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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

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

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|Euler–Lagrange equation]]) of  $  ( M  ^ {n} ,F ) $
describe geodesics (cf. [[Geodesic line|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|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.

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

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|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:

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|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 [[#References|[a1]]], [[#References|[a2]]], [[#References|[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|Lyapunov stability]]); if  $  K \leq  0 $
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>