Difference between revisions of "Berwald space"
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+ | $#C+1 = 43 : ~/encyclopedia/old_files/data/B110/B.1100400 Berwald space | ||
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− | + | The [[Berwald connection|Berwald connection]] $ B \Gamma $ | |
+ | satisfies | ||
− | + | $$ | |
+ | \nabla _ {k} ^ {\textrm{ H } } g _ {ij } = - 2 \nabla _ {l} ^ {\textrm{ H } } C _ {ijk } y ^ {l} , | ||
+ | $$ | ||
− | Clearly, | + | 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|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|Riemannian space]]; [[Minkowski 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 $: | ||
− | 2) | + | $$ |
+ | F ^ {2} = ( \beta ^ {2} + \gamma ^ {2} ) { \mathop{\rm exp} } \left \{ { | ||
+ | \frac{2I }{J} | ||
+ | } { \mathop{\rm tan} } ^ {-1 } { | ||
+ | \frac \gamma \beta | ||
+ | } \right \} , | ||
+ | $$ | ||
− | + | $$ | |
+ | J = \sqrt {4 - I ^ {2} } ; | ||
+ | $$ | ||
− | + | 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|Berwald connection]]). | ||
Applications of Berwald spaces in biology, physics and stochastic processes can be found in [[#References|[a1]]], [[#References|[a2]]]. | Applications of Berwald spaces in biology, physics and stochastic processes can be found in [[#References|[a1]]], [[#References|[a2]]]. |
Latest revision as of 10:58, 29 May 2020
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) |
Berwald space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Berwald_space&oldid=46031