The Finsler geometry of the Teichmüller metric

2017 ◽  
Vol 3 (4) ◽  
pp. 1045-1057
Author(s):  
Weixu Su ◽  
Youliang Zhong
Keyword(s):  
Finsler Geometry ◽  
Teichmüller Metric

Teichmuller Geometry

2017 ◽  
Author(s):  
Benson Farb ◽  
Dan Margalit
Keyword(s):  
Uniqueness Theorem ◽  
Existence Theorems ◽  
Quadratic Differentials ◽  
Complex Structures ◽  
Metric Geometry ◽  
Quasiconformal Maps ◽  
Hyperbolic Structures ◽  
Teichmüller Metric ◽  
Uniqueness And Existence ◽  
Teichmüller Mapping

This chapter focuses on the metric geometry of Teichmüller space. It first explains how one can think of Teich(Sɡ) as the space of complex structures on Sɡ. To this end, the chapter defines quasiconformal maps between surfaces and presents a solution to the resulting Teichmüller's extremal problem. It also considers the correspondence between complex structures and hyperbolic structures, along with the Teichmüller mapping, Teichmüller metric, and the proof of Teichmüller's uniqueness and existence theorems. The fundamental connection between Teichmüller's theorems, holomorphic quadratic differentials, and measured foliations is discussed as well. Finally, the chapter describes the Grötzsch's problem, whose solution is tied to the proof of Teichmüller's uniqueness theorem.

scholarly journals Finsler geometry modeling of complex fluids: reduction in viscous resistance

2021 ◽  
Vol 1730 (1) ◽  
pp. 012036
Author(s):  
Masahiko Okumura ◽  
Ippei Homma ◽  
Shuta Noro ◽  
Hiroshi Koibuchi
Keyword(s):  
Complex Fluids ◽  
Finsler Geometry ◽  
Viscous Resistance ◽  
Geometry Modeling

scholarly journals Finsler geometry modeling for anisotropic diffusion in Turing patterns

2021 ◽  
Vol 1730 (1) ◽  
pp. 012035
Author(s):  
Hiroshi Koibuchi ◽  
Masahiko Okumura ◽  
Shuta Noro
Keyword(s):  
Anisotropic Diffusion ◽  
Finsler Geometry ◽  
Turing Patterns ◽  
Geometry Modeling

scholarly journals Finsler Geometry for Two-Parameter Weibull Distribution Function

2017 ◽  
Vol 2017 ◽  
pp. 1-6 ◽  
Author(s):  
Emrah Dokur ◽  
Salim Ceyhan ◽  
Mehmet Kurban
Keyword(s):  
Probability Density Function ◽  
Weibull Distribution ◽  
Probability Density ◽  
Density Function ◽  
Finsler Space ◽  
Finsler Geometry ◽  
Cumulative Probability ◽  
Two Dimensional ◽  
Energy Potential ◽  
Two Parameter

To construct the geometry in nonflat spaces in order to understand nature has great importance in terms of applied science. Finsler geometry allows accurate modeling and describing ability for asymmetric structures in this application area. In this paper, two-dimensional Finsler space metric function is obtained for Weibull distribution which is used in many applications in this area such as wind speed modeling. The metric definition for two-parameter Weibull probability density function which has shape (k) and scale (c) parameters in two-dimensional Finsler space is realized using a different approach by Finsler geometry. In addition, new probability and cumulative probability density functions based on Finsler geometry are proposed which can be used in many real world applications. For future studies, it is aimed at proposing more accurate models by using this novel approach than the models which have two-parameter Weibull probability density function, especially used for determination of wind energy potential of a region.

ON THE GAUSS–BONNET FORMULA IN RIEMANN–FINSLER GEOMETRY

2002 ◽  
Vol 34 (3) ◽  
pp. 329-340 ◽  
Author(s):  
BRAD LACKEY
Keyword(s):  
Finsler Space ◽  
Finsler Geometry ◽  
Euler Class ◽  
Constant Volume ◽  
Finsler Spaces ◽  
Chern Connection ◽  
Civita Connection ◽  
Levi Civita Connection ◽  
Riemannian Case ◽  
Torsion Free

Using Chern's method of transgression, the Euler class of a compact orientable Riemann–Finsler space is represented by polynomials in the connection and curvature matrices of a torsion-free connection. When using the Chern connection (and hence the Christoffel–Levi–Civita connection in the Riemannian case), this result extends the Gauss–Bonnet formula of Bao and Chern to Finsler spaces whose indicatrices need not have constant volume.

GEOMETRY OF SPACETIME AND FINSLER GEOMETRY

2009 ◽  
Vol 24 (08n09) ◽  
pp. 1678-1685 ◽  
Author(s):  
REZA TAVAKOL
Keyword(s):  
General Relativity ◽  
String Theory ◽  
Riemannian Geometry ◽  
Lorentz Invariance ◽  
Finsler Geometry ◽  
Spacetime Geometry ◽  
Test Particles ◽  
Light Rays ◽  
Recent Developments ◽  
Underlying Geometry

A central assumption in general relativity is that the underlying geometry of spacetime is pseudo-Riemannian. Given the recent attempts at generalizations of general relativity, motivated both by theoretical and observational considerations, an important question is whether the spacetime geometry can also be made more general and yet still remain compatible with observations? Here I briefly summarize some earlier results which demonstrate that there are special classes of Finsler geometry, which is a natural metrical generalization of the Riemannian geometry, that are strictly compatible with the observations regarding the motion of idealised test particles and light rays. I also briefly summarize some recent attempts at employing Finsler geometries motivated by more recent developments such as those in String theory, whereby Lorentz invariance is partially broken.

scholarly journals Fiberwise convexity of Hill’s lunar problem

2017 ◽  
Vol 09 (04) ◽  
pp. 571-630 ◽  
Author(s):  
Junyoung Lee
Keyword(s):  
Energy Level ◽  
Contact Structure ◽  
Finsler Geometry ◽  
Critical Energy ◽  
Critical Value ◽  
Finsler Metrics ◽  
Tight Contact ◽  
Systolic Inequality ◽  
Geodesic Problem ◽  
Hill's Lunar Problem

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on [Formula: see text] with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on [Formula: see text]. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

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