Finsler connection preserving the two-vector angle under the indicatrix-inhomogeneous treatment

TL;DR

This paper introduces a new Finsler connection that preserves the two-vector angle even when the indicatrix curvature varies smoothly across the manifold, extending previous angle-preserving connections.

Contribution

It explicitly constructs a covariant, metric Finsler connection that maintains the two-vector angle in indicatrix-inhomogeneous spaces with variable curvature.

Findings

The connection is metric and covariant-constant.

The associated curvature tensor is derived from covariant derivative commutators.

Explicit representations and the Finsleroid space are provided.

Abstract

The Finsler spaces in which the tangent Riemannian spaces are conformally flat prove to be characterized by the condition that the indicatrix is a space of constant curvature. In such spaces the Finslerian normalized two-vector angle can be explicated from the respective two-vector angle of the associated Riemannian space. Therefore the way is opening to propose explicitly the connection preserving the angle even at the indicatrix-inhomogeneous level, that is, when the indicatrix curvature value ${\mathcal C}_{\text{Ind.}} $ is permitted to be an arbitrary smooth function of the indicatrix position point $x$. The connection obtained is metrical with the deflection part which is proportional to the gradient of the function $H(x)$ entering the equality ${\mathcal C}_{\text{Ind.}} \equiv H^2.$ Also the connection is covariant-constant. When the transitivity of covariant derivative is used, from the commutators of covariant derivatives the associated curvature tensor is found. Various useful representations have been developed. The Finsleroid space has been explicitly outlined.