Berwald space
The Berwald connection satisfies
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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
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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) ,
:
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2) ,
:
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3) ,
:
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4) :
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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) |
Berwald space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=46031