# Berwald space

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].

How to Cite This Entry:
Berwald space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=46031
This article was adapted from an original article by P.L. Antonelli (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article