Finsler metrics and semi-symmetric compatible linear connections

TL;DR

This paper investigates semi-symmetric compatible linear connections on Finsler manifolds, providing a new proof of their uniqueness and explicit formulas for solutions, thus advancing understanding of generalized Berwald manifolds.

Contribution

It offers a new linear algebra-based proof for the uniqueness of semi-symmetric compatible linear connections and derives explicit solution formulas without integration.

Findings

Semi-symmetric compatible linear connections are uniquely determined.

Explicit formulas for solutions are derived without integration.

Necessary and sufficient conditions for solvability are formulated.

Abstract

Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds. Compatible linear connections are the solutions of the so-called compatibility equations containing the torsion components as unknowns. Although there are some theoretical results for the solvability of the compatibility equations (monochromatic Finsler metrics \cite{BM}, extremal compatible linear connections, algorithmic solutions \cite{V14}), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique. Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved \cite{V10} (see also \cite{V11}) that such a compatible linear connection must be uniquely determined. The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. We present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies and an explicit formula for the solution without integration. Necessary conditions of the solvability are also formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.