Finsler geometry

A metric generalization of Riemannian geometry, where the general definition of the length of a vector is not necessarily given in the form of the square root of a quadratic form as in the Riemannian case. Such a generalization was first developed in the paper by P. Finsler [1].

The object studied in Finsler geometry is a real $ N $- dimensional differentiable manifold $ M $( of class at least $ C ^ {3} $) with a system of local coordinates $ x ^ {i} $, on which a real non-negative scalar function $ F ( x, y) $ in $ 2N $ independent variables $ x ^ {i} $ and $ y ^ {i} $ is given, where $ y ^ {i} $ are the components of the contravariant vectors tangent to $ M $ at the point $ x ^ {i} $. Suppose that $ F ( x, y) $ belongs to the class $ C ^ {3} $ in $ x ^ {i} $, and that in each tangent space $ M _ {x} $ to $ M $ there is a domain $ M _ {x} ^ {*} $ such that, first, it is conical (in the sense that if any vector $ y ^ {i} $ tangent at some point $ x ^ {i} $ belongs to $ M _ {x} ^ {*} $, then every other tangent vector that is collinear with $ y ^ {i} $ and tangent at the same point $ x ^ {i} $ also belongs to $ M _ {x} ^ {*} $), and secondly, $ F ( x, y) $ belongs to the class $ C ^ {5} $ in $ y ^ {i} \in M _ {x} ^ {*} $. Non-zero vectors $ y ^ {i} \in M _ {x} ^ {*} $ are called admissible. Suppose further that for every admissible $ y ^ {i} $ and every point $ x ^ {i} $:

$$ F ( x, y) > 0,\ \ \mathop{\rm det} \frac{\partial ^ {2} F ^ { 2 } ( x, y) }{ \partial y ^ {i} \partial y ^ {j} } \neq 0, $$

and also that $ F ( x, y) $ is positively homogeneous of degree one in $ y ^ {i} $, that is, $ F ( x, ky) = kF ( x, y) $ for every $ k > 0 $ and all $ x ^ {i} $ and admissible $ y ^ {i} $. Under these conditions the triple $ ( M, M _ {x} ^ {*} , F ( x, y)) $ is called a Finsler space, and $ F $ a Finsler metric. The value of $ F ( x, y) $ is interpreted as the length of the vector $ y ^ {i} $ tangent at $ x ^ {i} $.

If a Finsler space admits a coordinate system $ x ^ {i} $ such that $ F $ does not depend on these $ x $, then it is called a Minkowski space. The latter is related to a Finsler space in the same way as a Euclidean space is related to a Riemannian space. A Finsler space is called positive definite if one imposes a condition on $ F $ that ensures that the quadratic form $ z ^ {i} z ^ {j} \{ \partial ^ {2} F ^ { 2 } ( x, y)/ \partial y ^ {i} \partial y ^ {j} \} $ is positive definite for all $ x ^ {i} $ and non-zero $ y ^ {j} $.

Imposing the condition of homogeneity in $ y ^ {i} $ on $ F $ has a clear geometrical meaning from the point of view of invariant concepts in centro-affine spaces, the tangent spaces $ M _ {x} $ being such spaces. Namely, the ratio of the lengths of any two collinear vectors $ y _ {1} ^ {i} $ and $ y _ {2} ^ {i} = ky _ {1} ^ {i} $ in $ M _ {x} ^ {*} $ can be invariantly defined in the following way: $ y _ {1} ^ {1} /y _ {2} ^ {1} = y _ {1} ^ {2} /y _ {2} ^ {2} = \dots = k $, which does not include any metric functions. Thus, the homogeneity condition imposed on $ F $ is a condition that the Finslerian definition of length is consistent with the particular centro-affine definition; the Finsler metric is needed to compare the lengths of non-collinear vectors.

The tensor

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

is called the Finsler metric tensor. By Euler's theorem on homogeneous functions,

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

where, by definition, $ y _ {i} = g _ {ij} ( x, y) y ^ {j} $. These formulas are an immediate generalization of their Riemannian analogues, and follow from just the homogeneity condition. Finsler geometry reduces to Riemannian geometry in the case when the metric tensor $ g _ {ij} ( x, y) $ is assumed to be independent of $ y ^ {n} $. The last condition can be written in the form $ C _ {ijk} = 0 $, where

$$ C _ {ijk} ( x, y) = \ { \frac{1}{2} } \frac{\partial g _ {ij} ( x, y) }{\partial y ^ {k} } \equiv \ { \frac{1}{4} } \frac{\partial ^ {3} F ^ { 2 } ( x, y) }{\partial y ^ {i} \partial y ^ {j} \partial y ^ {k} } $$

is called the Cartan torsion tensor. It satisfies the identity $ y ^ {i} C _ {ijk} = 0 $. All Finsler relations can be turned into their Riemannian analogues by setting $ C _ {ijk} = 0 $. The Christoffel symbols $ \gamma _ {ij} ^ {k} ( x, y) $, which are constructed from the Finsler metric tensor by the same formula as in Riemannian geometry, do not obey the transformation law of the coefficients of a connection. Nevertheless, one can construct the coefficients of a connection from the first derivatives of the Finsler metric tensor so that (as also in Riemannian geometry) the covariant derivative of the metric tensor vanishes. They are called the Cartan connection coefficients and have the form

$$ \Gamma _ {ij} ^ {k} ( x, y) = \ \gamma _ {ij} ^ {k} - C _ {in} ^ {k} G _ {j} ^ {n} - C _ {jn} ^ {k} G _ {i} ^ {n} + C _ {ijn} G ^ {kn} , $$

where

$$ G _ {j} ^ {n} = \ - 2C _ {jm} ^ {n} G ^ {m} + y ^ {m} \gamma _ {mj} ^ {n} ,\ \ 2G ^ {m} = \ y ^ {n} y ^ {k} \gamma _ {nk} ^ {m} . $$

From the commutators of various covariant derivatives one can find expressions for the Finsler curvature tensors.

In each tangent space $ M _ {x} $ the Finsler metric defines an $ ( N - 1) $- dimensional hypersurface $ F ( x, y) = 1 $( where the $ x ^ {i} $ are regarded as fixed and the $ y ^ {i} $ as varying), called the indicatrix. The indicatrix is formed by the ends of the unit tangent vectors $ l ^ {i} = y ^ {i} /F ( x, y) $ tangent at the point $ x ^ {i} $. The fundamental significance of the concept of the indicatrix is already evident from the fact that, because the Finsler metric is homogeneous, the indicatrix at $ x ^ {i} $ uniquely determines the form of $ F ( x, y) $ at this point $ x ^ {i} $. In the Riemannian case the indicatrix is a sphere. Generally speaking, the indicatrix of a Finsler space can be a surface of a rather general form. The Finsler metric tensor induces a Riemannian metric on the indicatrix, converting it into a Riemannian space. For each fixed $ x $ the Finsler metric tensor is Riemannian in the variables $ y $. The pair $ ( M _ {x} ^ {*} , g _ {ij} ( x, y)) $, where the $ x ^ {n} $ are fixed and the $ y ^ {n} $ are variable, is called the tangent Riemannian space at $ x $( a Euclidean space in the case of Riemannian geometry); the Riemannian curvature tensor of this space reduces to the expression $ C _ {mh} ^ {j} C _ {ik} ^ {m} - C _ {mk} ^ {j} C _ {ih} ^ {m} $. The indicatrix is a hypersurface that is imbedded in the tangent Riemannian space. The most immediate example of a Finsler metric function is the $ f $- th root of a form of order $ f $.

Let $ f ( x) $ and $ r ^ {A} ( x) $ be real scalar functions of class $ C ^ {3} $ satisfying at each point $ x $ the conditions $ f \neq 0, 1 $ or 2, and $ r ^ {A} \neq 0 $, and let $ S _ {i} ^ {A} ( x) $ be $ N $ linearly independent real covariant vector fields of class $ C ^ {3} $, $ A = 1 \dots N $. Then for

$$ F _ {1} ( x, y) = \ \left [ \sum _ {A = 1 } ^ { N } r ^ {A} ( x) \cdot ( S _ {m} ^ {A} ( x) y ^ {m} ) ^ {f ( x) } \right ] ^ {1/f ( x) } $$

the curvature of the indicatrix is constant and equal to $ f ^ { 2 } /4 ( f - 1 ) $, and for

$$ F _ {2} ( x, y) = \ \prod _ {A = 1 } ^ { N } ( S ^ {A} ( x) y ^ {m} ) ^ {r ^ {A} ( x) } , \ \sum _ {A = 1 } ^ { N } r ^ {A} ( x) = 1 , $$

the curvature tensor of the indicatrix is zero. The determinant of the Finsler metric tensor is independent of $ y ^ {i} $ if and only if $ C _ {i} = 0 $, where $ C _ {i} = C _ {in} ^ {n} $. If a Finsler space is positive definite and the indicatrix is a convex surface, then $ C _ {i} \neq 0 $. The function $ F _ {2} $ is the only known example (1984) of a Finsler metric for which $ C _ {i} = 0 $( not counting the proper Riemannian case).

One can select special types of Finsler spaces by postulating some special form of the characteristic Finsler tensors. If the base manifold $ M $ admits a field of frames $ S _ {i} ^ {A} ( x) $ globally, and $ F ^ {*} ( y ^ {A} ) $ is the metric function of some Minkowski space, then one can introduce a Finsler metric on $ M $:

$$ F ( x ^ {n} , y ^ {i} ) = \ F ^ {*} ( S _ {i} ^ {A} ( x ^ {n} ) y ^ {i} ) . $$

In this case the Finsler space and the metric are called $ 1 $- form. The functions $ F _ {1} $ and $ F _ {2} $ are $ 1 $- form when $ f $ and $ r ^ {A} $ are constants. $ 1 $- form spaces may be reckoned to be the simplest from the point of view of the way the variables $ x ^ {n} $ enter in the metric. A Finsler space is called $ C $- reducible if it is not Riemannian, if $ N > 2 $ and if the Cartan torsion tensor can be represented in the form

$$ C _ {ijm} = \ { \frac{1}{N + 1 } } ( h _ {ij} C _ {m} + h _ {jm} C _ {i} + h _ {mi} C _ {j} ), $$

where $ h _ {ij} = g _ {ij} - l _ {i} l _ {j} $. $ C $- reducible spaces can have metrics of only two types: either the Kropina metric $ F _ {3} = \alpha ^ {2} / \beta $, or the Randers metric $ F _ {4} = \alpha + \beta $, where $ \beta = b _ {i} ( x) y ^ {i} $, $ \alpha ^ {2} = a _ {ij} y ^ {i} y ^ {j} $, $ b _ {i} ( x) $ is a covariant vector field, and $ a _ {ij} ( x) $ is a Riemannian metric tensor. For example, the Lagrange function of an electric test charge in a gravitational or electromagnetic field is a Randers metric. The Finsler metric tensor corresponding to $ F _ {2} $ has signature $ (+ - - \dots ) $, which makes it of interest in developing a Finslerian generalization of the general theory of relativity; this signature is also encountered in the case of the choice of metric tensors of the form $ F _ {1} $. Such a generalization can be based on the concept of an oscillating Riemannian space to a Finsler space, according to which the Finsler metric tensor associates with each vector field $ y ^ {i} ( x) $ the so-called osculating Riemannian metric tensor $ g _ {mn} ( x, y( x)) $. Choosing tensor fields $ z ^ {A} ( x) $ depending only on the $ x _ {i} $, from which one constructs the Finsler metric according to $ F ( x, y) = v ( z ^ {A} ( x), y) $, where $ v $ is a scalar function, one can regard the $ z ^ {A} $ as genuine gravitational field variables. The Finslerian geometrization of space-time also makes it possible to develop a theory of physical fields with various internal symmetries, relying on the concept of the group of transformations of the tangent vectors $ y ^ {i} $ that leave the Finsler metric invariant.

References

[1]	P. Finsler, "Ueber Kurven und Flächen in allgemeinen Räumen" , Göttingen (1918) (Dissertation)
[2]	H. Rund, "The differential geometry of Finsler spaces" , Springer (1959)
[3]	G.S. Asanov, "Finsler geometry, relativity and gauge theories" , Reidel (1985) (Translated from Russian)
[4]	M. Matsumoto, "Foundations of Finsler geometry and special Finsler spaces" , Kaiseisha Press (1986)