[Paper Review] Finsler Metrics with K=0 and S=0

Zhongmin Shen|ArXiv.org|Sep 10, 2001

Advanced Differential Geometry Research5 references4 citations

TL;DR

This paper constructs non-projectively flat Finsler metrics with zero flag curvature (K=0) and zero S-curvature (S=0) in all dimensions ≥2 using a Randers-type metric F = α + β, where α is a Riemannian metric and β is a 1-form with ‖β‖_α < 1. The key contribution is a family of such metrics that are not locally Minkowskian, contradicting a prior claim that all K=0 Randers metrics must be locally Minkowskian, and showing that positive completeness is essential for that result.

ABSTRACT

In Finsler geometry, there are infinitely many models of constant curvature. The Funk metrics, the Hilbert-Klein metrics and the Bryant metrics are projectively flat with non-zero constant curvature. A recent example constructed by the author is projectively flat with zero curvature. In this paper, we introduce a technique to construct non-projectively flat Finsler metrics with zero curvature in each dimension. The technique can be used to construct many non-projectively flat Finsler metrics of constant curvature.

Motivation & Objective

• To construct explicit examples of Finsler metrics with zero flag curvature (K=0) that are not locally Minkowskian or projectively flat.
• To resolve a contradiction with a prior claim that all Randers metrics with K=0 must be locally Minkowskian.
• To demonstrate that the assumption of positive completeness is essential for the classification of K=0 Randers metrics as locally Minkowskian.
• To provide a geometric construction motivated by the shortest time problem in a rotating medium.
• To show that bounded Cartan torsions do not imply local Minkowskian structure when K=0, even for Randers metrics.

Proposed method

• The paper constructs a Randers metric F = α + β on a domain Ω ⊂ ℝⁿ defined by F(y) = [√((-yu + xv)² + |y|²(1 - x² - y²)) - (-yu + xv)] / (1 - x² - y²), where y = (u, v, y̅) ∈ TₚΩ.
• It uses the geodesic coefficients Gⁱ derived from the spray coefficients, computing them explicitly via the formula Gⁱ = G̃ⁱ + Hⁱ, where G̃ⁱ are the coefficients of the Riemannian metric α and Hⁱ depend on the 1-form β and its associated tensor sᵢⱼ.
• The S-curvature is computed using the trace of the Hessian of the spray coefficients, and it is shown to vanish via the identity ∂Gⁱ/∂yⁱ = 0.
• The flag curvature is computed by verifying that the Riemann curvature tensor Rⁱₖ satisfies Rⁱₖ = 0 for all i, k, using the condition ∂Gⁱ/∂x + ∂Gⁱ/∂y + ∂Gⁱ/∂yⁱ ∂Gʲ/∂yʲ - ∂Gⁱ/∂yʲ ∂Gʲ/∂yⁱ = 0.
• The construction is generalized to higher dimensions by extending the metric α and β to include additional coordinates, preserving the curvature and S-curvature conditions.
• The geometric motivation comes from the shortest time problem in a rotating fluid, where β represents a rotational velocity field and α is the Euclidean metric.

Experimental results

Research questions

• RQ1 Can non-projectively flat Finsler metrics with K=0 and S=0 be constructed in all dimensions ≥2?
• RQ2 Is it possible to have a Randers metric with K=0 that is not locally Minkowskian, contradicting prior claims?
• RQ3 What role does positive completeness play in the classification of K=0 Randers metrics?
• RQ4 Can bounded Cartan torsions coexist with K=0 in non-Minkowskian Finsler metrics?
• RQ5 Does the S-curvature condition S=0 imply Berwald or locally Minkowskian structure under K=0?

Key findings

• The paper constructs a family of Finsler metrics F = α + β on Ω ⊂ ℝⁿ with K=0 and S=0 that are not locally Minkowskian, providing a counterexample to the claim that all K=0 Randers metrics must be locally Minkowskian.
• The constructed metrics are not projectively flat, as evidenced by the non-vanishing spray coefficients Gⁱ that do not satisfy the projective flatness condition.
• The S-curvature vanishes identically, as shown by the trace condition ∂Gⁱ/∂yⁱ = 0 and the volume form matching the Euclidean one.
• The flag curvature is zero, verified by showing all components of the Riemann curvature tensor Rⁱₖ vanish, including R¹₁ = R¹₂ = R²₁ = R²₂ = 0.
• The metric is not positively complete, as the domain Ω is bounded and geodesics cannot be extended to infinity, which is consistent with the necessity of positive completeness in the classification theorem.
• The construction generalizes to higher dimensions by adding coordinates with G³ = 0 and preserving the curvature and S-curvature conditions.