Finsler connection preserving angle in dimensions $N\ge3$

TL;DR

This paper explores angle-preserving connections in Finsler spaces conformally related to Riemannian spaces, providing explicit formulas and examples, especially for Finsleroid-type spaces, revealing their geometric structure and curvature properties.

Contribution

It introduces explicit angle-preserving connections in Finsler spaces conformally automorphic to Riemannian spaces, with detailed formulas and examples for Finsleroid spaces.

Findings

The indicatrix of such Finsler spaces has constant curvature.

The angle-preserving connection is explicitly constructed and is metrical.

Explicit examples for Finsleroid spaces are provided.

Abstract

We show that if a Finsler space is conformally automorphic to a Riemannian space and the automorphism is positively homogeneous with respect to tangent vectors, then the indicatrix of the Finsler space is a space of constant curvature. In this case, the Finslerian two-vector angle can explicitly be found, which gives rise to simple and explicit representation for the connection preserving the angle in the indicatrix-homogeneous case. The connection is metrical and the Finsler space is obtainable from the Riemannian space by means of the parallel deformation. Since also the transitivity of covariant derivative holds, in such Finsler spaces the metrical non-linear angle-preserving connection is the respective export of the metrical linear Riemannian connection. From the commutators of covariant derivatives the associated curvature tensor is found. In case of the ${\cal FS}$-space, the explicit example of the conformally automorphic transformation can be developed, which entails the explicit connection coefficients and the metric function of the Finsleroid type.