# 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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multiplication loops of locally compact topological translation planes; Lie groups which are the groups topologically generated by all left and right translations of topological loops; the inverse problem of the calculus of variations for second order ordinary differential equations: existence of variational multipliers, in particular, of multipliers satisfying the Finsler homogeneity conditions, and Riemannian and Finsler metrizability; metric structures associated with Lagrangians and Finsler functions variational structures in Finsler geometry and applications in physics (general relativity, Feynmam integral); Hamiltonian structures for homogeneous Lagrangians.
Bejancu: Finsler Geometry and Applications, Ellis Horwood Ltd.
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.
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 [24].
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.
All of those materials have established the pseudo-manifold geometry and combinatorially Finsler geometry or Riemannian geometry.
Our primary aim is to develop a new foundational world-geometry based on the intuitive notion of a novel, fully naturalized kind of Finsler geometry, which extensively mimics the Eulerian description of the mechanics of continuous media with special emphasis on the world-velocity field, in the sense that the whole space-time continuum itself is taken to be globally dynamic on both microscopic and macroscopic scales.
Finsler geometry is the most natural generalization of Riemannian geometry.
Let us pass to a new Finsler geometry on the base of the space of non-degenerated polynumbers [P.
which is equivalent to the torsion-free condition of the Chern-Rund connection in natural coordinates (see [3], [7], [8] and [9] for another connections of Finsler Geometry, and [2], [3] and [8] for the interesting application of Finsler Geometry).
A thorough study of Finsler geometry and Clifford algebras has been undertaken by Vacaru [81] where Clifford/spinor structures were defined with respect to Nonlinear connections associated with certain nonholonomic modifications of Riemann-Cartan gravity.

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