View source for Berwald connection

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103901.png" /> denote the [[Tangent bundle|tangent bundle]] of a smooth <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103902.png" />-dimensional [[Manifold|manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103903.png" />, with zero-section removed. In [[Finsler geometry|Finsler geometry]], one starts with a smooth metric function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103904.png" /> and its associated metric tensor, given locally by

<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/b/b110/b110390/b1103905.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103906.png" /> are the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103907.png" /> coordinates (positions and velocities) and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103908.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b1103909.png" /> denote partial differentials with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039011.png" />, respectively. It is assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039012.png" /> is non-singular on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039013.png" /> and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039014.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039015.png" /> extend continuously to the entire tangent bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039016.png" />. The pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039017.png" /> is called a Finsler space. The Euler–Lagrange equations (cf. [[Euler–Lagrange equation|Euler–Lagrange equation]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039018.png" /> describe geodesics (cf. [[Geodesic line|Geodesic line]]) and have the local description

<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/b/b110/b110390/b11039019.png" /></td> </tr></table>

where the differential of arc length is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039021.png" /> are the usual Levi-Cività (or Christoffel) symbols (cf. [[Christoffel symbol|Christoffel symbol]]) in terms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039022.png" />, its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039023.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039024.png" />. Note that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039025.png" /> depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039026.png" />. This is not the case in [[Riemannian geometry|Riemannian geometry]], where they are the coefficients of a unique, metric compatible, symmetric connection. In Finsler geometry there are several important connections, but <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039027.png" /> itself is not a [[Connection|connection]]. One way to proceed is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039028.png" /> and form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039030.png" />. It can be readily proved that the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039031.png" /> transform like a classical [[Affine connection|affine connection]], in spite of their dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039032.png" />, i.e.

<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/b/b110/b110390/b11039033.png" /></td> </tr></table>

Also, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039034.png" /> have a transformation law induced from that of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039035.png" />, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039036.png" />, by the [[Euler theorem|Euler theorem]] on homogeneous functions. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039037.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039039.png" /> are positively homogeneous in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039040.png" /> of degree two, one and zero, respectively. The triple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039041.png" /> is an example of a pre-Finsler connection [[#References|[a1]]], <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039042.png" />, meaning that:

1) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039043.png" /> transform just like the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039044.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039045.png" /> above (they are called the coefficients of the pre-Finsler connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039046.png" />);

2) the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039047.png" /> functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039048.png" /> transform just like <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039049.png" /> (they are called the coefficients of a non-linear connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039050.png" />) and

3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039051.png" /> is a tensor (cf. [[Tensor calculus|Tensor calculus]]) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039052.png" />.

Using these local expressions one can further introduce the vertical covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039053.png" /> and the horizontal covariant derivative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039054.png" />, as follows: for any contravariant vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039055.png" />, set

1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039056.png" /> and

2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039057.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039058.png" /> is the Finsler delta-derivative operator on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039059.png" /> corresponding to the non-linear connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039060.png" />. The important thing is that for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039062.png" /> is a covariant vector. Similar rules for higher-order tensors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039063.png" /> are just what one expects and all of the above have global descriptions.

The Okada theorem states that for a pre–Finsler connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039064.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039065.png" /> such that:

<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/b/b110/b110390/b11039066.png" /></td> </tr></table>

<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/b/b110/b110390/b11039067.png" /></td> </tr></table>

one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039068.png" />. The pre-Finsler connection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039069.png" /> is the so-called Berwald connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039070.png" />.

==Curvature of the Berwald connection.==
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039071.png" /> is a contravariant vector, then

<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/b/b110/b110390/b11039072.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039073.png" /> defines the so-called (HV)-curvature, also known as the spray curvature or Douglas tensor [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]) of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039074.png" />. Also,

<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/b/b110/b110390/b11039075.png" /></td> </tr></table>

where the Berwald curvature tensor is

<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/b/b110/b110390/b11039076.png" /></td> </tr></table>

and the VH-torsion tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039077.png" /> is

<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/b/b110/b110390/b11039078.png" /></td> </tr></table>

Here, the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039079.png" /> denotes that the entire expression before it is to be rewritten with the indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039081.png" /> interchanged.

A fundamental result in Berwald geometry is that both <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039083.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039084.png" /> is locally Minkowski. (Being locally Minkowski means that there is an admissible change of coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039085.png" /> so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039086.png" /> is actually independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039087.png" />.) Consequently, the geodesics in such a space have the local expression <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039088.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039089.png" />.

Now, generally, in Berwald theory one has

<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/b/b110/b110390/b11039090.png" /></td> </tr></table>

whereas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039091.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/b/b110/b110390/b11039092.png" /></td> </tr></table>

so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039093.png" /> is completely determined by the so-called Berwald–Gauss curvature <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039094.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039095.png" />. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039096.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039097.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039098.png" /> is positive definite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039099.png" /> otherwise. The pair of contravariant vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390100.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390101.png" />, is called the Berwald frame. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390102.png" /> are normal vectors and are oriented. They are both of unit length and orthogonal relative to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390103.png" />. Of course, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390105.png" />. The scalar invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390106.png" /> is positively homogeneous of degree zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390107.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390108.png" /> everywhere, then the geodesics of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390109.png" /> are Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]); if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b110390110.png" /> everywhere, they are unstable [[#References|[a1]]], [[#References|[a4]]].

See also [[Berwald space|Berwald space]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.L. Antonelli,   R.S. Ingarden,   M. Matsumoto,   "The theory of sprays and Finsler spaces with applications in physics and biology" , Kluwer Acad. Publ.  (1993)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  P.L. Antonelli,   T. (eds.) Zastawniak,   "Lagrange geometry, Finsler spaces and noise applied in biology and physics"  ''Math. and Comput. Mod. (Special Issue)'' , '''20'''  (1994)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  M. Matsumoto,   "Foundations of Finsler geometry and special Finsler spaces" , Kaiseisha Press  (1986)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H. Rund,   "The differential geometry of Finsler spaces" , Springer  (1959)</TD></TR></table>