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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
+
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<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>
+
{{TEX|auto}}
 +
{{TEX|done}}
  
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
+
Let  $  {\widetilde{T}  } M  ^ {n} $
 +
denote the [[Tangent bundle|tangent bundle]] of a smooth  $  n $-
 +
dimensional [[Manifold|manifold]] $  M  ^ {n} $,
 +
with zero-section removed. In [[Finsler geometry|Finsler geometry]], one starts with a smooth metric function  $  F : { {\widetilde{T}  } M  ^ {n} } \rightarrow {\mathbf R  ^ {1} } $
 +
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/b11039019.png" /></td> </tr></table>
+
$$
 +
g _ {ij }  ( x,y ) = {
 +
\frac{1}{2}
 +
} {\dot \partial  } _ {i} {\dot \partial  } _ {j} F  ^ {2} , \quad i,j = 1 \dots n,
 +
$$
  
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.
+
where $  ( x  ^ {i} ,y  ^ {i} ) $
 +
are the $  2n $
 +
coordinates (positions and velocities) and  $  \partial  _ {i} $
 +
and  $  {\dot \partial  } _ {j} $
 +
denote partial differentials with respect to  $  x  ^ {i} $
 +
and  $  y  ^ {j} $,
 +
respectively. It is assumed that  $  ( g _ {ij }  ) $
 +
is non-singular on  $  {\widetilde{T}  } M  ^ {n} $
 +
and that $  F $
 +
and  $  g _ {ij }  $
 +
extend continuously to the entire tangent bundle  $  TM  ^ {n} $.  
 +
The pair  $  ( M  ^ {n} ,F ) $
 +
is called a Finsler space. The Euler–Lagrange equations (cf. [[Euler–Lagrange equation|Euler–Lagrange equation]]) of $  ( M  ^ {n} ,F ) $
 +
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/b11039033.png" /></td> </tr></table>
+
$$
 +
{
 +
\frac{dy  ^ {i} }{ds }
 +
} + \gamma _ {jk }  ^ {i} ( x,b ) y  ^ {j} y  ^ {k} = 0, \quad {
 +
\frac{dx  ^ {i} }{ds }
 +
} = y  ^ {i} ,
 +
$$
  
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:
+
where the differential of arc length is  $  ds = F ( x,dx ) $
 +
and  $  \gamma _ {jk }  ^ {i} ( x,y ) $
 +
are the usual Levi-Cività (or Christoffel) symbols (cf. [[Christoffel symbol|Christoffel symbol]]) in terms of  $  g _ {ij }  ( x,y ) $,
 +
its inverse  $  g ^ {ij } ( x,y ) $
 +
and  $  \partial  _ {i} g _ {kl }  $.  
 +
Note that the  $  \gamma _ {jk }  ^ {i} $
 +
depend on  $  y $.  
 +
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  $  \gamma _ {jk }  ^ {i} ( x,y ) $
 +
itself is not a [[Connection|connection]]. One way to proceed is as follows. Let  $  G  ^ {i} = ( {1 / 2 } ) \gamma _ {jk }  ^ {i} y  ^ {j} y  ^ {k} $
 +
and form  $  G _ {j}  ^ {i} ( x,y ) = {\dot \partial  } _ {j} G  ^ {i} ( x,y ) $
 +
and $  G _ {jk }  ^ {i} ( x,y ) = {\dot \partial  } _ {k} G _ {j}  ^ {i} ( x,y ) $.  
 +
It can be readily proved that the  $  G _ {jk }  ^ {i} ( x,y ) $
 +
transform like a classical [[Affine connection|affine connection]], in spite of their dependence on  $  y $,
 +
i.e.
  
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" />);
+
$$
 +
G _ {jk }  ^ {i} X _ {b}  ^ {j} X _ {c}  ^ {k} = {\overline{G}\; } _ {bc }  ^ {a} X _ {a}  ^ {i} + {
 +
\frac{\partial  X  ^ {i} _ {b} }{\partial  {\overline{x}\; }  ^ {c} }
 +
} , \quad X  ^ {i} _ {b} = {
 +
\frac{\partial  X  ^ {i} }{\partial  {\overline{x}\; }  ^ {b} }
 +
} .
 +
$$
  
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
+
Also, the  $  G _ {j}  ^ {i} ( x,y ) $
 +
have a transformation law induced from that of  $  G _ {jk }  ^ {i} ( x,y ) $,
 +
because  $  G _ {j}  ^ {i} = G _ {jk }  ^ {i} y  ^ {k} $,
 +
by the [[Euler theorem|Euler theorem]] on homogeneous functions. Note that  $  G  ^ {i} $,
 +
$  G _ {j}  ^ {i} $
 +
and  $  G _ {jk }  ^ {i} $
 +
are positively homogeneous in  $  y  ^ {k} $
 +
of degree two, one and zero, respectively. The triple  $  B \Gamma = ( G _ {jk }  ^ {i} ( x,y ) ,G _ {j}  ^ {i} ( x,y ) ,0 ) $
 +
is an example of a pre-Finsler connection [[#References|[a1]]],  $  F \Gamma = ( F _ {jk }  ^ {i} ( x,y ) , N _ {j}  ^ {i} ( x,y ) ,V _ {jk }  ^ {i} ( x,y ) ) $,
 +
meaning that:
  
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" />.
+
1) the  $  F _ {jk }  ^ {i} ( x,y ) $
 +
transform just like the  $  n  ^ {3} $
 +
functions  $  G _ {jk }  ^ {i} ( x,y ) $
 +
above (they are called the coefficients of the pre-Finsler connection on  $  ( M  ^ {n} ,F ) $);
  
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
+
2) the $  n  ^ {2} $
 +
functions  $  N _ {j}  ^ {i} ( x,y ) $
 +
transform just like  $  G _ {j}  ^ {i} ( x,y ) $(
 +
they are called the coefficients of a non-linear connection on  $  {\widetilde{T}  } M  ^ {n} $)
 +
and
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039056.png" /> and
+
3) $  V _ {jk }  ^ {i} ( x,y ) $
 +
is a tensor (cf. [[Tensor calculus|Tensor calculus]]) on  $  M  ^ {n} $.
  
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.
+
Using these local expressions one can further introduce the vertical covariant derivative $  \nabla ^ {\textrm{ V } } $
 +
and the horizontal covariant derivative  $  \nabla ^ {\textrm{ H } } $,
 +
as follows: for any contravariant vector  $  A  ^ {r} ( x,y ) $,  
 +
set
  
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:
+
1)  $  \nabla _ {j} ^ {\textrm{ H } } A  ^ {i} = \delta _ {j} A  ^ {i} + A  ^ {r} F _ {rj }  ^ {i} $
 +
and
  
<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>
+
2)  $  \nabla _ {j} ^ {\textrm{ V } } A  ^ {i} = {\dot \partial  } _ {j} A  ^ {i} + A  ^ {r} V _ {rj }  ^ {i} $,
 +
where  $  \delta _ {i} = \partial  _ {i} - N _ {j}  ^ {r} {\dot \partial  } _ {r} $
 +
is the Finsler delta-derivative operator on  $  ( M  ^ {n} ,F ) $
 +
corresponding to the non-linear connection  $  N _ {j}  ^ {i} ( x,y ) $.
 +
The important thing is that for any function  $  f : { {\widetilde{T}  } M  ^ {n} } \rightarrow {\mathbf R  ^ {1} } $,
 +
$  \delta _ {i} f $
 +
is a covariant vector. Similar rules for higher-order tensors  $  A ( x,y ) $
 +
are just what one expects and all of the above have global descriptions.
  
<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>
+
The Okada theorem states that for a pre–Finsler connection  $  F \Gamma = ( F _ {jk }  ^ {i} , N _ {j}  ^ {i} , V _ {jk }  ^ {i} ) $
 +
on  $  ( M  ^ {n} ,F ) $
 +
such that:
  
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" />.
+
$$
 +
\nabla ^ {\textrm{ H } } F = 0, \quad F _ {jk }  ^ {i} = F _ {kj }  ^ {i} , \quad N _ {j}  ^ {i} = F _ {rj }  ^ {i} y  ^ {r} ,
 +
$$
 +
 
 +
$$
 +
{\dot \partial  } _ {k} N _ {j}  ^ {i} = F _ {kj }  ^ {i} , \quad V _ {jk }  ^ {i} = 0,
 +
$$
 +
 
 +
one has $  F \Gamma = B \Gamma = ( G _ {jk }  ^ {i} ,G _ {j}  ^ {i} ,0 ) $.  
 +
The pre-Finsler connection $  B \Gamma $
 +
is the so-called Berwald connection on $  ( M  ^ {n} ,F ) $.
  
 
==Curvature of the Berwald connection.==
 
==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
+
If $  A  ^ {i} ( x,y ) $
 +
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>
+
$$
 +
\nabla _ {k} ^ {\textrm{ V } } \nabla _ {j} ^ {\textrm{ H } } A  ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ V } } A  ^ {i} = A  ^ {r} G _ {rjk }  ^ {i} ,
 +
$$
  
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,
+
where $  G _ {rjk }  ^ {i} = \nabla _ {k} ^ {\textrm{ V } } G _ {rj }  ^ {i} $
 +
defines the so-called (HV)-curvature, also known as the spray curvature or Douglas tensor [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]) of $  B \Gamma $.  
 +
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>
+
$$
 +
\nabla _ {k} ^ {\textrm{ H } } \nabla _ {j} ^ {\textrm{ H } } A  ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ H } } A  ^ {i} = A  ^ {h} B _ {hjk }  ^ {i} - ( \nabla _ {l} ^ {\textrm{ V } } A  ^ {i} ) R _ {jk }  ^ {l} ,
 +
$$
  
 
where the Berwald curvature tensor is
 
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>
+
$$
 +
B _ {hjk }  ^ {i} = \partial  _ {k} G _ {hj }  ^ {i} - G _ {k}  ^ {r} ( {\dot \partial  } _ {r} G _ {hj }  ^ {i} ) + G _ {hj }  ^ {r} G _ {rk }  ^ {i} - ( {j / k } )
 +
$$
  
and the VH-torsion tensor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039077.png" /> is
+
and the VH-torsion tensor of $  B \Gamma $
 +
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>
+
$$
 +
R _ {jk }  ^ {l} = \partial  _ {k} G _ {j}  ^ {l} - G _ {jr }  ^ {l} G _ {k}  ^ {r} - ( {j / k } ) .
 +
$$
  
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.
+
Here, the symbol $  ( {j / k } ) $
 +
denotes that the entire expression before it is to be rewritten with the indices $  j $
 +
and $  k $
 +
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" />.
+
A fundamental result in Berwald geometry is that both $  B _ {hjk }  ^ {i} = 0 $
 +
and $  G _ {jkl }  ^ {i} = 0 $
 +
if and only if $  ( M  ^ {n} ,F ) $
 +
is locally Minkowski. (Being locally Minkowski means that there is an admissible change of coordinates $  x \rightarrow {\overline{x}\; } $
 +
so that $  F ( {\overline{x}\; } , {\overline{y}\; } ) $
 +
is actually independent of $  {\overline{x}\; }  ^ {i} $.)  
 +
Consequently, the geodesics in such a space have the local expression $  {\overline{x}\; }  ^ {i} = a  ^ {i} s + b ^ {i} $,  
 +
$  i = 1 \dots n $.
  
 
Now, generally, in Berwald theory one has
 
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>
+
$$
 +
R _ {jk }  ^ {i} = B _ {hjk }  ^ {i} y  ^ {h} ,
 +
$$
  
whereas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110390/b11039091.png" />,
+
whereas for $  n = 2 $,
  
<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>
+
$$
 +
R _ {jk }  ^ {i} = \epsilon FK m  ^ {i} ( l _ {j} m _ {k} - l _ {k} m _ {j} ) ,
 +
$$
  
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]]].
+
so that $  B _ {hjk }  ^ {i} $
 +
is completely determined by the so-called Berwald–Gauss curvature $  K ( x,y ) $
 +
of $  ( M  ^ {n} ,F ) $.  
 +
The number $  \epsilon $
 +
equals $  + 1 $
 +
if $  g _ {ij }  $
 +
is positive definite and $  - 1 $
 +
otherwise. The pair of contravariant vectors $  ( l  ^ {i} ,m  ^ {j} ) $,  
 +
where $  l  ^ {i} = { {y  ^ {i} } / F } $,  
 +
is called the Berwald frame. The $  m  ^ {i} $
 +
are normal vectors and are oriented. They are both of unit length and orthogonal relative to $  g _ {ij }  ( x,y ) $.  
 +
Of course, $  l _ {i} = g _ {ij }  l  ^ {j} $
 +
and $  m _ {i} = g _ {ij }  m  ^ {j} $.  
 +
The scalar invariant $  K ( x,y ) $
 +
is positively homogeneous of degree zero in $  y  ^ {i} $.  
 +
If  $  K > 0 $
 +
everywhere, then the geodesics of $  ( M  ^ {n} ,F ) $
 +
are Lyapunov stable (cf. [[Lyapunov stability|Lyapunov stability]]); if $  K \leq  0 $
 +
everywhere, they are unstable [[#References|[a1]]], [[#References|[a4]]].
  
 
See also [[Berwald space|Berwald space]].
 
See also [[Berwald space|Berwald space]].

Latest revision as of 10:58, 29 May 2020


Let $ {\widetilde{T} } M ^ {n} $ denote the tangent bundle of a smooth $ n $- dimensional manifold $ M ^ {n} $, with zero-section removed. In Finsler geometry, one starts with a smooth metric function $ F : { {\widetilde{T} } M ^ {n} } \rightarrow {\mathbf R ^ {1} } $ and its associated metric tensor, given locally by

$$ g _ {ij } ( x,y ) = { \frac{1}{2} } {\dot \partial } _ {i} {\dot \partial } _ {j} F ^ {2} , \quad i,j = 1 \dots n, $$

where $ ( x ^ {i} ,y ^ {i} ) $ are the $ 2n $ coordinates (positions and velocities) and $ \partial _ {i} $ and $ {\dot \partial } _ {j} $ denote partial differentials with respect to $ x ^ {i} $ and $ y ^ {j} $, respectively. It is assumed that $ ( g _ {ij } ) $ is non-singular on $ {\widetilde{T} } M ^ {n} $ and that $ F $ and $ g _ {ij } $ extend continuously to the entire tangent bundle $ TM ^ {n} $. The pair $ ( M ^ {n} ,F ) $ is called a Finsler space. The Euler–Lagrange equations (cf. Euler–Lagrange equation) of $ ( M ^ {n} ,F ) $ describe geodesics (cf. Geodesic line) and have the local description

$$ { \frac{dy ^ {i} }{ds } } + \gamma _ {jk } ^ {i} ( x,b ) y ^ {j} y ^ {k} = 0, \quad { \frac{dx ^ {i} }{ds } } = y ^ {i} , $$

where the differential of arc length is $ ds = F ( x,dx ) $ and $ \gamma _ {jk } ^ {i} ( x,y ) $ are the usual Levi-Cività (or Christoffel) symbols (cf. Christoffel symbol) in terms of $ g _ {ij } ( x,y ) $, its inverse $ g ^ {ij } ( x,y ) $ and $ \partial _ {i} g _ {kl } $. Note that the $ \gamma _ {jk } ^ {i} $ depend on $ y $. This is not the case in Riemannian geometry, where they are the coefficients of a unique, metric compatible, symmetric connection. In Finsler geometry there are several important connections, but $ \gamma _ {jk } ^ {i} ( x,y ) $ itself is not a connection. One way to proceed is as follows. Let $ G ^ {i} = ( {1 / 2 } ) \gamma _ {jk } ^ {i} y ^ {j} y ^ {k} $ and form $ G _ {j} ^ {i} ( x,y ) = {\dot \partial } _ {j} G ^ {i} ( x,y ) $ and $ G _ {jk } ^ {i} ( x,y ) = {\dot \partial } _ {k} G _ {j} ^ {i} ( x,y ) $. It can be readily proved that the $ G _ {jk } ^ {i} ( x,y ) $ transform like a classical affine connection, in spite of their dependence on $ y $, i.e.

$$ G _ {jk } ^ {i} X _ {b} ^ {j} X _ {c} ^ {k} = {\overline{G}\; } _ {bc } ^ {a} X _ {a} ^ {i} + { \frac{\partial X ^ {i} _ {b} }{\partial {\overline{x}\; } ^ {c} } } , \quad X ^ {i} _ {b} = { \frac{\partial X ^ {i} }{\partial {\overline{x}\; } ^ {b} } } . $$

Also, the $ G _ {j} ^ {i} ( x,y ) $ have a transformation law induced from that of $ G _ {jk } ^ {i} ( x,y ) $, because $ G _ {j} ^ {i} = G _ {jk } ^ {i} y ^ {k} $, by the Euler theorem on homogeneous functions. Note that $ G ^ {i} $, $ G _ {j} ^ {i} $ and $ G _ {jk } ^ {i} $ are positively homogeneous in $ y ^ {k} $ of degree two, one and zero, respectively. The triple $ B \Gamma = ( G _ {jk } ^ {i} ( x,y ) ,G _ {j} ^ {i} ( x,y ) ,0 ) $ is an example of a pre-Finsler connection [a1], $ F \Gamma = ( F _ {jk } ^ {i} ( x,y ) , N _ {j} ^ {i} ( x,y ) ,V _ {jk } ^ {i} ( x,y ) ) $, meaning that:

1) the $ F _ {jk } ^ {i} ( x,y ) $ transform just like the $ n ^ {3} $ functions $ G _ {jk } ^ {i} ( x,y ) $ above (they are called the coefficients of the pre-Finsler connection on $ ( M ^ {n} ,F ) $);

2) the $ n ^ {2} $ functions $ N _ {j} ^ {i} ( x,y ) $ transform just like $ G _ {j} ^ {i} ( x,y ) $( they are called the coefficients of a non-linear connection on $ {\widetilde{T} } M ^ {n} $) and

3) $ V _ {jk } ^ {i} ( x,y ) $ is a tensor (cf. Tensor calculus) on $ M ^ {n} $.

Using these local expressions one can further introduce the vertical covariant derivative $ \nabla ^ {\textrm{ V } } $ and the horizontal covariant derivative $ \nabla ^ {\textrm{ H } } $, as follows: for any contravariant vector $ A ^ {r} ( x,y ) $, set

1) $ \nabla _ {j} ^ {\textrm{ H } } A ^ {i} = \delta _ {j} A ^ {i} + A ^ {r} F _ {rj } ^ {i} $ and

2) $ \nabla _ {j} ^ {\textrm{ V } } A ^ {i} = {\dot \partial } _ {j} A ^ {i} + A ^ {r} V _ {rj } ^ {i} $, where $ \delta _ {i} = \partial _ {i} - N _ {j} ^ {r} {\dot \partial } _ {r} $ is the Finsler delta-derivative operator on $ ( M ^ {n} ,F ) $ corresponding to the non-linear connection $ N _ {j} ^ {i} ( x,y ) $. The important thing is that for any function $ f : { {\widetilde{T} } M ^ {n} } \rightarrow {\mathbf R ^ {1} } $, $ \delta _ {i} f $ is a covariant vector. Similar rules for higher-order tensors $ A ( x,y ) $ are just what one expects and all of the above have global descriptions.

The Okada theorem states that for a pre–Finsler connection $ F \Gamma = ( F _ {jk } ^ {i} , N _ {j} ^ {i} , V _ {jk } ^ {i} ) $ on $ ( M ^ {n} ,F ) $ such that:

$$ \nabla ^ {\textrm{ H } } F = 0, \quad F _ {jk } ^ {i} = F _ {kj } ^ {i} , \quad N _ {j} ^ {i} = F _ {rj } ^ {i} y ^ {r} , $$

$$ {\dot \partial } _ {k} N _ {j} ^ {i} = F _ {kj } ^ {i} , \quad V _ {jk } ^ {i} = 0, $$

one has $ F \Gamma = B \Gamma = ( G _ {jk } ^ {i} ,G _ {j} ^ {i} ,0 ) $. The pre-Finsler connection $ B \Gamma $ is the so-called Berwald connection on $ ( M ^ {n} ,F ) $.

Curvature of the Berwald connection.

If $ A ^ {i} ( x,y ) $ is a contravariant vector, then

$$ \nabla _ {k} ^ {\textrm{ V } } \nabla _ {j} ^ {\textrm{ H } } A ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ V } } A ^ {i} = A ^ {r} G _ {rjk } ^ {i} , $$

where $ G _ {rjk } ^ {i} = \nabla _ {k} ^ {\textrm{ V } } G _ {rj } ^ {i} $ defines the so-called (HV)-curvature, also known as the spray curvature or Douglas tensor [a1], [a2], [a3]) of $ B \Gamma $. Also,

$$ \nabla _ {k} ^ {\textrm{ H } } \nabla _ {j} ^ {\textrm{ H } } A ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ H } } A ^ {i} = A ^ {h} B _ {hjk } ^ {i} - ( \nabla _ {l} ^ {\textrm{ V } } A ^ {i} ) R _ {jk } ^ {l} , $$

where the Berwald curvature tensor is

$$ B _ {hjk } ^ {i} = \partial _ {k} G _ {hj } ^ {i} - G _ {k} ^ {r} ( {\dot \partial } _ {r} G _ {hj } ^ {i} ) + G _ {hj } ^ {r} G _ {rk } ^ {i} - ( {j / k } ) $$

and the VH-torsion tensor of $ B \Gamma $ is

$$ R _ {jk } ^ {l} = \partial _ {k} G _ {j} ^ {l} - G _ {jr } ^ {l} G _ {k} ^ {r} - ( {j / k } ) . $$

Here, the symbol $ ( {j / k } ) $ denotes that the entire expression before it is to be rewritten with the indices $ j $ and $ k $ interchanged.

A fundamental result in Berwald geometry is that both $ B _ {hjk } ^ {i} = 0 $ and $ G _ {jkl } ^ {i} = 0 $ if and only if $ ( M ^ {n} ,F ) $ is locally Minkowski. (Being locally Minkowski means that there is an admissible change of coordinates $ x \rightarrow {\overline{x}\; } $ so that $ F ( {\overline{x}\; } , {\overline{y}\; } ) $ is actually independent of $ {\overline{x}\; } ^ {i} $.) Consequently, the geodesics in such a space have the local expression $ {\overline{x}\; } ^ {i} = a ^ {i} s + b ^ {i} $, $ i = 1 \dots n $.

Now, generally, in Berwald theory one has

$$ R _ {jk } ^ {i} = B _ {hjk } ^ {i} y ^ {h} , $$

whereas for $ n = 2 $,

$$ R _ {jk } ^ {i} = \epsilon FK m ^ {i} ( l _ {j} m _ {k} - l _ {k} m _ {j} ) , $$

so that $ B _ {hjk } ^ {i} $ is completely determined by the so-called Berwald–Gauss curvature $ K ( x,y ) $ of $ ( M ^ {n} ,F ) $. The number $ \epsilon $ equals $ + 1 $ if $ g _ {ij } $ is positive definite and $ - 1 $ otherwise. The pair of contravariant vectors $ ( l ^ {i} ,m ^ {j} ) $, where $ l ^ {i} = { {y ^ {i} } / F } $, is called the Berwald frame. The $ m ^ {i} $ are normal vectors and are oriented. They are both of unit length and orthogonal relative to $ g _ {ij } ( x,y ) $. Of course, $ l _ {i} = g _ {ij } l ^ {j} $ and $ m _ {i} = g _ {ij } m ^ {j} $. The scalar invariant $ K ( x,y ) $ is positively homogeneous of degree zero in $ y ^ {i} $. If $ K > 0 $ everywhere, then the geodesics of $ ( M ^ {n} ,F ) $ are Lyapunov stable (cf. Lyapunov stability); if $ K \leq 0 $ everywhere, they are unstable [a1], [a4].

See also Berwald space.

References

[a1] 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)
[a2] 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)
[a3] M. Matsumoto, "Foundations of Finsler geometry and special Finsler spaces" , Kaiseisha Press (1986)
[a4] H. Rund, "The differential geometry of Finsler spaces" , Springer (1959)
How to Cite This Entry:
Berwald connection. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_connection&oldid=11677
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article