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 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 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).

STABLE HARMONIC MAPS BETWEEN FINSLER MANIFOLDS AND SSU MANIFOLDS

2012 ◽  
Vol 14 (03) ◽  
pp. 1250015 ◽  
Author(s):  
JINTANG LI
Keyword(s):  
Harmonic Maps ◽  
Harmonic Map ◽  
American Mathematical Society ◽  
Variational Formula ◽  
Finsler Manifold ◽  
Finsler Manifolds ◽  
Recent Developments ◽  
Riemannian Case ◽  
The Stability ◽  
The Second Variation

Using the properties of Cartan tensor, we rewrite the second variation formula for harmonic maps between Finsler manifolds, and we prove that there is no non-degenerate stable harmonic map from a compact SSU manifold to any Finsler manifold, which is obtained by Howard and Wei for the Riemannian case. We also include a proof of a theorem of Shen–Wei which states that there is no non-degenerate stable harmonic map from a compact Finsler manifold to any SSU manifold, by the same second variational formula (see Eq. (2.1) in [Y. B. Shen and S. W. Wei, The stability of harmonic maps on Finster manifolds, Houston J. Math. 34 (2008) 1049–1056]) and the same method [S. W. Wei, An extrinsic average variational method, in Recent Developments in Geometry, Contemporary Mathematics, Vol. 101 (American Mathematical Society, Providence, RI, 1989), pp. 55–78].

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.

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 Projectively flat Finsler manifolds with infinite dimensional holonomy

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Zoltán Muzsnay ◽  
Péter T. Nagy
Keyword(s):  
Lie Group ◽  
Holonomy Group ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Compact Lie Group ◽  
Finsler Manifolds ◽  
Infinite Dimensional ◽  
Constant Flag Curvature ◽  
Projectively Flat

AbstractRecently, we developed a method for the study of holonomy properties of non-Riemannian Finsler manifolds and obtained that the holonomy group cannot be a compact Lie group if the Finsler manifold of dimension >2 has non-zero constant flag curvature. The purpose of this paper is to move further, exploring the holonomy properties of projectively flat Finsler manifolds of non-zero constant flag curvature. We prove in particular that projectively flat Randers and Bryant–Shen manifolds of non-zero constant flag curvature have infinite dimensional holonomy group.

SOME RESULTS ON FUNDAMENTAL GROUPS AND BETTI NUMBERS OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (06) ◽  
pp. 1250063 ◽  
Author(s):  
YIBING SHEN ◽  
WEI ZHAO
Keyword(s):  
Ricci Curvature ◽  
Betti Number ◽  
Betti Numbers ◽  
Finsler Manifold ◽  
Fundamental Groups ◽  
Finsler Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Finiteness Theorems ◽  
The Relationship

In this paper the relationship between the Ricci curvature and the fundamental groups of Finsler manifolds are studied. We give an estimate of the first Betti number of a compact Finsler manifold. Two finiteness theorems for fundamental groups of compact Finsler manifolds are proved. Moreover, the growth of fundamental groups of Finsler manifolds with almost-nonnegative Ricci curvature are considered.

HORIZONTAL LAPLACIAN ON TANGENT BUNDLE OF FINSLER MANIFOLD WITH g-NATURAL METRIC

2012 ◽  
Vol 09 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI ◽  
CHUNPING ZHONG
Keyword(s):  
Tangent Bundle ◽  
Vector Fields ◽  
Vector Bundles ◽  
Killing Vector ◽  
Finsler Manifold ◽  
Order Differential Operator ◽  
Finsler Manifolds ◽  
Scalar And Vector Fields ◽  
Matsumoto Metric ◽  
Weitzenböck Formula

Recently the third author studied horizontal Laplacians in real Finsler vector bundles and complex Finsler manifolds. In this paper, we introduce a class of g-natural metrics Ga,b on the tangent bundle of a Finsler manifold (M, F) which generalizes the associated Sasaki–Matsumoto metric and Miron metric. We obtain the Weitzenböck formula of the horizontal Laplacian associated to Ga,b, which is a second-order differential operator for general forms on tangent bundle. Using the horizontal Laplacian associated to Ga,b, we give some characterizations of certain objects which are geometric interest (e.g. scalar and vector fields which are horizontal covariant constant) on the tangent bundle. Furthermore, Killing vector fields associated to Ga,b are investigated.

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