Finsler Space as a Locally Minkowskian Space: Caught Between Curvature and Torsion Tensors.

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The extension of Riemannian “point”-space {xi} into a “line-space” {xi, dxi} make things clearer but not easier: how do you explain to a physicist a geometry supporting at least 3 curvature tensors and five torsion tensors? Not to speak of its usefulness for physics! Fortunately, the “impenetrable forest” by now has become a real, enjoyable park: through the application of the concepts of fibre bundle and non-linear connection. The different curvatures and torsion tensors result from vertical and horizontal parts of geometric objects in the tangent bundle, or in the Finsler bundle of the underlying manifold.

In essence, Finsler geometry is analogous to Riemannian geometry: there, the tangent space in a point p is euclidean space; here, the tangent space is just a normed space, i.e., Minkowski Space. Put differently: A Finsler metric for a differentiable manifold M is a map that assigns to each point x ∈ M a norm on the tangent space TxM. When referred to the almost exclusive use of methods from Riemannian geometry it means that this norm is demanded to derive from the length of a smooth path γ : [a, b] → M defined by ∫ab ∥ dγ(t)/dt ∥ dt. Then Finsler space becomes an example for the class of length spaces.

Starting from the length of the curve,

dγ(p, q):= ∫pq Lx(t), dx(t)/dt dt

the variational principle δdγ(p, q) = 0 leads to the Euler-Lagrange equation

d/dt(∂L/∂x ̇i) – ∂L/∂xi = 0

which may be rewritten into

d2xi/dt2 + 2Gi(xl, x ̇m) = 0

with Gi(xl, x ̇m) = 1/4gkl(-∂L/∂xl + ∂2L/∂xl∂x ̇m), and 2gik = ∂2L/∂x ̇l∂x ̇m, gilgjl = δij. The theory then is developed from the Lagrangian defined in this way. This involves an important object Nil := ∂Gi/∂yl, the geometrical meaning of which is a non-linear connection.

In general, a Finsler structure L(x, y) with y := dx(t))/dt = x ̇ and homogeneous degree 1 in y is introduced, from which the Finsler metric follows as:

fij = fji = ∂(1/2L2)/∂yi∂yj, fijyiyj = L2, yl∂L/∂yl = L, fijyj = L∂L/∂yi

A further totally symmetric tensor Cijk ensues:

Cijk := ∂(1/2L2)/∂yi∂yj∂yk

which will be interpreted as a torsion tensor. As an example of a Finsler metric is the Randers metric.

L(x.y) = bi(x)yi + √(aij(x)yiyj)

The Finsler metric following is

fik = bibk + aik + 2b(iak)lyˆl − aillakmm(bnn)

with yˆk := yk(alm(x)ylym)−1/2. Setting aij = ηij, yk = x ̇k, and identifying bi with the electromagnetic 4-potential eAi leads back to the Lagrangian for the motion of a charged particle.

In this context, a Finsler space thus is called a locally Minkowskian space if there exists a coordinate system, in which the Finsler structure is a function of yi alone. The use of the “element of support” (xi, dxk ≡ yk) essentially amounts to a step towards working in the tangent bundle TM of the manifold M.

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