[Paper Review] Finsleroid gives rise to the angle-preserving connection

G. S. Asanov|ArXiv.org|Oct 6, 2009

Advanced Differential Geometry Research5 references3 citations

TL;DR

This paper introduces the Finsleroid Finsler space, defined by a covering indicatrix of constant curvature, and demonstrates that such spaces admit an angle-preserving, isometric connection derived from a conformal transformation to a Riemannian space. The resulting connection is non-symmetric but metric-compatible, enabling the direct induction of Levi-Civita-type connections and curvature tensors, with explicit evaluation for the axial Finsleroid of FF P D g-type.

ABSTRACT

The Finslerian unit ball is called the {\it Finsleroid} if the covering indicatrix is a space of constant curvature. We prove that Finsler spaces with such indicatrices possess the remarkable property that the tangent spaces are conformally flat with the conformal factor of the power dependence on the Finsler metric function. It is amazing but the fact that in such spaces the notion of the two-vector angle defined by the geodesic arc on the indicatrix can readily be induced from the Riemannian space obtained upon the conformal transformation, which opens up the straightforward way to induce also the connection coefficients and the concomitant curvature tensor. Thus, we are successfully inducing the Levi-Civita connection from the Riemannian space into the Finsleroid space, obtaining the isometric connection. The resultant connection coefficients are not symmetric. However, the metricity condition holds fine, that is, the produced covariant derivative of the Finsleroid metric tensor vanishes identically. The particular case underlined by the axial Finsleroid of the ${\mathbf\cF\cF^{PD}_{g}}$-type is explicitly evaluated in detail. Keywords: Finsler metrics, connection, curvature, conformal properties.

Motivation & Objective

• To establish a Finslerian connection that preserves angles between tangent vectors, mirroring the Levi-Civita property in Riemannian geometry.
• To explore whether the angle notion defined via geodesic arcs on the indicatrix can be used to induce a canonical connection in Finsler geometry.
• To demonstrate that Finsler spaces with indicatrices of constant curvature (Finsleroids) allow for a conformal transformation to a Riemannian space, enabling the transfer of the Levi-Civita connection.
• To derive and explicitly compute the connection coefficients and curvature tensor for the axial Finsleroid of FF P D g-type.

Proposed method

• Define the Finsleroid as a Finsler space whose indicatrix has constant curvature, ensuring conformal flatness of tangent spaces with power-law conformal factors.
• Construct the geodesic-arc angle α{x}(y1, y2) as the Riemannian length of the geodesic arc joining unit vectors on the indicatrix, using the induced Riemannian metric.
• Apply a conformal transformation to map the Finsleroid tangent space to a Riemannian space, then induce the Levi-Civita connection and curvature tensor from this Riemannian structure.
• Derive the induced connection coefficients in the Finsleroid space by lifting the Riemannian connection, showing they are non-symmetric but satisfy the metricity condition.
• Explicitly compute the connection and curvature components for the axial Finsleroid of the FF P D g-type using frame formalism and tensor transformations.
• Verify the metric compatibility of the induced connection by confirming the covariant derivative of the Finsleroid metric tensor vanishes identically.

Experimental results

Research questions

• RQ1 Can the angle between two vectors in a Finsler space be defined via the geodesic arc length on the indicatrix, and does this definition lead to a canonical, angle-preserving connection?
• RQ2 Does a Finsler space with an indicatrix of constant curvature (a Finsleroid) allow for a conformal mapping to a Riemannian space that preserves geometric structure?
• RQ3 Can the Levi-Civita connection of the conformally related Riemannian space be successfully induced into the Finsleroid space to yield a well-defined, metric-compatible connection?
• RQ4 What are the explicit forms of the connection coefficients and curvature tensor for the axial Finsleroid of the FF P D g-type?
• RQ5 Is the induced connection in the Finsleroid space non-symmetric yet still metric-compatible, and how does this affect its geometric interpretation?

Key findings

• The Finsleroid space, defined by a constant-curvature indicatrix, is conformally flat with a conformal factor depending on the Finsler metric function as a power law.
• The geodesic-arc angle α{x}(y1, y2) is defined as the Riemannian length of the geodesic arc on the indicatrix, enabling a consistent angle notion in Finsler geometry.
• The induced connection in the Finsleroid space is non-symmetric but satisfies the metricity condition, meaning the covariant derivative of the Finsleroid metric tensor vanishes identically.
• The curvature tensor of the Finsleroid space is successfully induced from the Riemannian space via conformal transformation, preserving geometric consistency.
• For the axial Finsleroid of the FF P D g-type, the connection coefficients and curvature components are explicitly computed using a specialized orthonormal frame and tensor transformation rules.
• The method confirms that isometric, angle-preserving parallel transport is possible in Finsleroid spaces, even though the connection is nonlinear and non-symmetric.