On the extremal compatible linear connection of a generalized Berwald manifold

Generalized Berwald manifolds are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). It is known (Vincze in J AMAPN 21:199–204, 2005) that 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 (preserving the Finslerian length of tangent vectors) is uniquely determined by its torsion. If the torsion is zero then we have a classical Berwald manifold. Otherwise, the torsion is some strange data we need to express in terms of the intrinsic quantities of the Finsler manifold. The paper presents the idea of the extremal compatible linear connection of a generalized Berwald manifold by minimizing the pointwise length of its torsion tensor. It is uniquely determined because the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Using the reference element method, the extremal compatible linear connection can be expressed in terms of the canonical data as well. It is an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.


Introduction
The notion of generalized Berwald manifolds goes back to Wagner [6]. They are Finsler manifolds admitting linear connections such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). We are interested in the unicity of the compatible linear connection and its expression in terms of the canonical data of the Finsler manifold (intrinsic characterization). If the torsion is zero (classical Berwald manifolds), the intrinsic characterization is the vanishing of the mixed curvature tensor of the canonical horizontal distribution. The problem of intrinsic characterization is solved in the more general case of Finsler manifolds admitting semi-symmetric compatible linear connections [3], see also [5]. We also have a unicity statement because the torsion tensor of the semi-symmetric compatible linear connection can be explicitly expressed in terms of metrics and differential forms given by averaging. Especially, the integration of the Riemann-Finsler metric on the indicatrix hypersurfaces (the so-called averaged Riemannian metric) gives a Riemannian environment for the investigations. The fundamental result of the theory [2] states that a linear connection satisfying the compatibility condition must be metrical with respect to the averaged Riemannian metric. Therefore the compatible linear connection is uniquely determined by its torsion tensor. Unfortunately, the unicity statement for the compatible linear connection of a generalized Berwald manifold is false in general [4]. To avoid the difficulties coming from different possible solutions, the idea is to look for the extremal solution in some sense: the extremal compatible linear connection of a generalized Berwald manifold keeps its torsion as close to the zero as possible. It is a conditional extremum problem involving functions defined on a local neighbourhood of the tangent manifold. In the case of a given point of the manifold, the reference element method guarantees that the number of the Lagrange multipliers is equal to the number of the equations for the compatibility of the linear connection with the Finslerian metric. Therefore the solution of the conditional extremum problem with a reference element can be expressed in terms of the canonical data. The solution of the conditional extremum problem regardless of the reference elements can be constructed algorithmically at each point of the manifold. If the pointwise solutions constitute a continuous section of the torsion tensor bundle then the continuity of the components of the torsion tensor implies the continuity of the connection parameters in the linear system of first order ordinary differential equations of parallel vector fields. Using parallel translations with respect to such a connection we can conclude that the Finslerian metric is monochromatic. By the fundamental result of the theory [1] it is sufficient and necessary for a Finslerian metric to be a generalized Berwald metric. Therefore we have an intrinsic algorithm to check the existence of compatible linear connections on a Finsler manifold because it is equivalent to the existence of the extremal compatible linear connection.

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 . . , n and π : T M → M is the canonical projection. A Finsler metric is a continuous function F : T M → R satisfying the following conditions: F is smooth on the complement of the zero section (regularity), F (tv) = tF (v) for all t > 0 (positive homogeneity) and the Hessian g ij = ∂ 2 E ∂y i ∂y j of the energy function E = F 2 /2 is positive definite at all nonzero elements v ∈ T p M (strong convexity), p ∈ M . The so-called Riemann-Finsler metric g is constituted by the components g ij . It is defined on the complement of the zero section. The Riemann-Finsler metric makes the complement of the origin a Riemannian manifold in each tangent space. The canonical objects are the volume form dμ = det g ij dy 1 ∧ . . . ∧ dy n , the Liouville vector field C := y 1 ∂/∂y 1 + . . . + y n ∂/∂y n and the induced volume form The averaged Riemannian metric is defined by 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 : [0, 1] → M . We have that where (x k • X) = c k and X k = −c i X j Γ k ij • c because of the differential equation for parallel vector fields. Therefore This means that the parallel transports with respect to ∇ preserve the Finslerian length of tangent vectors (compatibility condition) if and only if

C. Vincze AEM
where i = 1, . . . , n. The vector fields of type span the horizontal distribution belonging to ∇. In a similar way, we can introduce the horizontal vector fields X h * i (i = 1, . . . , n) with respect to the Lévi-Civita connection of the averaged Riemannian metric F * (v) := γ p (v, v).

Theorem 1. [2] If a linear connection on the base manifold is compatible with the Finslerian metric function then it must be metrical with respect to the averaged Riemannian metric.
In what follows we are going to substitute the connection parameters with the components of the torsion tensor in the equations of the compatibility condition (4). Since the torsion tensor bundle can be equipped with a Riemannian metric in a natural way, we can measure the length of the torsion to formulate an extremum problem for the compatible linear connection keeping its torsion as close to the origin as possible.

The extremal compatible linear connection of a generalized Berwald manifold
Let F be the Finslerian metric of a connected generalized Berwald manifold and suppose that ∇ is a compatible linear connection. Taking vector fields with pairwise vanishing Lie brackets on a local neighbourhood of the base manifold, the Christoffel process implies that and, consequently, where ∇ * denotes the Lévi-Civita connection of the averaged Riemannian metric γ and T is the torsion tensor of ∇. In terms of the connection parameters and the compatibility condition (4) can be written into the form Vol. 96 (2022) On the extremal compatible linear connection 57 Formula (6) shows that the correspondence ∇ T preserves the affine combinations of the linear connections, i.e. for any real number λ ∈ R we have Additionally, if ∇ 1 and ∇ 2 satisfy the compatibility condition (4) then so does This means that the set containing the restrictions of the torsion tensors of the compatible linear connections to the Cartesian product As the point is varying we have an affine distribution of the torsion tensor bundle In terms of local coordinates, the bundle is spanned by and its dimension is n 2 n.

Definition 2. The products
. . , n) form an orthonormal basis at the point p ∈ M if the coordinate vector fields ∂/∂u 1 , . . . , ∂/∂u n form an orthonormal basis with respect to the averaged Riemannian metric at the point p ∈ M . The norm of the torsion tensor is defined by ∂ ∂u k and the products form an orthonormal basis at the point p ∈ M . The corresponding inner product is Let a point p ∈ M be given and consider the affine subspace where i = 1, . . . , n and v ∈ T p M . Note that A p is nonempty because it contains the restrictions of the torsion tensors of the compatible linear connections to the Cartesian product T p M × T p M . If T p ∈ A p then we can write A p as the where i = 1, . . . , n and v ∈ T p M .
Since the identities Q i a Q j b γ ij (p) = γ ab (p) and P a i P b j γ ij (p) = γ ab (p) give that Q j b γ jl (p) = P j l γ jb (p) and P b j γ jk (p) = Q k j γ jb (p), we have where i = 1, . . . , n and v ∈ T p M . Taking the product with the matrix Q i a the equivalent system of equations is Remark 1. The previous argument obviously works for any element of the group G containing orthogonal transformations of the tangent space T p M with respect to the averaged Riemannian metric such that the Finslerian indicatrix is invariant: F • ϕ = F (ϕ ∈ G). It is also clear that G is a compact subgroup in the orthogonal group and Hol∇ ⊂ G. Finally, the proof of Lemma 1 also works for the invariance of the Finslerian metric under ϕ := τ pq , where τ pq is a parallel translation induced by a linear connection on the base manifold. The invariance implies that (ϕT ) q ∈ H q , where T p ∈ H p and

is a smooth affine distribution of constant rank of the torsion tensor bundle.
Proof. Let ∇ be a compatible linear connection, T be its torsion tensor and the point p ∈ M be given. According to Lemma 1 (see also Remark 1) we have that A q = T q + τ pq (H p ) for any q ∈ M , where τ pq is the parallel transport along an arbitrary curve joining p and q.
In what follows we introduce the extremal compatible linear connection of a generalized Berwald manifold in terms of its torsion T 0 by taking the closest point T 0 q ∈ A q to the origin for any q ∈ M . Since d(∇ 1 , ∇ 2 ) := T 1 − T 2 is a pointwise distance function on the set of metric linear connections, the minimal torsion also minimizes the difference between a compatible connection and the Lévi-Civita connection.

Definition 3. The extremal compatible linear connection of a generalized
Berwald manifold is the uniquely determined compatible linear connection minimizing the norm of its torsion by taking the values of the pointwise minima.

A conditional extremum problem for the extremal compatible linear connection
Let a point p ∈ M be given and consider the conditional extremum problem where the affine subspace where the index i = 1, . . . , n refers to the corresponding equation in (8). The symmetric differences in formula (13) are due to a < b. If the coordinate vector fields ∂/∂u 1 , . . . , ∂/∂u n form an orthonormal basis at p ∈ M with respect to the averaged Riemannian metric γ, then Since the vector fields come from the Liouville vector field (the outer unit normal to the Finslerian indicatrix) by an Euclidean quarter rotation in the corresponding 2-planes, their actions on F are automatically zero at the contact points of the Finslerian and the Riemannian spheres.

Vertical and horizontal contact points
is called a vertical contact point of the Finslerian and the averaged Riemannian metric functions. A nonzero element v ∈ T p M is a horizontal contact point of the Finslerian and the averaged Riemannian metric functions if X h * i F (v) = 0 (i = 1, . . . , n). The tangent space at p ∈ M is vertical/horizontal contact if all nonzero elements v ∈ T p M are vertical/horizontal contact.
The vertical/horizontal contact vector fields can also be defined in a similar way: the vector field X on the base manifold is vertical/horizontal contact if either X(p) ∈ T p M is vertical/horizontal contact or X(p) = 0.

Remark 2.
First of all note that the vertical contact points are independent of the choice of the coordinate system. Geometrically, the tangent hyperplanes of the Finslerian and the Riemannian spheres passing through a vertical contact point are the same in the corresponding tangent space. This is because their Euclidean gradient vectors in T p M are proportional: Vol. 96 (2022) On the extremal compatible linear connection 61 where the coordinate vector fields ∂/∂u 1 , . . . , ∂/∂u n form an orthonormal basis at p ∈ M with respect to the averaged Riemannian metric γ.

Corollary 2. The vertical contact points of a generalized Berwald manifold are horizontal contact points.
Proof. Suppose that the coordinate vector fields ∂/∂u 1 , . . . , ∂/∂u n form an orthonormal basis at p ∈ M with respect to the averaged Riemannian metric γ. It can be easily seen that Therefore, by formula (15),  Proof. The statement is trivial because the zero element in ∧ 2 T * p M ⊗ T p M solves the equations in (10) under the conditions X h * i F (v) = 0 (i = 1, . . . , n) for any nonzero v ∈ T p M . Remark 3. If we have a classical Berwald manifold then any nonzero element v ∈ T M is a horizontal contact point of the Finslerian and the averaged Riemannian metric functions and vice versa. The extremal compatible linear connection is ∇ * (the Lévi-Civita connection of the averaged Riemannian metric) with vanishing torsion.

The reference element method
In what follows we are going to find the Lagrange multipliers for the torsion tensor of the extremal compatible linear connection. They transform the compatibility condition to a system of linear equations containing at most n unknown parameters instead of n 2 n. Let a point p ∈ M and the reference element v ∈ T p M \ {0} be given and consider the conditional extremum problem where i = 1, . . . , n. It is clear that

Lemma 2. Introducing the notations
This means that v is a vertical contact point and our assumption is false. In addition, the subsequent rows contain zeros in the corresponding positions: