View source for Finsler geometry

A metric generalization of [[Riemannian geometry|Riemannian geometry]], where the general definition of the length of a vector is not necessarily given in the form of the square root of a quadratic form as in the Riemannian case. Such a generalization was first developed in the paper by P. Finsler [[#References|[1]]].

The object studied in Finsler geometry is a real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403901.png" />-dimensional differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403902.png" /> (of class at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403903.png" />) with a system of local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403904.png" />, on which a real non-negative scalar function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403905.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403906.png" /> independent variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403907.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403908.png" /> is given, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f0403909.png" /> are the components of the contravariant vectors tangent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039010.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039011.png" />. Suppose that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039012.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039013.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039014.png" />, and that in each tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039015.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039016.png" /> there is a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039017.png" /> such that, first, it is conical (in the sense that if any vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039018.png" /> tangent at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039019.png" /> belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039020.png" />, then every other tangent vector that is collinear with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039021.png" /> and tangent at the same point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039022.png" /> also belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039023.png" />), and secondly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039024.png" /> belongs to the class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039025.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039026.png" />. Non-zero vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039027.png" /> are called admissible. Suppose further that for every admissible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039028.png" /> and every point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039029.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039030.png" /></td> </tr></table>

and also that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039031.png" /> is positively homogeneous of degree one in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039032.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039033.png" /> for every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039034.png" /> and all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039035.png" /> and admissible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039036.png" />. Under these conditions the triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039037.png" /> is called a Finsler space, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039038.png" /> a Finsler metric. The value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039039.png" /> is interpreted as the length of the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039040.png" /> tangent at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039041.png" />.

If a Finsler space admits a coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039042.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039043.png" /> does not depend on these <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039044.png" />, then it is called a Minkowski space. The latter is related to a Finsler space in the same way as a Euclidean space is related to a Riemannian space. A Finsler space is called positive definite if one imposes a condition on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039045.png" /> that ensures that the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039046.png" /> is positive definite for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039047.png" /> and non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039048.png" />.

Imposing the condition of homogeneity in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039049.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039050.png" /> has a clear geometrical meaning from the point of view of invariant concepts in centro-affine spaces, the tangent spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039051.png" /> being such spaces. Namely, the ratio of the lengths of any two collinear vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039052.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039053.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039054.png" /> can be invariantly defined in the following way: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039055.png" />, which does not include any metric functions. Thus, the homogeneity condition imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039056.png" /> is a condition that the Finslerian definition of length is consistent with the particular centro-affine definition; the Finsler metric is needed to compare the lengths of non-collinear vectors.

The tensor

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039057.png" /></td> </tr></table>

is called the Finsler metric tensor. By Euler's theorem on homogeneous functions,

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039058.png" /></td> </tr></table>

where, by definition, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039059.png" />. These formulas are an immediate generalization of their Riemannian analogues, and follow from just the homogeneity condition. Finsler geometry reduces to Riemannian geometry in the case when the metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039060.png" /> is assumed to be independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039061.png" />. The last condition can be written in the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039062.png" />, where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039063.png" /></td> </tr></table>

is called the Cartan torsion tensor. It satisfies the identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039064.png" />. All Finsler relations can be turned into their Riemannian analogues by setting <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039065.png" />. The Christoffel symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039066.png" />, which are constructed from the Finsler metric tensor by the same formula as in Riemannian geometry, do not obey the transformation law of the coefficients of a connection. Nevertheless, one can construct the coefficients of a connection from the first derivatives of the Finsler metric tensor so that (as also in Riemannian geometry) the covariant derivative of the metric tensor vanishes. They are called the Cartan connection coefficients and have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039067.png" /></td> </tr></table>

where

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039068.png" /></td> </tr></table>

From the commutators of various covariant derivatives one can find expressions for the Finsler curvature tensors.

In each tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039069.png" /> the Finsler metric defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039070.png" />-dimensional hypersurface <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039071.png" /> (where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039072.png" /> are regarded as fixed and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039073.png" /> as varying), called the indicatrix. The indicatrix is formed by the ends of the unit tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039074.png" /> tangent at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039075.png" />. The fundamental significance of the concept of the indicatrix is already evident from the fact that, because the Finsler metric is homogeneous, the indicatrix at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039076.png" /> uniquely determines the form of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039077.png" /> at this point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039078.png" />. In the Riemannian case the indicatrix is a sphere. Generally speaking, the indicatrix of a Finsler space can be a surface of a rather general form. The Finsler metric tensor induces a Riemannian metric on the indicatrix, converting it into a Riemannian space. For each fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039079.png" /> the Finsler metric tensor is Riemannian in the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039080.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039081.png" />, where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039082.png" /> are fixed and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039083.png" /> are variable, is called the tangent Riemannian space at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039084.png" /> (a Euclidean space in the case of Riemannian geometry); the Riemannian curvature tensor of this space reduces to the expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039085.png" />. The indicatrix is a hypersurface that is imbedded in the tangent Riemannian space. The most immediate example of a Finsler metric function is the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039086.png" />-th root of a form of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039087.png" />.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039089.png" /> be real scalar functions of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039090.png" /> satisfying at each point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039091.png" /> the conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039092.png" /> or 2, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039093.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039094.png" /> be <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039095.png" /> linearly independent real covariant vector fields of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039097.png" />. Then for

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039098.png" /></td> </tr></table>

the curvature of the indicatrix is constant and equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f04039099.png" />, and for

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390100.png" /></td> </tr></table>

the curvature tensor of the indicatrix is zero. The determinant of the Finsler metric tensor is independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390101.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390103.png" />. If a Finsler space is positive definite and the indicatrix is a convex surface, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390104.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390105.png" /> is the only known example (1984) of a Finsler metric for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390106.png" /> (not counting the proper Riemannian case).

One can select special types of Finsler spaces by postulating some special form of the characteristic Finsler tensors. If the base manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390107.png" /> admits a field of frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390108.png" /> globally, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390109.png" /> is the metric function of some Minkowski space, then one can introduce a Finsler metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390110.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390111.png" /></td> </tr></table>

In this case the Finsler space and the metric are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390114.png" />-form. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390115.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390116.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390117.png" />-form when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390118.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390119.png" /> are constants. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390120.png" />-form spaces may be reckoned to be the simplest from the point of view of the way the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390121.png" /> enter in the metric. A Finsler space is called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390123.png" />-reducible if it is not Riemannian, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390124.png" /> and if the Cartan torsion tensor can be represented in the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390125.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390126.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390127.png" />-reducible spaces can have metrics of only two types: either the Kropina metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390128.png" />, or the Randers metric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390129.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390131.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390132.png" /> is a covariant vector field, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390133.png" /> is a Riemannian metric tensor. For example, the Lagrange function of an electric test charge in a gravitational or electromagnetic field is a Randers metric. The Finsler metric tensor corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390134.png" /> has signature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390135.png" />, which makes it of interest in developing a Finslerian generalization of the general theory of relativity; this signature is also encountered in the case of the choice of metric tensors of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390136.png" />. Such a generalization can be based on the concept of an oscillating Riemannian space to a Finsler space, according to which the Finsler metric tensor associates with each vector field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390137.png" /> the so-called osculating Riemannian metric tensor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390138.png" />. Choosing tensor fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390139.png" /> depending only on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390140.png" />, from which one constructs the Finsler metric according to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390141.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390142.png" /> is a scalar function, one can regard the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390143.png" /> as genuine gravitational field variables. The Finslerian geometrization of space-time also makes it possible to develop a theory of physical fields with various internal symmetries, relying on the concept of the group of transformations of the tangent vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f040/f040390/f040390144.png" /> that leave the Finsler metric invariant.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  P. Finsler,   "Ueber Kurven und Flächen in allgemeinen Räumen" , Göttingen  (1918)  (Dissertation)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  H. Rund,   "The differential geometry of Finsler spaces" , Springer  (1959)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  G.S. Asanov,   "Finsler geometry, relativity and gauge theories" , Reidel  (1985)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  M. Matsumoto,   "Foundations of Finsler geometry and special Finsler spaces" , Kaiseisha Press  (1986)</TD></TR></table>