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The [[Berwald connection|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|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|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

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

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|Berwald connection]]).

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

====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></table>