Holonomy Theory of Finsler Manifolds

Abstract

The holonomy group of a Riemannian or Finslerian manifold can be introduced in a very natural way: it is the group generated by parallel translations along loops with respect to the canonical connection. The Riemannian holonomy groups have been extensively studied and by now their complete classification is known. On the Finslerian holonomy, however, only few results are known and, as our results show, it can be essentially different from the Riemannian one.

In recent papers we have developed a method for the investigation of holonomy properties of non-Riemannian Finsler manifolds by constructing tangent Lie algebras to the holonomy group: the curvature algebra, the infinitesimal holonomy algebra, and the holonomy algebra. In this book chapter we present this method and give a unified treatment of our results. In particular we show that the dimension of these tangent algebras is usually greater than the possible dimensions of Riemannian holonomy groups and in many cases is infinite. We prove that the holonomy group of a locally projectively flat Finsler manifold of constant curvature is finite dimensional if and only if it is a Riemannian manifold or a flat Finsler manifold. We also show that the topological closure of the holonomy group of a certain class of simply connected, projectively flat Finsler 2-manifolds of constant curvature (spherically symmetric Finsler 2-manifolds) is not a finite dimensional Lie group, and we prove that its topological closure is the connected component of the full diffeomorphism group of the circle.