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The [[Berwald connection|Berwald connection]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104001.png" /> satisfies
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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/b110400/b1104002.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104003.png" /> denotes the so-called Cartan torsion tensor. Clearly, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104004.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104005.png" /> is Riemannian (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104006.png" /> has no dependence on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104007.png" />). So, generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104008.png" /> nor is, generally, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b1104009.png" /> vanishing.
+
The [[Berwald connection|Berwald connection]]  $  B \Gamma $
 +
satisfies
  
A Berwald space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040010.png" /> is a Finsler manifold (cf. [[Finsler space|Finsler space]]) such that its Berwald coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040011.png" /> are independent of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040012.png" />. In fact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040013.png" /> is a Berwald space if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040014.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040015.png" />.
+
$$
 +
\nabla _ {k} ^ {\textrm{ H } } g _ {ij }  = - 2 \nabla _ {l} ^ {\textrm{ H } } C _ {ijk }  y  ^ {l} ,
 +
$$
  
Clearly, all Riemannian and locally Minkowski spaces are Berwald spaces (cf. also [[Riemannian space|Riemannian space]]; [[Minkowski space|Minkowski space]]). L. Berwald gave a complete characterization of such spaces. He used the frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040016.png" /> and noted that
+
where  $  C _ {ijk }  = {
 +
\frac{1}{2}
 +
} \nabla _ {k} ^ {\textrm{ V } } g _ {ij }  = {
 +
\frac{1}{2}
 +
} {\dot \partial  } _ {k} g _ {ij }  $
 +
denotes the so-called Cartan torsion tensor. Clearly, $  C _ {ijk }  = 0 $
 +
if and only if  $  ( M  ^ {n} ,F ) $
 +
is Riemannian (i.e.,  $  g _ {ij }  $
 +
has no dependence on  $  y $).  
 +
So, generally,  $  \nabla _ {k} ^ {\textrm{ V } } g _ {ij }  \neq 0 $
 +
nor is, generally,  $  \nabla _ {l} ^ {\textrm{ H } } C _ {ijk }  $
 +
vanishing.
  
<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/b110400/b11040017.png" /></td> </tr></table>
+
A Berwald space  $  ( M  ^ {n} ,F ) $
 +
is a Finsler manifold (cf. [[Finsler space|Finsler space]]) such that its Berwald coefficients  $  G _ {jk }  ^ {i} $
 +
are independent of  $  y  ^ {i} $.
 +
In fact,  $  ( M  ^ {n} ,F ) $
 +
is a Berwald space if and only if  $  G _ {jkl }  ^ {i} = 0 $
 +
if and only if  $  \nabla _ {l} ^ {\textrm{ H } } C _ {ijk }  = 0 $.
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040018.png" /> is the so-called principal scalar invariant.
+
Clearly, all Riemannian and locally Minkowski spaces are Berwald spaces (cf. also [[Riemannian space|Riemannian space]]; [[Minkowski space|Minkowski space]]). L. Berwald gave a complete characterization of such spaces. He used the frame  $  ( l  ^ {i} ,m  ^ {j} ) $
 +
and noted that
  
Berwald's theorem, slightly rephrased, reads as follows. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040019.png" /> is a Berwald space which is not locally Minkowski (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040020.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040021.png" /> is a constant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040022.png" /> has one of the following four forms:
+
$$
 +
F \cdot C _ {ijk }  = I m _ {i} m _ {j} m _ {k} ,
 +
$$
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040023.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040024.png" />:
+
where  $  I $
 +
is the so-called principal scalar invariant.
  
<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/b110400/b11040025.png" /></td> </tr></table>
+
Berwald's theorem, slightly rephrased, reads as follows. If  $  ( M  ^ {2} ,F ) $
 +
is a Berwald space which is not locally Minkowski (i.e.,  $  K \neq 0 $),
 +
then  $  I $
 +
is a constant and  $  F ( x,y ) $
 +
has one of the following four forms:
  
<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/b110400/b11040026.png" /></td> </tr></table>
+
1)  $  \epsilon = + 1 $,
 +
$  I  ^ {2} < 4 $:
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040028.png" />:
+
$$
 +
F  ^ {2} = ( \beta  ^ {2} + \gamma  ^ {2} ) { \mathop{\rm exp} } \left \{ {
 +
\frac{2I }{J}
 +
} { \mathop{\rm tan} } ^ {-1 } {
 +
\frac \gamma  \beta
 +
} \right \} ,
 +
$$
  
<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/b110400/b11040029.png" /></td> </tr></table>
+
$$
 +
J = \sqrt {4 - I  ^ {2} } ;
 +
$$
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040031.png" />:
+
2) $  \epsilon = + 1 $,  
 +
$  I  ^ {2} = 4 $:
  
<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/b110400/b11040032.png" /></td> </tr></table>
+
$$
 +
F  ^ {2} = \beta  ^ {2} { \mathop{\rm exp} } \left \{ {
 +
\frac{I \gamma } \beta
 +
} \right \} ;
 +
$$
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040033.png" />:
+
3) $  \epsilon = + 1 $,
 +
$  I  ^ {2} > 4 $:
  
<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/b110400/b11040034.png" /></td> </tr></table>
+
$$
 +
F  ^ {2} = \beta \gamma \left \{ {
 +
\frac \gamma  \beta
 +
} \right \} ^ { {I / J } } ,  J = \sqrt {I  ^ {2} - 4 } ;
 +
$$
  
Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040035.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040036.png" /> are independent <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040037.png" />-forms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040038.png" /> that depend on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040039.png" /> and where the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040040.png" /> equals <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040041.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040042.png" /> is positive definite and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110400/b11040043.png" /> otherwise (cf. also [[Berwald connection|Berwald connection]]).
+
4)  $  \epsilon = - 1 $:
 +
 
 +
$$
 +
F  ^ {2} = \beta \gamma \left \{ {
 +
\frac \gamma  \beta
 +
} \right \} ^ { {I / J } } ,  J = \sqrt {I  ^ {2} + 4 } .
 +
$$
 +
 
 +
Here  $  \beta $
 +
and $  \gamma $
 +
are independent $  1 $-
 +
forms in $  y  ^ {i} $
 +
that depend on $  x $
 +
and where the number $  \epsilon $
 +
equals $  + 1 $
 +
if $  g _ {ij }  $
 +
is positive definite and $  - 1 $
 +
otherwise (cf. also [[Berwald connection|Berwald connection]]).
  
 
Applications of Berwald spaces in biology, physics and stochastic processes can be found in [[#References|[a1]]], [[#References|[a2]]].
 
Applications of Berwald spaces in biology, physics and stochastic processes can be found in [[#References|[a1]]], [[#References|[a2]]].

Latest revision as of 10:58, 29 May 2020


The Berwald connection $ B \Gamma $ satisfies

$$ \nabla _ {k} ^ {\textrm{ H } } g _ {ij } = - 2 \nabla _ {l} ^ {\textrm{ H } } C _ {ijk } y ^ {l} , $$

where $ C _ {ijk } = { \frac{1}{2} } \nabla _ {k} ^ {\textrm{ V } } g _ {ij } = { \frac{1}{2} } {\dot \partial } _ {k} g _ {ij } $ denotes the so-called Cartan torsion tensor. Clearly, $ C _ {ijk } = 0 $ if and only if $ ( M ^ {n} ,F ) $ is Riemannian (i.e., $ g _ {ij } $ has no dependence on $ y $). So, generally, $ \nabla _ {k} ^ {\textrm{ V } } g _ {ij } \neq 0 $ nor is, generally, $ \nabla _ {l} ^ {\textrm{ H } } C _ {ijk } $ vanishing.

A Berwald space $ ( M ^ {n} ,F ) $ is a Finsler manifold (cf. Finsler space) such that its Berwald coefficients $ G _ {jk } ^ {i} $ are independent of $ y ^ {i} $. In fact, $ ( M ^ {n} ,F ) $ is a Berwald space if and only if $ G _ {jkl } ^ {i} = 0 $ if and only if $ \nabla _ {l} ^ {\textrm{ H } } C _ {ijk } = 0 $.

Clearly, all Riemannian and locally Minkowski spaces are Berwald spaces (cf. also Riemannian space; Minkowski space). L. Berwald gave a complete characterization of such spaces. He used the frame $ ( l ^ {i} ,m ^ {j} ) $ and noted that

$$ F \cdot C _ {ijk } = I m _ {i} m _ {j} m _ {k} , $$

where $ I $ is the so-called principal scalar invariant.

Berwald's theorem, slightly rephrased, reads as follows. If $ ( M ^ {2} ,F ) $ is a Berwald space which is not locally Minkowski (i.e., $ K \neq 0 $), then $ I $ is a constant and $ F ( x,y ) $ has one of the following four forms:

1) $ \epsilon = + 1 $, $ I ^ {2} < 4 $:

$$ F ^ {2} = ( \beta ^ {2} + \gamma ^ {2} ) { \mathop{\rm exp} } \left \{ { \frac{2I }{J} } { \mathop{\rm tan} } ^ {-1 } { \frac \gamma \beta } \right \} , $$

$$ J = \sqrt {4 - I ^ {2} } ; $$

2) $ \epsilon = + 1 $, $ I ^ {2} = 4 $:

$$ F ^ {2} = \beta ^ {2} { \mathop{\rm exp} } \left \{ { \frac{I \gamma } \beta } \right \} ; $$

3) $ \epsilon = + 1 $, $ I ^ {2} > 4 $:

$$ F ^ {2} = \beta \gamma \left \{ { \frac \gamma \beta } \right \} ^ { {I / J } } , J = \sqrt {I ^ {2} - 4 } ; $$

4) $ \epsilon = - 1 $:

$$ F ^ {2} = \beta \gamma \left \{ { \frac \gamma \beta } \right \} ^ { {I / J } } , J = \sqrt {I ^ {2} + 4 } . $$

Here $ \beta $ and $ \gamma $ are independent $ 1 $- forms in $ y ^ {i} $ that depend on $ x $ and where the number $ \epsilon $ equals $ + 1 $ if $ g _ {ij } $ is positive definite and $ - 1 $ otherwise (cf. also Berwald connection).

Applications of Berwald spaces in biology, physics and stochastic processes can be found in [a1], [a2].

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)
How to Cite This Entry:
Berwald space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=12731
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article