On compatible linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors. By the fundamental result of the theory (Vincze in J AMAPN 21:199–204, 2005) such a linear connection must be metrical with respect to the averaged Riemannian metric given by integration of the Riemann–Finsler metric on the indicatrix hypersurfaces. Therefore the linear connection is uniquely determined by its torsion tensor. If the torsion is zero then we have a classical Berwald manifolds. Otherwise the torsion is a strange data we need to express in terms of quantities of the Finsler manifold. In the paper we are going to give explicit formulas for the linear connections with totally anti-symmetric torsion tensor of three-dimensional generalized Berwald manifolds (Theorem 2). The results are based on averaging of (intrinsic) Finslerian quantities by integration over the indicatrix surfaces. They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric (Theorems 3 and 4). The possible cases are Riemannian spaces of constant zero curvature, constant positive curvature or Riemannian spaces admitting Killing vector fields of constant Riemannian length.


Introduction
The notion of generalized Berwald manifolds goes back to Wagner (1943). They are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). The basic questions of the theory are the unicity of the compatible linear connection and its expression in terms of the canonical data of the Finsler manifold (intrinsic characterization). In case of a classical Berwald manifold admitting a compatible linear connection with zero torsion, the intrinsic characterization is the vanishing of the mixed curvature tensor of the canonical horizontal distribution. In general the intrinsic characterization of the compatible linear connection is based on the so-called averaged Riemannian metric given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. By the fundamental result of the theory (Vincze 2005a) such a linear connection must be metrical with respect to the averaged Riemannian metric. Therefore the linear connection is uniquely determined by its torsion tensor. Following Agricola and Friedrich (2004) consider the decomposition T is the trace tensor of the torsion and In case of 2D the torsion tensor is automatically of the form (1); see Vincze et al. (2018). If the dimension is at least three then the trace-less part can be divided into two further components by separating the totally anti-symmetric/axial part A 1 . Therefore we have eight possible classes of generalized Berwald manifolds depending on the surviving terms such as classical Berwald manifolds admitting torsion-free compatible linear connections (Szabó 1981) (we have no surviving terms) or Finsler manifolds admitting semisymmetric compatible linear connections (we have no trace-less part) (Vincze 2005b(Vincze , 2008(Vincze , 2013 and (Vincze 2017). In the paper we are going to give explicit formulas for the linear connections with totally anti-symmetric torsion preserving the Finslerian length of tangent vectors in case of three-dimensional Finsler manifolds (Theorem 2). The results are based on averaging of (intrinsic) Finslerian quantities by integration over the indicatrix surfaces. They imply some consequences for the base manifold as a Riemannian space with respect to the averaged Riemannian metric. The possible cases are Riemannian space forms of constant zero curvature, constant positive curvature or Riemannian spaces admitting Killing vector fields of constant Riemannian length (Theorems 3 and 4). The Riemannian consequences are simple to prove but they have essential influence on the differential topology of the base manifold (Remarks 2 and 3). However, the results are dominated by Theorem 2 as the explicit expression of the only possible compatible linear connection with totally anti-symmetric torsion for a Finsler metric in 3D.

Notations and terminology
Let M be a differentiable manifold with local coordinates u 1 , . . . , u n . The induced coordinate system of the tangent manifold T M consists of the functions x 1 , . . . , x n and y 1 , . . . , y n . For any v ∈ T p M, . . , n and π : T M → M is the canonical projection.

Finsler metrics
A Finsler metric is a continuous function F : T M → R satisfying the following conditions: (F1) F is smooth on the complement of the zero section (regularity), (F2) F(tv) = t F(v) for all t > 0 (positive homogenity), (F3) the Hessian is positive definite at all nonzero elements v ∈ T p M, p ∈ M (strong convexity).
The so-called Riemann-Finsler metric g is constituted by the components g i j . It is defined on the complement of the zero section because the second order partial differentiability of the energy function at the origin does not follow automatically: if E is of class C 2 on the entire tangent manifold T M then, by the positively homogeneity of degree two, it follows that E is quadratic on the tangent spaces, i.e. the space is Riemannian. The Riemann-Finsler metric makes each tangent space (except at the origin) a Riemannian manifold with standard canonical objects such as the volume form dμ = det g i j dy 1 ∧ · · · ∧ dy n , the Liouville vector field C := y 1 ∂/∂ y 1 + · · · + y n ∂/∂ y n and the induced volume form

Generalized Berwald manifolds
Definition 1 A linear connection ∇ on the base manifold M is called compatible to the Finslerian metric if the parallel transports with respect to ∇ preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds.
Suppose that the parallel transports with respect to ∇ (a linear connection on the base manifold) preserve the Finslerian length of tangent vectors and let X be a parallel vector field along the curve c : because of the differential equation for parallel vector fields. If F is the Finslerian metric then and, by formula (2), This means that the parallel transports with respect to ∇ preserve the Finslerian length of tangent vectors (compatibility condition) if and only if where the vector fields of type span the associated horizontal distribution belonging to ∇.

Theorem 1 (Vincze 2005a) If a linear connection on the base manifold is compatible to the Finslerian metric function then it must be metrical with respect to the averaged Riemannian metric
Using the so-called Binet-Legendre metric is another possibility to provide a Riemannian environment for the investigations. It has been succesfully used by Troyanov and Matveev (2012), see also Bartelmeß and Matveev (2018).

Three-dimensional Finsler manifolds admitting compatible linear connections with totally anti-symmetric torsion tensor
Suppose that ∇ is a compatible linear connection of a three-dimensional generalized Berwald manifold. By Theorem 1, such a linear connection must be metrical with respect to the averaged Riemannian metric (7) given by integration of the Riemann-Finsler metric on the indicatrix hypersurfaces. Therefore ∇ is uniquely determined by its torsion tensor. Taking vector fields with pairwise vanishing Lie brackets on the neighbourhood U of the base manifold, the Christoffel process implies that and, consequently, where ∇ * denotes the Lévi-Civita connection of the averaged Riemannian metric.

Definition 2
The torsion tensor is totally anti-symmetric if its lowered tensor

Corollary 1 If ∇ is a metrical linear connection with totally anti-symmetric torsion then
and the geodesics of ∇ * and ∇ coincide.
If dim M = 3 then dim ∧ 3 M = 1 and, consequently, for some local function f : U → R, where the orientation is choosen such that the coordinate vector fields represent a positive basis. This means that Taking the Riemannian energy E * (v) := γ (v, v)/2, the Riemann-Finsler metric is g * i j = γ i j • π and the cross product of vertical vector fields is defined by with bilinear extension. Formula (10) can be written in terms of the induced horizontal structures as follows. Since the horizontal distributions induced by ∇ * and ∇ are spanned by the vector fields respectively, we have, by formula (10), that

Three-dimensional Finsler manifolds admitting compatible linear connections with totally anti-symmetric torsion tensor
Let M be a three-dimensional Finsler manifold admitting a compatible linear connection with totally anti-symmetric torsion tensor. Using the comparison formula (11), the compatibility condition (5) gives that where the vector field V is defined by the formula V :

then the Finslerian indicatrix ∂ K p is a sphere with respect to the averaged Riemannian metric.
Proof Since the infinitesimal rotation represented by the matrix implies that the vector fields V 1 , V 2 and V 3 span the tangent plane to the Finslerian indicatrix at any v ∈ ∂ K p . Therefore its Euclidean normal vector field (with respect to g * ) is proportional to C. Taking a curve c : [0, 1] → ∂ K p we have i.e. the Euclidean norm of c(t) is constant. Since the Finslerian indicatrix surface is arcwise connected this means that it is a sphere with respect to the averaged Riemannian metric.

Theorem 2 For a three-dimensional non-Riemannian Finsler manifold, the compatible linear connection with totally anti-symmetric torsion tensor must be of the form
where ∇ * is the Lévi-Civita connection of the averaged Riemannian metric γ and the function f is given by If the integrand on the right hand side is zero then, by Lemma 1, ∂ K p is a Euclidean sphere in T p M with respect to γ . In case of a generalized Berwald manifolds we have linear parallel transports between the tangent spaces. Since the translates of a quadratic surface are quadratic, this means that the manifold is Riemannian. Otherwise we can divide equation (15) to express the function f .

Curvature properties
Let a point p ∈ M be fixed and consider the subgroup G of orthogonal transformations with respect to the averaged inner product leaving the indicatrix ∂ K p invariant in T p M. Such a group is obviously closed in O(3) and, consequently, it is compact. If we have a generalized Berwald manifold then the group G is essentially independent of the choice of p because the parallel translations with respect to the compatible linear connection ∇ makes them isomorphic provided that the manifold is connected. On the other hand G must be finite or reducible unless the manifold is Riemannian; see (Vincze 2018, Remark 5). According to Theorem 1 it follows that Hol ∇ ⊂ G, i.e. the holonomy group of a compatible linear connection is finite or reducible in case of a a non-Riemannian generalized Berwald manifold. Using vector fields X , Y and Z with pairwise vanishing Lie-brackets on a neighbourhood U ⊂ M, the comparison formula (13) says that because of the Jacobi identity and the product rule Using the vector triple product extension formula Remark 1 The curvature tensor of ∇ obviously satisfies the curvature property Property also holds because ∇ is metrical with respect to the averaged Riemannian metric γ . The curvature tensor of ∇ satisfies the Jacobi identity if and only if the function f is constant because Taking the inner product of both sides with Z (for example) it can be easily seen that the left hand side is zero if and only if Z f = 0 for any vector field Z on the base manifold. To complete the list of the classical curvature properties note that the blocksymmetry and the Jacobi identity are equivalent to each other in 3D for any curvature tensor satisfying (19). It follows by a pure algebraic way (Vincze 2017, Remark 3.11).

The case of finite holonomy group
Suppose that G is finite and, consequently, the holonomy group of the compatible linear connection is also finite, i.e. its curvature is zero. respect to ∇ we can extend (21) to each point of the manifold. Note that ±β p in the focal set provide that (21) is invariant under not only the restricted holonomy group but the entire one including possibly reflections about the two-dimensional invariant subspace. Such a smoothly varying family of convex bodies induces a (non-Riemannian) fundamental function F such that it is invariant under the parallel transport with respect to ∇.

Remark 3
Killing vector fields of constant length naturally appear in different geometric constructions such as K -contact and Sasakian manifolds (Blair 1976;Belgun et al. 2003;Boyer and Galicki 2000). There are many restrictions to the existence of Killing vector fields of constant length on a Riemannian manifold; for a comprehensive survey see (Berestovskii and Nikonorov 2008): for example, if a compact Riemannian manifold admits such a vector field then its Euler characteristic must be zero in the sense of a Theorem due to Hopf (Berestovskii and Nikonorov 2008, Section 1).