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We study the different types of Finsler space with

The common Finsler idea used by the physicists Beil and Holland is the existence of a nonholonomic frame on the vertical subbundle

We denote the tangent space at

A Finsler structure of

Regularity:

Positive homogeneity:

Strong convexity:

Throughout the project, the lowering and raising of indices are carried out by the fundamental tensor

The Cartan tensor vanishes if and only if

Using Euler’s theorem on homogeneous function, we can get useful property of the fundamental tensor

The Finsler space

An

In order to define

We start with a real

Denote by

A Generalized Lagrange metric (GL-metric) is a metric

In local coordinates, we denote

The quadratic form

If with respect to another system of local coordinates

Consider

Consider a

Consider also

If we take

A Finsler space

(1^{0}) If

(2^{0}) If

These classes of Finsler spaces play an important role in Finsler geometry and they are dual in the sense of Hrimiuc and Shimada [

For a Finsler space with

Now we will consider particular Finsler

Consider a Finsler space

If a particle in a space time moves along a curved, nongeodesic path, then it is said that the particle is under the influence of some external force. In such a case, an external force term is added to the equation of motions to explain the path of motion. Alternative point of view is that motion can be explained by a new metric, which would result from a gauge transformation. In this way, physical force fields can be geometrized, and general relativistic idea of space time curvature determining the path of the particle will also include fields other than gravitation. For this purpose, a class of gauge transformations which act on tangent space is considered. There are actually several ways to introduce Finsler geometry. Probably the most common way is just to assume a certain form for the metric function

The Gauge transformation is defined as follows:

There is a way around this homogeneity problem [

The gauge transformation is used to get the Finsler space. The connections given so far describe the transition to that space. The coordinate transformation deals with the properties of the resulting Finsler space. It describes the translation (sometimes called the transplantation) as one moves from one point to another in the space. The coordinate basis of

The behavior of the metric under the coordinate transformation is, in the adapted basis,

It is now time to get some specific physics using the above developments. There are several gauge transformations which might give useful results. One of them is now examined and compared. They are given by

It will be assumed that the vector

It is not difficult to show how a transformation like (

The potential

In general,

The transformation connection is easy to derive:

This means that a purely geometric derivation of electromagnetism has been developed.

In the present theory, all potentials are included in a metric which transforms like a metric. All fields are included in a connection which transforms like a connection. The equations of motion are geodesic equations. The energy-momentum for all fields is derived from a curvature.

By way of comparison, consider three other gauge transformations which produce Finsler metrics which can be related to the one just studied. One transformation is

In 1982, Holland studied a unified formalism that uses nonholonomic frames on space time arising from consideration of a charged particle moving in an external electromagnetic field [

The authors declare that they have no competing interests.