scholarly journals On the extremal compatible linear connection of a generalized Berwald manifold

2021 ◽  
Author(s):  
Csaba Vincze
Keyword(s):  
Compatibility Condition ◽  
Torsion Tensor ◽  
Linear Connection ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Reference Element ◽  
Finsler Manifolds ◽  
Linear Connections ◽  
Tangent Vectors

AbstractGeneralized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

scholarly journals Geodesic Random Walks, Diffusion Processes and Brownian Motion on Finsler Manifolds

2021 ◽  
Author(s):  
Tianyu Ma ◽  
Vladimir S. Matveev ◽  
Ilya Pavlyukevich
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Random Walks ◽  
Diffusion Processes ◽  
Riemannian Metric ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Bounded Geometry ◽  
And Brownian Motion

AbstractWe show that geodesic random walks on a complete Finsler manifold of bounded geometry converge to a diffusion process which is, up to a drift, the Brownian motion corresponding to a Riemannian metric.

FINSLER LAPLACIANS AND MINIMAL-ENERGY MAPS

2000 ◽  
Vol 11 (01) ◽  
pp. 1-13 ◽  
Author(s):  
PAUL CENTORE
Keyword(s):  
Riemannian Metric ◽  
Energy Functional ◽  
Mean Value ◽  
Finsler Metric ◽  
Harmonic Mappings ◽  
Laplacian Operator ◽  
Minimal Energy ◽  
Finsler Manifolds ◽  
The Mean ◽  
Energy Maps

For any Finsler manifold, there is a geometrically natural Laplacian operator, called the mean-value Laplacian, which generalizes the Riemannian Laplacian. We show that, like the Riemannian Laplacian (for functions), we can see the vanishing of the mean-value Laplacian at some function f as the minimizing of an energy functional e(f) by f. This energy functional e depends on a Riemannian metric canonically associated to the Finsler metric and on a canonically associated volume form. We relate this construction to a more general construction of Jost, and define a notion of harmonic mappings between Finsler manifolds.

scholarly journals Harmonic Coordinates for the Nonlinear Finsler Laplacian and Some Regularity Results for Berwald Metrics

Axioms ◽  
2019 ◽  
Vol 8 (3) ◽  
pp. 83 ◽  
Author(s):  
Erasmo Caponio ◽  
Antonio Masiello
Keyword(s):  
Finsler Manifold ◽  
Finsler Metric ◽  
Optimal Regularity ◽  
Finsler Manifolds ◽  
Fundamental Tensor ◽  
Harmonic Coordinates ◽  
Riemannian Case ◽  
Regularity Results

We prove existence of harmonic coordinates for the nonlinear Laplacian of a Finsler manifold and apply them in a proof of the Myers–Steenrod theorem for Finsler manifolds. Different from the Riemannian case, these coordinates are not suitable for studying optimal regularity of the fundamental tensor, nevertheless, we obtain some partial results in this direction when the Finsler metric is Berwald.

scholarly journals Some Liouville Theorems on Finsler Manifolds

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 351 ◽  
Author(s):  
Minqiu Wang ◽  
Songting Yin
Keyword(s):  
Recent Literature ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Liouville Type ◽  
Finsler Manifolds ◽  
Liouville Theorems ◽  
Superharmonic Functions ◽  
Liouville Type Theorems ◽  
Geometric Relationship

We give some Liouville type theorems of L p harmonic (resp. subharmonic, superharmonic) functions on a complete noncompact Finsler manifold. Using the geometric relationship between a Finsler metric and its reverse metric, we remove some restrictions on the reversibility. These improve the recent literature (Zhang and Xia, 2014).

Vertical Chern Type Classes on Complex Finsler Bundles

2011 ◽  
Vol 57 (2) ◽  
pp. 377-386
Author(s):  
Cristian Ida
Keyword(s):  
Differential Forms ◽  
Linear Connection ◽  
Curvature Form ◽  
Cohomology Groups ◽  
Finsler Manifolds ◽  
Invariance Properties ◽  
Type Classes

Vertical Chern Type Classes on Complex Finsler BundlesIn the present paper, we define vertical Chern type classes on complex Finsler bundles, as an extension of thev-cohomology groups theory on complex Finsler manifolds. These classes are introduced in a classical way by using closed differential forms with respect to the conjugated vertical differential in terms of the vertical curvature form of Chern-Finsler linear connection. Also, some invariance properties of these classes are studied.

2-ρ-DERIVATIONS ON A ρ-ALGEBRA AND APPLICATIONS TO THE QUATERNIONIC ALGEBRA

2007 ◽  
Vol 04 (03) ◽  
pp. 457-469 ◽  
Author(s):  
CĂTĂLIN CIUPALĂ
Keyword(s):  
Linear Connection ◽  
Linear Connections

In this paper, we introduce 2-ρ-derivations on a ρ-algebra A, and define 2-linear connections on a ρ-bimodule M over A using these 2-derivations. Then we introduce and study the curvature of a linear connection. Our results are applied to the particular case of the quaternionic algebra ℍ.

ON PROJECTIVELY RELATED WARPED PRODUCT FINSLER MANIFOLDS

2011 ◽  
Vol 08 (05) ◽  
pp. 953-967 ◽  
Author(s):  
M. M. REZAII ◽  
Y. ALIPOUR-FAKHRI
Keyword(s):  
Warped Product ◽  
Sufficient Conditions ◽  
Linear Connection ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifolds ◽  
Non Linear ◽  
Necessary And Sufficient

Let 𝔽1 = (M1,F1) and 𝔽2 = (M2,F2) be two Finsler manifolds and let M = M1 × M2 and S is a spray in M. Also 𝔽 = (M1 × f M2, F) is a warped product Finsler manifolds, such that the function f : M1 → ℝ+ is not constant. In this paper, we define a non-linear connection on warped product 𝔽, and finally, we have presented some necessary and sufficient conditions under which the spray manifold (M1 × M2, S) is projectively equivalent to the warped product Finsler manifolds (M1 × f M2, F).

scholarly journals Prolongations of linear connections to the frame bundle

1983 ◽  
Vol 28 (3) ◽  
pp. 367-381
Author(s):  
Luis A. Cordero ◽  
Manuel de Leon
Keyword(s):  
Linear Connection ◽  
Frame Bundle ◽  
Linear Connections ◽  
Complete Lift

In this paper we construct the prolongation of a linear connection Γ on a manifold Μ to the bundle space of its frame bundle, and show that such prolongated connection coincides with the so-called complete lift of Γ to .

LAPLACIAN COMPARISON ON COMPLEX FINSLER MANIFOLDS AND ITS APPLICATIONS

2013 ◽  
Vol 24 (05) ◽  
pp. 1350034
Author(s):  
JINXIU XIAO ◽  
CHUNHUI QIU ◽  
QUN HE ◽  
ZHIHUA CHEN
Keyword(s):  
Maximum Principle ◽  
Distance Function ◽  
Finsler Metric ◽  
Finsler Manifolds ◽  
Complex Finsler Metric ◽  
Bonnet Theorem

By defining the Rund Laplacian, we obtain the first and the second holomorphic variation formulas for the strongly pseudoconvex complex Finsler metric. Using the holomorphic variation formulas, we get an estimate for the Levi forms of distance function on complex Finsler manifolds. Further, an estimate for the Rund Laplacians of distance function on strongly pseudoconvex complex Finsler manifolds is obtained. As its applications, we get the Bonnet theorem and maximum principle on complex Finsler manifolds.

scholarly journals On the Holonomy Groups of Linear Connections

1956 ◽  
Vol 10 ◽  
pp. 97-100 ◽  
Author(s):  
Jun-Ichi Hano ◽  
Hideki Ozeki
Keyword(s):  
Linear Group ◽  
Affine Space ◽  
General Linear Group ◽  
Affine Connection ◽  
Holonomy Group ◽  
Linear Connection ◽  
General Linear ◽  
Linear Connections ◽  
Holonomy Groups ◽  
Lie Subgroup

In this note we show in § 1, as the main result, that any connected Lie subgroup of the general linear group GL(n, R) can be realized as the holonomy group of a linear connection, i.e. the homogeneous holonomy group of the associeted affine connection, defined on an affine space of dimension n (n ≧ 2).

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