New Methods in Finsler Geometry

On the holonomy of Finsler manifolds

speaker: Zoltan Muzsnay (University of Debrecen)

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. Riemannian holonomy groups have been extensively studied and by now, their complete classification is known. On Finslerian holonomy, however, only a few results are known and they show, that it can be essentially different from the Riemannian one. In recent papers, a method was developed to investigate the holonomy through Lie algebras, tangent to the holonomy group. The holonomy algebra, the curvature algebra, and the infinitesimal holonomy algebra can provide valuable information about the holonomy. As a working example, we show that the holonomy group of a locally projectively flat Finsler manifold of constant curvature is finite dimensional if and only if it is flat or Riemannian. We also show that the holonomy group of a projectively flat simply connected Randers surface of non-zero constant curvature is maximal and its closure is isomorphic to the orientation preserving diffeomorphism group of the circle. These results are surprising because they show that even in the case when the geodesic structure is simple (the geodesics are straight lines), the holonomy group can still be a very large group.

timetable:
Thu 24 May, 14:10 - 15:00, Aula Dini
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