Berwald connection

Let $ {\widetilde{T} } M ^ {n} $ denote the tangent bundle of a smooth $ n $- dimensional manifold $ M ^ {n} $, with zero-section removed. In Finsler geometry, one starts with a smooth metric function $ F : { {\widetilde{T} } M ^ {n} } \rightarrow {\mathbf R ^ {1} } $ and its associated metric tensor, given locally by

$$ g _ {ij } ( x,y ) = { \frac{1}{2} } {\dot \partial } _ {i} {\dot \partial } _ {j} F ^ {2} , \quad i,j = 1 \dots n, $$

where $ ( x ^ {i} ,y ^ {i} ) $ are the $ 2n $ coordinates (positions and velocities) and $ \partial _ {i} $ and $ {\dot \partial } _ {j} $ denote partial differentials with respect to $ x ^ {i} $ and $ y ^ {j} $, respectively. It is assumed that $ ( g _ {ij } ) $ is non-singular on $ {\widetilde{T} } M ^ {n} $ and that $ F $ and $ g _ {ij } $ extend continuously to the entire tangent bundle $ TM ^ {n} $. The pair $ ( M ^ {n} ,F ) $ is called a Finsler space. The Euler–Lagrange equations (cf. Euler–Lagrange equation) of $ ( M ^ {n} ,F ) $ describe geodesics (cf. Geodesic line) and have the local description

$$ { \frac{dy ^ {i} }{ds } } + \gamma _ {jk } ^ {i} ( x,b ) y ^ {j} y ^ {k} = 0, \quad { \frac{dx ^ {i} }{ds } } = y ^ {i} , $$

where the differential of arc length is $ ds = F ( x,dx ) $ and $ \gamma _ {jk } ^ {i} ( x,y ) $ are the usual Levi-Cività (or Christoffel) symbols (cf. Christoffel symbol) in terms of $ g _ {ij } ( x,y ) $, its inverse $ g ^ {ij } ( x,y ) $ and $ \partial _ {i} g _ {kl } $. Note that the $ \gamma _ {jk } ^ {i} $ depend on $ y $. This is not the case in Riemannian geometry, where they are the coefficients of a unique, metric compatible, symmetric connection. In Finsler geometry there are several important connections, but $ \gamma _ {jk } ^ {i} ( x,y ) $ itself is not a connection. One way to proceed is as follows. Let $ G ^ {i} = ( {1 / 2 } ) \gamma _ {jk } ^ {i} y ^ {j} y ^ {k} $ and form $ G _ {j} ^ {i} ( x,y ) = {\dot \partial } _ {j} G ^ {i} ( x,y ) $ and $ G _ {jk } ^ {i} ( x,y ) = {\dot \partial } _ {k} G _ {j} ^ {i} ( x,y ) $. It can be readily proved that the $ G _ {jk } ^ {i} ( x,y ) $ transform like a classical affine connection, in spite of their dependence on $ y $, i.e.

$$ G _ {jk } ^ {i} X _ {b} ^ {j} X _ {c} ^ {k} = {\overline{G}\; } _ {bc } ^ {a} X _ {a} ^ {i} + { \frac{\partial X ^ {i} _ {b} }{\partial {\overline{x}\; } ^ {c} } } , \quad X ^ {i} _ {b} = { \frac{\partial X ^ {i} }{\partial {\overline{x}\; } ^ {b} } } . $$

Also, the $ G _ {j} ^ {i} ( x,y ) $ have a transformation law induced from that of $ G _ {jk } ^ {i} ( x,y ) $, because $ G _ {j} ^ {i} = G _ {jk } ^ {i} y ^ {k} $, by the Euler theorem on homogeneous functions. Note that $ G ^ {i} $, $ G _ {j} ^ {i} $ and $ G _ {jk } ^ {i} $ are positively homogeneous in $ y ^ {k} $ of degree two, one and zero, respectively. The triple $ B \Gamma = ( G _ {jk } ^ {i} ( x,y ) ,G _ {j} ^ {i} ( x,y ) ,0 ) $ is an example of a pre-Finsler connection [a1], $ F \Gamma = ( F _ {jk } ^ {i} ( x,y ) , N _ {j} ^ {i} ( x,y ) ,V _ {jk } ^ {i} ( x,y ) ) $, meaning that:

1) the $ F _ {jk } ^ {i} ( x,y ) $ transform just like the $ n ^ {3} $ functions $ G _ {jk } ^ {i} ( x,y ) $ above (they are called the coefficients of the pre-Finsler connection on $ ( M ^ {n} ,F ) $);

2) the $ n ^ {2} $ functions $ N _ {j} ^ {i} ( x,y ) $ transform just like $ G _ {j} ^ {i} ( x,y ) $( they are called the coefficients of a non-linear connection on $ {\widetilde{T} } M ^ {n} $) and

3) $ V _ {jk } ^ {i} ( x,y ) $ is a tensor (cf. Tensor calculus) on $ M ^ {n} $.

Using these local expressions one can further introduce the vertical covariant derivative $ \nabla ^ {\textrm{ V } } $ and the horizontal covariant derivative $ \nabla ^ {\textrm{ H } } $, as follows: for any contravariant vector $ A ^ {r} ( x,y ) $, set

1) $ \nabla _ {j} ^ {\textrm{ H } } A ^ {i} = \delta _ {j} A ^ {i} + A ^ {r} F _ {rj } ^ {i} $ and

2) $ \nabla _ {j} ^ {\textrm{ V } } A ^ {i} = {\dot \partial } _ {j} A ^ {i} + A ^ {r} V _ {rj } ^ {i} $, where $ \delta _ {i} = \partial _ {i} - N _ {j} ^ {r} {\dot \partial } _ {r} $ is the Finsler delta-derivative operator on $ ( M ^ {n} ,F ) $ corresponding to the non-linear connection $ N _ {j} ^ {i} ( x,y ) $. The important thing is that for any function $ f : { {\widetilde{T} } M ^ {n} } \rightarrow {\mathbf R ^ {1} } $, $ \delta _ {i} f $ is a covariant vector. Similar rules for higher-order tensors $ A ( x,y ) $ are just what one expects and all of the above have global descriptions.

The Okada theorem states that for a pre–Finsler connection $ F \Gamma = ( F _ {jk } ^ {i} , N _ {j} ^ {i} , V _ {jk } ^ {i} ) $ on $ ( M ^ {n} ,F ) $ such that:

$$ \nabla ^ {\textrm{ H } } F = 0, \quad F _ {jk } ^ {i} = F _ {kj } ^ {i} , \quad N _ {j} ^ {i} = F _ {rj } ^ {i} y ^ {r} , $$

$$ {\dot \partial } _ {k} N _ {j} ^ {i} = F _ {kj } ^ {i} , \quad V _ {jk } ^ {i} = 0, $$

one has $ F \Gamma = B \Gamma = ( G _ {jk } ^ {i} ,G _ {j} ^ {i} ,0 ) $. The pre-Finsler connection $ B \Gamma $ is the so-called Berwald connection on $ ( M ^ {n} ,F ) $.

Curvature of the Berwald connection.

If $ A ^ {i} ( x,y ) $ is a contravariant vector, then

$$ \nabla _ {k} ^ {\textrm{ V } } \nabla _ {j} ^ {\textrm{ H } } A ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ V } } A ^ {i} = A ^ {r} G _ {rjk } ^ {i} , $$

where $ G _ {rjk } ^ {i} = \nabla _ {k} ^ {\textrm{ V } } G _ {rj } ^ {i} $ defines the so-called (HV)-curvature, also known as the spray curvature or Douglas tensor [a1], [a2], [a3]) of $ B \Gamma $. Also,

$$ \nabla _ {k} ^ {\textrm{ H } } \nabla _ {j} ^ {\textrm{ H } } A ^ {i} - \nabla _ {j} ^ {\textrm{ H } } \nabla _ {k} ^ {\textrm{ H } } A ^ {i} = A ^ {h} B _ {hjk } ^ {i} - ( \nabla _ {l} ^ {\textrm{ V } } A ^ {i} ) R _ {jk } ^ {l} , $$

where the Berwald curvature tensor is

$$ B _ {hjk } ^ {i} = \partial _ {k} G _ {hj } ^ {i} - G _ {k} ^ {r} ( {\dot \partial } _ {r} G _ {hj } ^ {i} ) + G _ {hj } ^ {r} G _ {rk } ^ {i} - ( {j / k } ) $$

and the VH-torsion tensor of $ B \Gamma $ is

$$ R _ {jk } ^ {l} = \partial _ {k} G _ {j} ^ {l} - G _ {jr } ^ {l} G _ {k} ^ {r} - ( {j / k } ) . $$

Here, the symbol $ ( {j / k } ) $ denotes that the entire expression before it is to be rewritten with the indices $ j $ and $ k $ interchanged.

A fundamental result in Berwald geometry is that both $ B _ {hjk } ^ {i} = 0 $ and $ G _ {jkl } ^ {i} = 0 $ if and only if $ ( M ^ {n} ,F ) $ is locally Minkowski. (Being locally Minkowski means that there is an admissible change of coordinates $ x \rightarrow {\overline{x}\; } $ so that $ F ( {\overline{x}\; } , {\overline{y}\; } ) $ is actually independent of $ {\overline{x}\; } ^ {i} $.) Consequently, the geodesics in such a space have the local expression $ {\overline{x}\; } ^ {i} = a ^ {i} s + b ^ {i} $, $ i = 1 \dots n $.

Now, generally, in Berwald theory one has

$$ R _ {jk } ^ {i} = B _ {hjk } ^ {i} y ^ {h} , $$

whereas for $ n = 2 $,

$$ R _ {jk } ^ {i} = \epsilon FK m ^ {i} ( l _ {j} m _ {k} - l _ {k} m _ {j} ) , $$

so that $ B _ {hjk } ^ {i} $ is completely determined by the so-called Berwald–Gauss curvature $ K ( x,y ) $ of $ ( M ^ {n} ,F ) $. The number $ \epsilon $ equals $ + 1 $ if $ g _ {ij } $ is positive definite and $ - 1 $ otherwise. The pair of contravariant vectors $ ( l ^ {i} ,m ^ {j} ) $, where $ l ^ {i} = { {y ^ {i} } / F } $, is called the Berwald frame. The $ m ^ {i} $ are normal vectors and are oriented. They are both of unit length and orthogonal relative to $ g _ {ij } ( x,y ) $. Of course, $ l _ {i} = g _ {ij } l ^ {j} $ and $ m _ {i} = g _ {ij } m ^ {j} $. The scalar invariant $ K ( x,y ) $ is positively homogeneous of degree zero in $ y ^ {i} $. If $ K > 0 $ everywhere, then the geodesics of $ ( M ^ {n} ,F ) $ are Lyapunov stable (cf. Lyapunov stability); if $ K \leq 0 $ everywhere, they are unstable [a1], [a4].

See also Berwald space.

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)
[a3]	M. Matsumoto, "Foundations of Finsler geometry and special Finsler spaces" , Kaiseisha Press (1986)
[a4]	H. Rund, "The differential geometry of Finsler spaces" , Springer (1959)