# Finsler Geometry

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## Finsler geometry

[′fin·slər jē′äm·ə·trē]
(mathematics)
The study of the geometry of a manifold in terms of the various possible metrics on it by means of Finsler structures.

## Finsler Geometry

the theory of Finsler spaces, in which the differential ds of the arc length of a curve depends on the point under consideration in the space and on the choice of direction at the point. In other words, Finsler geometry is a theory of spaces where lengths are measured in small steps and the scale of measurement depends on the point of the space and the choice of direction at the point. The concept of such spaces was first introduced by B. Riemann in 1854. The first detailed examination of the theory of the spaces was presented by the German mathematician P. Finsler in 1918. Finsler geometry is widely used in the calculus of variations and in theoretical physics.

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To prove Theorems 1, we need to obtain a Laplacian comparison theorem on the Finsler manifold and then follow the method of Schoen and Yau in  (see also ).
Shen, "A universal volume comparison theorem for Finsler manifolds and related results," Canadian Journal of Mathematics, vol.
The pair (M, F) is called Finsler manifold. In particular, if g does not depend on y, we recover the Riemannian geometry.
Mathematicians at the University of Malaga in Spain examine the subtleties of both the Cauch boundary for a generalized and possible non-symmetric distance, and the Gromov compactification for any, possibly incomplete, Finsler manifold. They also introduce the new Busemann compactification, relating it to the previous two completions, and give a full description of the causal boundary of any standard conformally stationary spacetime.
We say that the triple [F.sup.m] = (M, M' F) with satisfying ([F.sub.1]) and ([F.sup.2]) is a Finsler manifold and F is the fundamental function of [F.sup.m].
Let [F.sup.3] = (M, M, F) be a 3-dimensional Finsler manifold and C be a smooth curve in Ml given locally by the parametric Equations;
It is said to be a Finsler manifold if the length s of any curve t [right arrow] ([x.sup.1](t), ..., [x.sup.m](t)) (a [less than or equal to] t [less than or equal to] b) is given by an integral
A Finsler manifold M has a tangent bundle [pi]: TM [right arrow] M.
They cover differentiable manifolds, Finsler metrics, connections and curvatures, S-curvature, Riemann curvature, projective changes, comparison theorems, fundamental groups of Finsler manifolds, minimal immersions and harmonic maps, Einstein metrics, and miscellaneous topics.
Fueki: On Finsler manifolds with the Chern-Rund connection whose projectivised tangent bundle has the Sasaki type metric, Nonlinear Anal., Real World Appl., 10(2009), 191-202.
Contributors address such topics as two curvature-driven problems in Riemann-Finsler geometry, curvature properties of certain metrics, a connectiveness principle in positively curved Finsler manifolds, Riemann-Finsler surfaces, Finsler geometry in the tangent bundle, and topics in Finsler-inspired differential geometry such as perturbations of constant connection Wagner spaces, path geometries of almost-Grassmann structures, Ehresmann connections in relation to metrics and good metric derivatives and dynamical systems of the Lagrangian and Hamiltonian mechanical systems.

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