ABSTRACT Lagrange manifolds, as developed by R. Miron and M. Anastasiei, are generalizations of F... more ABSTRACT Lagrange manifolds, as developed by R. Miron and M. Anastasiei, are generalizations of Finsler manifolds. Diffusion theories of stochastic parallel transport and stochastic development, recently extended to Finsler manifolds, cannot be further broadened, in general, to Lagrange manifolds. However, for certain classes of Lagrange spaces some extension is possible. Here a certain connection in Lagrange space, called a conservative connection, is defined which allows progress in the extension of diffusion theory. A generalization of the Laplace-Beltrami operator for some of these same Lagrange spaces also results.
Proof: Let’s use mathematical induction. The inequality is true for n = 1. Now assume that it is ... more Proof: Let’s use mathematical induction. The inequality is true for n = 1. Now assume that it is true for n = k, and let’s show that it remains true for n = k + 1. Let x1, . . . , xk, xk+1 ∈ I and let λ1, . . . , λk, λk+1 ≥ 0 with λ1 + λ2 + . . . + λk + λk+1 = 1. At least one of λ1, λ2, . . . , λk+1 must be less then 1 (otherwise the inequality is trivial). Without loss of generality, let λk+1 < 1 and u = λ1 1−λk+1 x1 + . . . + λk 1−λk+1 xk. We have
The Theory of Finslerian Laplacians and Applications, 1998
A Lagrangian on n-dim C ∞ manifold M n is a real-valued function and continuous on the zero secti... more A Lagrangian on n-dim C ∞ manifold M n is a real-valued function and continuous on the zero section of π: TM n → M. Introduce a metric tensor by If we require max rank on then L is regular
Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated... more Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian L . In this paper we prove that ( M , L ) is of scalar flag curvature k if and only if the equation L ξ g + k λ L ξ g ˆ = 0 holds on Γ ( I λ M ) , the Lie algebra of tangent vector fields to the λ -indicatrix bundle I λ M , where g and g ˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.
Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated... more Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian L . In this paper we prove that ( M , L ) is of scalar flag curvature k if and only if the equation L ξ g + k λ L ξ g ˆ = 0 holds on Γ ( I λ M ) , the Lie algebra of tangent vector fields to the λ -indicatrix bundle I λ M , where g and g ˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.
The study of the geometry of Lagrange and Hamilton manifolds, using methods from Finsler geometry... more The study of the geometry of Lagrange and Hamilton manifolds, using methods from Finsler geometry, was initiated by R. Miron [An. Ştiint. Univ. Al. I. Cuza Iaşi, N. Ser., Sect. IA 32, 37-62 (1986; Zbl 0619.53021)]. He proved that Lagrange spaces L n (M,L(x,y)) and Hamilton spaces H n (M,H(x,p)) are locally related by Legendre transformations. This determines the notion of ℒ-duality between the manifolds L n and H n . In the paper under review, the authors exhaustively study the concept of ℒ-duality, obtaining the main geometrical object fields of the Hamilton manifold from those of the Lagrange manifold L n and conversely. The following topics are investigated: ℒ-duality of the Euler-Lagrange equations and Hamilton-Jacobi equations; ℒ-duality of canonical nonlinear connections; ℒ-duality of canonical metrical linear d-connections, etc. This idea is extremely useful for the study of the geometry of Cartan manifolds C n (which are not areal spaces) as ℒ-dual of the geometry of Finsler...
This is a continuation of the authors’ paper [D. Hrimiuc and H. Shimada, Nonlinear World 3, 613-6... more This is a continuation of the authors’ paper [D. Hrimiuc and H. Shimada, Nonlinear World 3, 613-641 (1996; Zbl 0894.53029)]. In the present paper they extend their theories to Cartan spaces, finding some properties of the Berwald connection, Berwald spaces, locally Minkowski spaces, etc., by using duality. The geometry of Cartan spaces can be used effectively to study Finsler spaces. For instance, it is shown that a Cartan space is of constant curvature if and only if every associated Finsler space is of constant curvature. The 𝔏-dual of a Finsler space with Kropina metric is a Cartan space with Randers metric.
ABSTRACT Lagrange manifolds, as developed by R. Miron and M. Anastasiei, are generalizations of F... more ABSTRACT Lagrange manifolds, as developed by R. Miron and M. Anastasiei, are generalizations of Finsler manifolds. Diffusion theories of stochastic parallel transport and stochastic development, recently extended to Finsler manifolds, cannot be further broadened, in general, to Lagrange manifolds. However, for certain classes of Lagrange spaces some extension is possible. Here a certain connection in Lagrange space, called a conservative connection, is defined which allows progress in the extension of diffusion theory. A generalization of the Laplace-Beltrami operator for some of these same Lagrange spaces also results.
Proof: Let’s use mathematical induction. The inequality is true for n = 1. Now assume that it is ... more Proof: Let’s use mathematical induction. The inequality is true for n = 1. Now assume that it is true for n = k, and let’s show that it remains true for n = k + 1. Let x1, . . . , xk, xk+1 ∈ I and let λ1, . . . , λk, λk+1 ≥ 0 with λ1 + λ2 + . . . + λk + λk+1 = 1. At least one of λ1, λ2, . . . , λk+1 must be less then 1 (otherwise the inequality is trivial). Without loss of generality, let λk+1 < 1 and u = λ1 1−λk+1 x1 + . . . + λk 1−λk+1 xk. We have
The Theory of Finslerian Laplacians and Applications, 1998
A Lagrangian on n-dim C ∞ manifold M n is a real-valued function and continuous on the zero secti... more A Lagrangian on n-dim C ∞ manifold M n is a real-valued function and continuous on the zero section of π: TM n → M. Introduce a metric tensor by If we require max rank on then L is regular
Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated... more Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian L . In this paper we prove that ( M , L ) is of scalar flag curvature k if and only if the equation L ξ g + k λ L ξ g ˆ = 0 holds on Γ ( I λ M ) , the Lie algebra of tangent vector fields to the λ -indicatrix bundle I λ M , where g and g ˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.
Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated... more Abstract Let ( M , L ) be a pseudo-Finsler manifold, ξ the geodesic spray vector field associated to the non-degenerate, 2-positively homogeneous Lagrangian L . In this paper we prove that ( M , L ) is of scalar flag curvature k if and only if the equation L ξ g + k λ L ξ g ˆ = 0 holds on Γ ( I λ M ) , the Lie algebra of tangent vector fields to the λ -indicatrix bundle I λ M , where g and g ˆ are pseudo-Riemannian metrics on the vertical and respectively on the horizontal subbundle. Also, we prove that any pseudo-Finsler manifold is of scalar flag curvature at any point of the light cone.
The study of the geometry of Lagrange and Hamilton manifolds, using methods from Finsler geometry... more The study of the geometry of Lagrange and Hamilton manifolds, using methods from Finsler geometry, was initiated by R. Miron [An. Ştiint. Univ. Al. I. Cuza Iaşi, N. Ser., Sect. IA 32, 37-62 (1986; Zbl 0619.53021)]. He proved that Lagrange spaces L n (M,L(x,y)) and Hamilton spaces H n (M,H(x,p)) are locally related by Legendre transformations. This determines the notion of ℒ-duality between the manifolds L n and H n . In the paper under review, the authors exhaustively study the concept of ℒ-duality, obtaining the main geometrical object fields of the Hamilton manifold from those of the Lagrange manifold L n and conversely. The following topics are investigated: ℒ-duality of the Euler-Lagrange equations and Hamilton-Jacobi equations; ℒ-duality of canonical nonlinear connections; ℒ-duality of canonical metrical linear d-connections, etc. This idea is extremely useful for the study of the geometry of Cartan manifolds C n (which are not areal spaces) as ℒ-dual of the geometry of Finsler...
This is a continuation of the authors’ paper [D. Hrimiuc and H. Shimada, Nonlinear World 3, 613-6... more This is a continuation of the authors’ paper [D. Hrimiuc and H. Shimada, Nonlinear World 3, 613-641 (1996; Zbl 0894.53029)]. In the present paper they extend their theories to Cartan spaces, finding some properties of the Berwald connection, Berwald spaces, locally Minkowski spaces, etc., by using duality. The geometry of Cartan spaces can be used effectively to study Finsler spaces. For instance, it is shown that a Cartan space is of constant curvature if and only if every associated Finsler space is of constant curvature. The 𝔏-dual of a Finsler space with Kropina metric is a Cartan space with Randers metric.
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