a generalization of the Generalized Field Theory (GFT) formulated by Mikhail and Wanas. The theory obtained is purely geometric. The horizontal (resp. vertical) field equations are derived by applying the Euler-Lagrange equations to an appropriate horizontal (resp. vertical) scalar Lagrangian. The symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Einstein's field equations in which the horizontal (resp. vertical) energy-momentum tensor is purely geometric. The skew-symmetric part of the resulting horizontal (resp. vertical) field equations gives rise to a generalized form of Maxwell equations in which the electromagnetic field is purely geometric. Some interesting special cases, which reveal the role of the nonlinear connection in the obtained field equations, are examined. Finally, the condition under which our constructed field equations reduce to the GFT is explicitly established.

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Section: Discussion and Concluding Remarksmentioning

confidence: 89%

Section: Discussion and Concluding Remarksmentioning

confidence: 99%

“…The material covered in the present section may be found in [19] and [20]. Some related topics may be found in [2].…”

Section: Basic Prelimenariesmentioning

confidence: 99%

Section: Motivation and Introductionmentioning

confidence: 99%

Section: Motivation and Introductionmentioning

confidence: 99%

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Teleparallel Lagrange geometry and a unified field theory

Wanas

Youssef

Sid-Ahmed

2010

Class. Quantum Grav.

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Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

Anco¹,

Vacaru²

2009

Journal of Geometry and Physics

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Methods in Riemann-Finsler geometry are applied to investigate bi-Hamiltonian structures and related mKdV hierarchies of soliton equations derived geometrically from regular Lagrangians and flows of non-stretching curves in tangent bundles. The total space geometry and nonholonomic flows of curves are defined by Lagrangian semisprays inducing canonical nonlinear connections (N-connections), Sasaki type metrics and linear connections. The simplest examples of such geometries are given by tangent bundles on Riemannian symmetric spaces G/SO(n) provided with an N-connection structure and an adapted metric, for which we elaborate a complete classification, and by generalized Lagrange spaces with constant Hessian. In this approach, bi-Hamiltonian structures are derived for geometric mechanical models and (pseudo) Riemannian metrics in gravity. The results yield horizontal/ vertical pairs of vector sine-Gordon equations and vector mKdV equations, with the corresponding geometric curve flows in the hierarchies described in an explicit form by nonholonomic wave maps and mKdV analogs of nonholonomic Schrödinger maps on a tangent bundle. * sanco@brocku.ca † Sergiu.Vacaru@gmail.com 1

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The entropy of Lagrange-Finsler spaces and Ricci flows

Vacaru

2009

Reports on Mathematical Physics

115

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We formulate a statistical analogy of regular Lagrange mechanics and Finsler geometry derived from Grisha Perelman's functionals and generalized for nonholonomic Ricci flows. Explicit constructions are elaborated when nonholonomically constrained flows of Riemann metrics result in Finsler like configurations, and inversely, when geometric mechanics is modelled on Riemann spaces with a preferred nonholonomic frame structure.

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Geometrical Theory of Embeddings of Vector Bundles

Cited by 9 publications

References 0 publications

Teleparallel Lagrange geometry and a unified field theory

Teleparallel Lagrange geometry and a unified field theory

Curve flows in Lagrange–Finsler geometry, bi-Hamiltonian structures and solitons

The entropy of Lagrange-Finsler spaces and Ricci flows

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