Geometry of Curves in Finsler Manifolds

Bejancu, Aurel; Farran, Hani Reda

doi:10.1007/978-94-015-9417-2_4

Cited by 7 publications

(65 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Finally, we note that a number of geometric ideas and methods applied in this section were considered in the approaches to the geometry of nonholonomic spaces and generalized Finsler-Lagrange geometry elaborated by the schools of G. Vranceanu and R. Miron and by A. Bejancu in Romania [30,31,2,3,23,24], see also Section 5. In these approaches it is possible to construct geometric models with metric compatible linear connections, which is important for developments connected with modern (non)commutative gravity and string theory [25,26].…”

Section: Example 21mentioning

confidence: 99%

“…The torsion (12) and curvature (71) (see Appendix) tensors in orthonormal component form with respect to (21) can be mapped into a distinguished orthonormal dual frame (23),…”

Section: X⌋γmentioning

confidence: 99%

“…This approach can be developed into a geometric formalism for mapping arbitrary (semi) Riemannian metrics [22] and regular Lagrange mechanical systems into bi-Hamiltonian structures and related solitonic equations following certain methods elaborated in the geometry of generalized Finsler and Lagrange spaces [2,3,23] and nonholonomic manifolds with applications in modern gravity [24,25,26].…”

Section: Introductionmentioning

confidence: 99%

“…7 This allows us to classify the N-anholonomic manifolds (and vector bundles) as having the same group and algebraic properties as pairs of symmetric Riemannian spaces of dimension n and m but nonholonomically soldered to the symmetric Riemannian space of dimension n + m. With respect to N-adapted orthonormal bases (23), with distinguished h-and vsubspaces, we obtain the same inner products and group and Lie algebra spaces as in (26).…”

mentioning

confidence: 98%

See 3 more Smart Citations

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

Anco¹,

Vacaru²

2009

Journal of Geometry and Physics

View full text Add to dashboard Cite

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

show abstract

Section: Example 21mentioning

confidence: 99%

“…The torsion (12) and curvature (71) (see Appendix) tensors in orthonormal component form with respect to (21) can be mapped into a distinguished orthonormal dual frame (23),…”

Section: X⌋γmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 98%

See 2 more Smart Citations

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

Anco¹,

Vacaru²

2009

Journal of Geometry and Physics

View full text Add to dashboard Cite

show abstract

“…It does not seem to us that the Asanov formulation of pseudo-Finsler geometries has anything to say about the situation at hand. While it is clear that in the general case we will want to invoke some form of pseudo-Finsler geometry, none of the extant formalisms are really suitable [9,10,11,12,13]. We hope that these brief comments might stimulate some interest in further developing this field.…”

Section: Multiple Metricsmentioning

confidence: 99%

Refringence, field theory and normal modes

Barceló

Liberati

Visser

2002

Class. Quantum Grav.

View full text Add to dashboard Cite

In a previous paper [gr-qc/0104001; Class. Quant. Grav. 18 (2001) [3595][3596][3597][3598][3599][3600][3601][3602][3603][3604][3605][3606][3607][3608][3609][3610] we have shown that the occurrence of curved spacetime "effective Lorentzian geometries" is a generic result of linearizing an arbitrary classical field theory around some non-trivial background configuration. This observation explains the ubiquitous nature of the "analog models" for general relativity that have recently been developed based on condensed matter physics. In the simple (single scalar field) situation analyzed in our previous paper, there is a single unique effective metric; more complicated situations can lead to bi-metric and multi-metric theories. In the present paper we will investigate the conditions required to keep the situation under control and compatible with experimenteither by enforcing a unique effective metric (as would be required to be strictly compatible with the Einstein Equivalence Principle), or at the worst by arranging things so that there are multiple metrics that are all "close" to each other (in order to be compatible with the Eötvös experiment). The algebraically most general situation leads to a physical model whose mathematical description requires an extension of the usual notion of Finsler geometry to a Lorentzian-signature pseudoFinsler geometry; while this is possibly of some interest in its own right, this particular case does not seem to be immediately relevant for either particle physics or gravitation. The key result is that wide classes of theories lend themselves to an effective metric description. This observation provides further evidence that the notion of "analog gravity" is rather generic.

show abstract

Radu Miron: The geometry of higher-order Hamilton spaces

Vacaru

2005

Gen Relativ Gravit

View full text Add to dashboard Cite

Radu Miron: The geometry of higher-order Hamilton spacesKluwer Academic Publishers, Dordrecht, 2003, 264 p., This monograph is the sixth one resulting from 50 years of research activity of the prominent Romanian school on Finsler geometry, Lagrange-Hamilton spaces, and their higher-order generalizations [1][2][3][4][5]. The new book presents an overview of the higher-order Hamilton spaces with applications to higher-order mechanics following the canonical non-linear connection (N-connection) and (semi-)spray formalism and the geometry of induced almost complex or contact Riemann-Cartan structures. The variational principles for higher-order Hamiltonians, HamiltonJacobi equations and related conservation laws and symmetries are described and investigated.In this review, I would like to discuss some relations of this original approach to geometry and physics and reflect on applications to modern gravity and string theory.There are two general directions towards geometrization of mechanics on the tangent or cotangent bundle and their higher-order extensions: The first one is based on the idea of describing classical mechanics and classical field theory in terms of symplectic geometry following certain procedures of geometrization of the Euler-Lagrange and Hamilton equations by using the (multi-)symplectic formalism, differential forms, jets, etc. (see, for instance, [6,7]). In a quite different way, in the second direction Lagrange and Hamilton mechanics was geometrized using Riemann-Finsler spaces and, in a more general context, certain types of non-holonomic manifolds enabled with non-integrable distributions defined by N-connections. The general ideas and methods originate from E. Cartan [8] and A. Kawaguchi [9, 10] who were the first to consider metric-compatible models of Finsler spaces and N-connection structures. In this way, the geometric constructions are not related to certain particular properties of the Euler-Lagrange, or

show abstract

Geometry of Curves in Finsler Manifolds

Cited by 7 publications

References 0 publications

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

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

Refringence, field theory and normal modes

Radu Miron: The geometry of higher-order Hamilton spaces

Contact Info

Product

Resources

About