# Finsler Geometry

## 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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So, people try to establish a modified gravitational theory described by Finsler geometry. Finsler geometry is the most general differential geometry, which regards Riemannian geometry as its special case, and it is just Riemannian geometry without quadratic restriction .
Up to now, Finsler geometry has developed rapidly in its global and analytic aspects.
In 1980 Professor Radu Miron initiates at the University of Brasov The First National Seminar on Finsler Geometry, which was held every two years ever since, and where Prof.
In this context, Finsler geometry is a very strong tool than well-known Riemann geometry for modeling physical phenomena that are genuinely asymmetric and/or nonisotropic [23-26].
Today, as per differential geometry and topology, both Riemannian and non-Riemannian geometry (such as Finsler geometry) can be used in General Relativity to understand better its geometric-folitional structure (such as Riemannian sub-manifolds and singular spaces) as well as its extensions (most ontologically and epistemologically unique, though, would be General Relativity's orthometric extensions--not just any extension--as I have alluded to elsewhere).
Projective Finsler geometry studies equivalent Finsler metrics on the same manifold with the same geodesics as points .
Bejancu: Finsler Geometry and Applications, Ellis Horwood Ltd., 1990.
Linfan Mao [4, 5] showed that SG are generalizations of Pseudo-Manifold Geometries, which in their turn are generalizations of Finsler Geometry, and which in its turn is a generalization of Riemann Geometry.
Matsumoto, Foundations of Finsler geometry and Special Finsler spaces, Kaiseisha press, Otsu, Saikawa 1986.
An important special case is when [F.sup.2] = [g.sub.ij] (x) [dx.sup.i][dx.sup.j] Historical developments have conferred the name Riemannian geometry to this case while the general case, Riemannian geometry without the quadratic restriction, has been known as Finsler geometry .
These proceedings of the September 2005 symposium reflect the respect participants felt for their late colleague Makato Matsumoto and his work in Finsler geometry. 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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