# Berwald connection

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].