[Paper Review] Finslerian angle-preserving connection in two-dimensional case. Regular realization

G. S. Asanov|ArXiv.org|Sep 9, 2009

Advanced Differential Geometry Research4 references3 citations

TL;DR

This paper introduces a novel Finslerian connection in two-dimensional Finsler spaces that preserves both vector lengths and angles during parallel transport, generalizing the Riemannian Levi-Civita connection. The connection is constructed via explicit coefficients $ N^k_i(x,y) $ and $ D^k_{in}(x,y) $ derived from the Finsler metric and an angle function $ \theta(x,y) $, yielding a curvature tensor $ \rho_{knij} = T(b_n n_k - b_k n_n)M_{ij} $, with global $ C^\infty $-regularity in the Finsleroid-regular case.

ABSTRACT

We show that the metrical connection can be introduced in the two-dimensional Finsler space such that entailed parallel transports along curves joining points of the underlying manifold keep the two-vector angle as well as the length of the tangent vector, thereby realizing isometries of tangent spaces under the parallel transports. The curvature tensor is found. In case of the Finsleroid-regular space, constructions possess the $C^{\infty}$-regular status globally regarding the dependence on tangent vectors. Many involved and important relations are explicitly derived. Keywords: Finsler metrics, angle, connection, curvature tensors

Motivation & Objective

• To extend the Riemannian property of isometric parallel transport to Finsler geometry by preserving both vector length and angle.
• To address the fundamental question of whether the angle structure in Finsler spaces can uniquely generate a compatible connection.
• To construct a globally $ C^\infty $-regular connection in the two-dimensional Finsleroid-regular space.
• To derive explicit expressions for the connection coefficients and curvature tensor under angle and length preservation constraints.

Proposed method

• Introduces a connection via coefficients $ N^k_i(x,y) $ that satisfy $ d_i F = 0 $ (length preservation) and $ d_i \theta = k_i $ (angle preservation), where $ \theta $ is the angle function measured via indicatrix arc length.
• Derives the coefficients $ D^k_{in} = -\partial N^k_i / \partial y^n $, which are not symmetric in $ i,n $, distinguishing the connection from the Riemannian Christoffel symbols.
• Uses the main scalar $ I $, the vector $ A_k = F C_k $, and the tensor $ A_{ijk} = I m_i m_j m_k $ to express curvature components.
• Applies the condition $ l_k N^{k}_{nmi} = 0 $ via $ F N^{k}_{nmi} = -A^k_{mi} \partial_n \ln|I| $, ensuring the metric tensor is preserved under covariant derivative.
• Constructs the curvature tensor via $ \rho_{knij} = E_{knij} - M^h_{ij} C^n_{hk} $, with $ E_{knij} = -\partial M^{n}_{ij}/\partial y^k $, leading to $ \rho_{knij} = T(b_n n_k - b_k n_n)M_{ij} $.
• Validates the construction in the Randers metric case, showing smoothness and regularity of $ \partial\theta/\partial x^n $ and $ \partial F/\partial x^n $ in $ y $, confirming global $ C^\infty $-regularity.

Experimental results

Research questions

• RQ1 Can a Finsler connection be constructed such that parallel transport preserves both vector length and the angle between two vectors in two-dimensional Finsler spaces?
• RQ2 Does the angle structure in Finsler geometry uniquely determine a compatible connection, analogous to how the metric determines the Levi-Civita connection in Riemannian geometry?
• RQ3 What is the explicit form of the connection coefficients $ N^k_i(x,y) $ and $ D^k_{in}(x,y) $ that ensure both length and angle preservation?
• RQ4 What is the curvature tensor of this angle-preserving connection, and how does it relate to the covariant derivative of the vector field $ k_i $?
• RQ5 Is the constructed connection globally $ C^\infty $-regular in the Finsleroid-regular space, particularly in the Randers metric case?

Key findings

• The connection coefficients $ N^k_i(x,y) $ are explicitly constructed to satisfy $ d_i F = 0 $ and $ d_i \theta = k_i $, ensuring preservation of vector length and angle under parallel transport.
• The curvature tensor is derived as $ \rho_{knij} = T(b_n n_k - b_k n_n)M_{ij} $, where $ M_{ij} = \partial k_j / \partial x^i - \partial k_i / \partial x^j $, showing dependence on the covariant derivative of the vector field $ k_i $.
• The connection is globally $ C^\infty $-regular in the Finsleroid-regular space, with all components, including $ \partial\theta/\partial x^n $, smooth in $ y $, as confirmed in the Randers metric case.
• The curvature tensor components are expressed via $ \bar{L}_{knij} = a_{nh} \bar{L}^h_{ij} $, with $ \rho_{knij} = \sqrt{(F/S)^3} \bar{L}_{knij} $ in the Randers case, showing consistent structure.
• The connection coefficients depend on both $ \partial F / \partial x^n $ and $ \partial \theta / \partial x^n $, distinguishing them from Riemannian Christoffel symbols and embedding geometric structure directly.
• The condition $ l_k N^{k}_{nmi} = 0 $ is satisfied via $ F N^{k}_{nmi} = -A^k_{mi} \partial_n \ln|I| $, ensuring the Finsler metric tensor is preserved under the covariant derivative.