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].

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)