Berwald space

The Berwald connection satisfies

where denotes the so-called Cartan torsion tensor. Clearly, if and only if is Riemannian (i.e., has no dependence on ). So, generally, nor is, generally, vanishing.

A Berwald space is a Finsler manifold (cf. Finsler space) such that its Berwald coefficients are independent of . In fact, is a Berwald space if and only if if and only if .

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 and noted that

where is the so-called principal scalar invariant.

Berwald's theorem, slightly rephrased, reads as follows. If is a Berwald space which is not locally Minkowski (i.e., ), then is a constant and has one of the following four forms:

1) , :

2) , :

3) , :

4) :

Here and are independent -forms in that depend on and where the number equals if is positive definite and 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)