Almost All Finsler Metrics have Infinite Dimensional Holonomy Group
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Almost All Finsler Metrics have Infinite Dimensional Holonomy Group B. Hubicska1 · V. S. Matveev2
· Z. Muzsnay1
Received: 21 July 2020 / Accepted: 4 September 2020 © The Author(s) 2020
Abstract We show that the set of Finsler metrics on a manifold contains an open everywhere dense subset of Finsler metrics with infinite-dimensional holonomy groups. Keywords Finsler geometry · Algebras of vector fields · Holonomy · Curvature Mathematics Subject Classification 53C29 · 53B40 · 22E65 · 17B66.
1 Introduction Finsler metrics appeared already in the inaugural lecture of Riemann in 1854 [19], under the name generalized metric. At the beginning of the XXth century, the intensive study of Finsler metrics was motivated by the optimal transport theory. A group of mathematicians lead by Cartheodory aimed to adapt mathematical tools which were effective in Riemannian geometry (such as affine connections, Jacobi vector fields, sectional curvature) for a more general situation. Finsler was a student of Cartheodory and his dissertation [5] is one of the important steps on this way. Riemannian geometry is one of the main sources of challenging problems in Finsler geometry: many Riemannian results are not valid in the Finslerian setup and one asks under what additional assumptions they are correct. Our paper studies the holonomy groups of Finsler manifolds. We give precise definitions later; at the present point let us recall that the Berwald connection (introduced
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Z. Muzsnay [email protected] B. Hubicska [email protected] V. S. Matveev [email protected]
1
University of Debrecen, Institute of Mathematics, Pf. 400, Debrecen 4002, Hungary
2
Friedrich-Schiller-Universität, Institut für Mathematik, 07737 Jena, Germany
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B.Hubicska et al.
by Berwald in 1926 [2]) can be viewed as an Ehresmann connection on the unit tangent bundle I M. Its holonomy group (at x ∈ M) is the subgroup of the group Di ff (Ix ) generated by the parallel transports along the loops starting and ending at x. For Riemannian metrics, the Berwald connection specifies to the Levi–Civita connection. Study of Riemannian holonomy groups is a prominent topic in Riemannian geometry and mathematical physics. It is known (see e.g. Borel and Lichnerowicz [3]) that the holonomy group is a subgroup of the orthogonal group; in particular it is always finite-dimensional. Moreover, all possible holonomy groups are described and classified due in particular to breakthrough results of Berger and Simons [1,20]. In the Finslerian case, the situation is very different and not much is known. By [21] (see also [12,13,22]) the so-called Berwald manifolds have finite-dimensional holonomy group. Also the so-called Landsberg manifolds have finite-dimensional holonomy group [10], but it is not jet known whether nonberwaldian Landsberg manifolds exist [11]. We are not aware of other examples of Finsler metrics with finite-dimensional holonomy group, it is an interesting problem to find such. From the other side, there are also not many explici
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