Metrizability of Affine Connections

Lajos Tamássy

Abstract

An affine connection Γ on a vector bundle η = (E, π, M, V) of a rank r is called Riemann metrizable if there exists on M a Riemann metric which preserves the scalar product of vector fields parallel displaced according to Γ. Γ determines a connection G in a bundle, where M is fibered by the manifold of the ellipsoids of R^{r} = π^{−1}, x ∈ M. We prove that Γ is Riemann metrizable iff G is integrable.

An analogous result is deduced in the case, where η is replaced by a Finsler vector bundle, Γ means a Finsler connection, and the metric is a Finsler metric.

Mathematics Subject Classification: 53B15, 53B40, 53C07, 53C60. Key words: Riemann metrizability of affine connections, Finsler metrizability of affine connections.

1 Introduction

We consider a vector bundle η = (E, π, M, V) over the n-dimensional base manifold M with an r-dimensional real vector space V as typical fiber, where E is the total space and π : E → M is the projection operator. An affine connection H_{η} in η is given by a special splitting T_{z}E = V_{z}E ⊕ H_{z}E, z ∈ E and it is determined locally by the connection coefficients Γ_{β}^{α}_{i}(x); α, β, … = 1, …, r; i, j, … = 1, …, n, where x ∈ M has the local coordinates x^{i}. H_{η} or Γ is called Riemann metrizable if there exists a Euclidean scalar product ⟨ , ⟩ in each fiber π^{−1}(x), i.e. a symmetrical bilinear form g(x), in local coordinates ⟨ξ, ζ⟩ = g_{αβ}(x)ξ^{α}(x)ζ^{β}(x), such that the length of the parallel translated ∥_{Γ}P_{C}ξ_{0}∥_{g} of a vector ξ_{0} ∈ π^{−1}(x_{0}) along any curve C(t) ⊂ M, C(t_{0}) = x_{0} is constant, i.e. if the connection Γ is compatible with the Riemannian metric g. g(x_{0}) is equivalent with an ellipsoid E(§_{′}) : }_{αβ}(§_{′})ξ^{α}ξ^{β} = ∞ in π^{−1}(x_{0}) called indicatrix. ΓP_{C} establishes a linear mapping π^{−1}(x(t_{0})) → π^{−1}(x(t)). Γ is metrizable if there exists a field E(§) such that from ξ_{0} ∈ E_{′} follows _{Γ}P_{C(t)}ξ_{0} ∈ E(§(⊔)), ∀C(⊔) ⊂ M. Indicatrices play the role of the unit sphere.

The most simple case is r = n. If Γ_{j}^{i}_{h}(x) is symmetrical and metrizable by a g(x),

then Γ is the Levi-Civita connection Γ^{g} of the Riemannian manifold V_{n} = (M, g).

Balkan Journal of Geometry and Its Applications, Vol.2, No.2, 1997, pp. 131-138 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Denoting the set of the Levi-Civita connections for the different g by {Γ^{g}} and suppos-

ing the symmetry Γ_{j}^{i}_{h}(x) = Γ_{h}^{i}_{j}(x) the question is whether Γ ∈ {Γ^{g}}. — Riemann metrizability of affine connections has been investigated by many authors from different points of view. I mention here only [1], [4], [5], [6], [9], [12].

A Finsler space F_{n} = (M, L) on the manifold M is given by the smooth fundamental function L : T M → R^{+}; (x, y) ↦ L(§, †), y ∈ T_{x}M which is supposed to be first order positively homogeneous: L(§, λ†) = |λ|L(§, †), λ ∈ R. Its indicatrix is given by I(x_{0}) = {y | L(§_{′}, †) = ∞} ⊂ T_{§_{′}}M (the convexity of I is mostly also supposed). Giving of F_{n} is equivalent to giving of {I(x)}. Then an affine metrical connection should satisfy that from y_{0} ∈ I(x_{0}) follows _{Γ}P_{C}y_{0} ∈ I(x_{1}), x_{1} ∈ C(t_{1}) (this could be denoted by _{Γ}P_{C}I(x_{0}) = I(x_{1})), while _{Γ}P_{C} is an affine mapping. However, this is impossible in general, e.g. if I(x_{0}) is an ellipsoid and I(x_{1}) is not so. This necessitates the introduction of the so called Finsler vector fields which are sections of a vector bundle ζ = (E, π, T M, V^{n}), in components ξ^{i}(x, y) with the property ξ^{i}(x, λy) = ξ^{i}(x, y), λ ∈ R, λy ≠ 0. The set {(x_{0}, λy_{0}) | λ ∈ R, λy_{0} ≠ 0} is geometrically a point x_{0} and the direction of y_{0} in T_{x_{0}}M; this is called a line-element. So Finsler vectors are defined in line-elements. The length (the norm) of such a vector

is defined by g_{ij}(x, y)ξ^{i}(x, y)ξ^{j}(x, y) := ∥ξ(x, y)∥^{2}, where g_{ij} := \frac{1}{2} \frac{∂^{2}L^{∈}}{∂y^{i}∂y^{j}} and hence

g_{ij}(x, λy) = g_{ij}(x, y). In an F_{n} = (M, L), g_{ij} is derived from L. A more general structure is F_{n} = (M, g), called generalized Finsler space, where we start directly with the metric tensor g_{ij}(x, y).

An affine connection Γ in the Finsler vector bundle ζ can be given locally by the connection coefficients F_{j}^{i}_{k}(x, y), V_{j}^{i}_{h}(x, y) in the form Γξ = ξ − d_{Γ}ξ, where

d_{Γ}ξ^{i}(x, y) = F_{j}^{i}_{k}(x, y)ξ^{j}(x, y)dx^{k} + V_{j}^{i}_{k}(x, y)ξ^{j}(x, y)dy^{k}. (1)

Γ is metrizable if there exists a scalar product g_{ij}(x, y) in each π^{−1}(x, y) such that ∥_{Γ}P_{C}ξ_{0}∥ = constant for any curve C(t) ⊂ M.

2 Connection in μ

We want to find a new, geometric condition for the Riemann metrizability of a vector bundle η = (E, π, M, V^{r}) endowed with the affine connection H_{η} given by Γ_{β}^{α}_{i}(x). First we derive from H_{η} an affine connection H_{μ} in μ = (E_{μ}, π_{μ}, M, V^{r^{2}}), and then from H_{μ} a connection H_{ν} in the bundle ν = (E_{ν}, π_{ν}, M, E), where E is the manifold of the ellipsoids in π^{−1}(x) ≅ V^{r} centered at the origin O of V^{r}.

Let us consider a canonical coordinate system (x^{i}, v^{α}) in π^{−1}(U) ⊂ E, where U ⊂ M is a coordinate neighbourhood of x ∈ M and v^{α} are components of v ∈ π^{−1} ≅ V^{r}. Similarly we have local coordinates (x^{i}, y^{a}) in π_{μ}^{−1}(U) ⊂ E_{μ}, where y^{a}, a = 1, …, r^{2}

are components of y ∈ π_{μ}^{−1}(x) ≅ V^{r^{2}}. Let v^{α}∈ π^{−1}(x) ≅ V^{r}, α, β = 1, …, r be r

vectors with components (v^{α})^{β}. Since any integer a (1 ≤ a ≤ r^{2}) can uniquely be represented in the form a = (α − 1)r + β, and conversely, any pair α, β uniquely determines such an a and thus

(2)  y^{a} = (v^{α})^{β}, a = (α − 1)r + β

determines a 1:1 mapping between π_{μ}^{−1}(x) and the vector r-tuples (v^{1}, …, v^{r}) which

can be considered as elements of ⊕^{r} π^{−1}(x) ≅⊕^{r} V^{r}.

Having an affine connection H_{η} in η with local connection coefficients Γ_{β}^{α}_{i}(x), we obtain for the parallel translated of v from x to x + dx

ΓP_{x,x+dx}v(x) = v(x) − d_{Γ}v(x), d_{Γ}v^{β}(x) = Γ_{σ}^{β}_{i}(x)v^{σ}dx^{i}.

Then we define an affine connection H_{μ} in μ with local coefficients G_{b}^{a}_{i}(x) by

(3)  d_{G}y := (d_{Γ} v^{1}, …, d_{Γ} v^{r}), \begin{matrix}y = (v^{1}, …, v^{r}) \\ d_{Γ}(v^{α})^{β} = Γ_{σ}^{β}_{i}(x)(v^{α})^{σ}dx^{i}. \end{matrix}

G_{b}^{a}_{i} can be expressed explicitely by Γ_{β}^{α}_{i} as follows:

(4)  \begin{aligned}d_{G}y^{a} & = & G_{b}^{a}_{i}(x)y^{b}dx^{i} \\ & = & d_{G}y^{(α−1)r+β} = G_{(κ−1)r+λ}^{(α−1)r+β}_{i}(x) y^{(κ−1)r+λ}dx^{i}, \end{aligned}

since a = (α − 1)r + β, b = (κ − 1)r + λ. By (3) and (2) we get

(5)  \begin{aligned}d_{G}y^{(α−1)r+β} & = & d_{Γ}(v^{α})^{β} = Γ_{σ}^{β}_{i}(x)(v^{α})^{σ}dx^{i} = \\ & = & Γ_{σ}^{β}_{i}(x)y^{(α−1)r+σ}dx^{i}. \end{aligned}

From (4) and (5) we obtain

\begin{aligned}G_{(κ−1)r+λ}^{(α−1)r+β}_{i}(x) y^{(κ−1)r+λ} & = & Γ_{σ}^{β}_{i}(x)δ_{κ}^{α}δ_{λ}^{σ}y^{(κ−1)r+λ} = \\ & = & Γ_{λ}^{β}_{i}(x)δ_{κ}^{α}y^{(κ−1)r+λ} \end{aligned}

and hence

G_{(κ−1)r+λ}^{(α−1)r+β}_{i}(x) = δ_{κ}^{α}Γ_{λ}^{β}_{i}(x).

3 Connection in ν

An ellipsoid E in π^{−1}(x) ≅ V^{r} centered at the origin O of V^{r} has the equation a_{αβ}v^{α}v^{β} = 1, a_{αβ} = a_{βα}, Det|a_{αβ}| > 0. The set {E} = E can be given a natural manifold structure, namely each E can be identified with the coefficients a_{αβ} which correspond to a point of R^{r^{2}}. Hence E can be identified with a variety of the Euclidean space R^{r^{2}}. Thus ν = (E_{ν}, π_{ν}, B, E) is a fiber bundle.

Now we want to derive from the H_{μ} determined by the affine connection H_{η} a

connection H_{ν} in ν : H_{η} ⇒ H_{μ} ⇒ H_{ν}. — Let y = (v^{1}, …, v^{r}) ∈ π_{μ}^{−1}(x) ⊂ E_{μ} be

such that v^{1}, …, v^{r} are linearly independent vectors in π^{−1}(x). From now on, in this section y denotes elements of E_{μ} with this independence property. The set of these

(x, y)-s will be denoted by E_{μ^{∗}} and the corresponding bundle by μ^{∗}= (E_{μ^{∗}}, π_{μ^{∗}}, M, V_{∗}^{r^{2}}).

We remark that V_{∗}^{r^{2}} is no vector space, and π_{μ^{∗}} is a restriction of π_{μ} to E_{μ}^{∗} ⊂ E_{μ}. H_{μ} is equivalent with the splitting T_{u}E_{μ} = V_{u}E_{μ} ⊕ H_{u}E_{μ}, u ∈ E_{μ}. The restriction of an affine connection H_{μ} to E_{μ^{∗}} ⊂ E_{μ} is also a connection in E_{μ^{∗}}, i.e. H_{μ} ⊂ E_{μ^{∗}} if u ∈ E_{μ^{∗}} ⊂ E_{μ}. This is so, because H_{η} takes by parallel translation linearly independent vectors of π^{−1}(x) into linearly independent vectors again. Also, H_{μ^{∗}} can be extended by continuity to a H_{μ}, and if H_{μ^{∗}} is a restriction of an affine connection H_{μ}, then its extension yields this H_{μ}.

The vectors v^{α} of a y can be considered as a system of conjugate axes of an ellipsoid E ∈ π_{ν}^{−∞}(§) centered at the origin O, and we order this E to y. Doing this with every (x, y) we obtain a strong bundle mapping

ρ : E_{μ^{∗}} → E_{ν}, π−1 ; μ^{∗}(x) → π_{ν}^{−1}(x), y ↦ E.

The inverse ρ^{−1}(E) = {†_{′}, †_{∞}, …, †, …} is an infinite set consisting of y_{0} = (v^{1}_{0}, …, v^{r}_{0}

), y_{1} = (v^{1}_{1}, …, v^{r}_{1}), …, y = (v^{1}, …, v^{r}), … such that every system v^{1}_{0}, …, v^{r}_{0}; v^{1}_{1}, …, v^{r}_{1}

; …; v^{1}, …, v^{r}; … forms conjugate axes of an ellipsoid E. Elements of ρ^{−1}(E) can be generated from a single element, e.g. from y_{0} as follows: Let V_{0}^{r} be a Euclidean vector space with an orthonormed base e^{α} and a : π^{−1}(x) → V_{0}^{r} an affine mapping taking v^{α}_{0} into e^{α}. Then the set {v^{α}= a^{−1} ∘ f ∘ a v^{α}_{0}, α = 1, …, r | f ∈ O(r)} produces all

vector systems y = (v^{1}, …, v^{r}) of ρ^{−1}(E), where O(r) denotes the group of rotations of V_{0}^{r}. This induces a classification of π−1 ; μ^{∗}(x) into equivalence classes, and ρ is a 1 : 1

mapping between the equivalence classes and the ellipsoids.

H_{μ} takes π_{μ}^{−1}(x) into π_{μ}^{−1}(x+dx) and so it takes y ∈ π−1 ; μ^{∗}(x) into ŷ ∈ π−1 ; μ^{∗}(x+dx).

However, according to (3), H_{μ} is defined via H_{η}, and in such a way that the images ŷ_{0}, ŷ_{1}, … ŷ, … by H_{μ} of the elements of an equivalence class {y_{0}, y_{1}, …, y, …} (i.e. of conjugate axes systems of an ellipsoid E) form again an equivalence class in π−1 ; μ^{∗}(x+dx)

(i.e. ŷ_{0}, ŷ_{1}, …, ŷ, … are conjugate axes systems of an ellipsoid again). This is shown on the diagram

(6)  \begin{matrix}ρ(x)\{y_{0}, y_{1}, …, y, …\} = E(§) ∈ π_{ν}^{−∞}(§) \\ ↓ H_{μ}  ↓ H_{ν} \\ ρ(x + dx)\{ŷ_{0}, ŷ_{1}, …, ŷ, …\} = Ê(x + dx) ∈ π_{ν}^{−1}(x + dx). \end{matrix}

It means that H_{μ} : π−1 ; μ^{∗}(x) → π−1 ; μ^{∗}(x + dx) preserves equivalence classes. Thus

ρ ∘ H_{μ} ∘ ρ^{−1} : π_{ν}^{−1}(x) → π_{ν}^{−1}(x + dx)

yields a connection H_{ν} in ν (This fact is discussed in more detail in [10], [11]).

If H_{ν} is integrable at least for one E_{′} ∈ π_{ν}^{−∞}(§_{′}) and E(§), E(§_{′}) = E_{′} is the integral manifold, then E(§) can be considered as indicatrix I(x) and g_{αβ}(x) in the equation g_{αβ}(x)v^{α}v^{β} = 1 of E(§) as metric tensor. Any v_{0} leading to a point of E_{≀} : ⊑_{′} ∈ E_{′} can be an axe of a conjugate axes system of E_{′}. Then, according to our construction, the parallel translated v of v_{0} according to H_{η} along a curve C ⊂ M from x_{0} to x is an element of E(§):

H_{η}P_{C;x_{0},x}v_{0} = v ∈ _{H_{ν}} P_{C;x_{0},x}E_{′} = E(§),

and hence

∥v_{0}∥_{g(x_{0})} = ∥v∥_{g(x)}.

We remark that v depends on the path C joining x_{0} and x, but E(§) does not. — This means: if H_{ν} is integrable, then H_{η} is metrizable.

The converse is obvious. If H_{η} is metrical with respect to g(x), then E(§) := I(§) is an integral manifold of H_{ν}.

Thus we obtain the Theorem. The affine connection H_{η} of a vector bundle η is Riemann metrizable iff the constructed connection H_{ν} in a bundle ν fibered with ellipsoids is integrable.

4 Coefficients of H_{ν}

We want to determine the connection coefficients of H_{ν}. H_{ν} orders to the ellipsoid E(§)

(7)  a_{αβ}(x)v^{α}v^{β} = 1 ∈ π_{ν}^{−1}(x)

the ellipsoid Ê(x + dx)

(8)  a_{αβ}(x + dx)v^{α}(x + dx)v^{β}(x + dx) = 1 ∈ π_{ν}^{−1}(x + dx).

According to the definition (construction) of H_{ν} this last equation is satisfied by the parallel translated with respect to H_{η} of v^{α}(x), i.e. by v^{α}(x + dx) = v^{α}(x) − Γ_{σ}^{α}_{i}(x)v^{σ}(x)dx^{i} + o(dx^{i}). (Since we work with linear connections, o(dx^{i}), i.e. higher order terms in dx^{i}, can be omitted.) Then the parallel translated of a_{αβ}(x) according to H_{ν} are the a_{αβ}(x + dx) appearing in (8). Denoting the connection coefficients of H_{ν} by M_{αβi}(x, a_{κλ}) we obtain from (8)

(a_{αβ} + M_{αβi}(x, a_{κλ})dx^{i})(v^{α} − Γ_{σ}^{α}_{i}v^{σ}dx^{i})(v^{β} − Γ_{σ}^{β}_{i}v^{σ}dx^{i}) = 1

or

a_{αβ}v^{α}v^{β} + [M_{αβi} − a_{κλ}(Γ_{β}^{λ}_{i}δ_{α}^{κ} + Γ_{α}^{κ}_{i}δ_{β}^{λ})] v^{α}v^{β}dx^{i} + o(dx^{i}) = 1.

By (7) the right hand side drops out with a_{αβ}v^{α}v^{β}. The remaining expression must vanish for every v ∈ E(§) and for every dx^{i}. Thus, omitting o(dx^{i}), we get

M_{αβi}(x, a_{κλ}) = (Γ_{β}^{λ}_{i}δ_{α}^{κ} + Γ_{α}^{κ}_{i}δ_{β}^{λ})a_{κλ}.

This means that M_{αβi}(x, a_{κλ}) is linear in a_{κλ}, i.e. H_{ν} is an affine connection and its connection coefficients are

(9)  M_{αβ}^{κλ}_{i}(x) = Γ_{α}^{κ}_{i}(x)δ_{β}^{λ} + Γ_{β}^{λ}_{i}(x)δ_{α}^{κ}.

We remark that these coefficients are symmetric in the sense that M_{αβ}^{κλ}_{i} = M_{βα}^{λκ}_{i}. Thus the symmetry of a_{αβ}(x) implies the symmetry of a_{αβ}(x + dx) = a_{αβ}(x) + M_{αβ}^{κλ}_{i}(x)a_{κλ}dx^{i} too, which are the coefficients of Ê(x + dx).

The condition of the absolute parallelism of a_{αβ}(x) with respect to H_{ν} is

\frac{∂a_{αβ}}{∂x^{i}} = −M_{αβ}^{κλ}_{i}(x)a_{κλ}(x).

This is integrable iff

T_{αβ}^{κλ}_{ij}(x)a_{κλ}(x) = 0

T_{αβ}^{κλ}_{ij} ≡ (\frac{∂M_{αβ}^{κλ}_{i}}{∂x^{j}} − M_{αβ}^{μν}_{i}M_{μν}^{κλ}_{j})_{[i,j]}

has a solution for a_{κλ} with positive determinant. We find that

T_{αβ}^{κλ}_{ij} = R_{α}^{κ}_{ij}δ_{β}^{λ} + R_{β}^{λ}_{ij}δ_{α}^{κ},

where R is the curvature tensor of Γ_{β}^{α}_{i}(x).

5 Finsler vector bundles

Considering a Finsler vector bundle ζ = (E, π, T M, V^{n}) and a connection Γ with connection coefficients F_{j}^{i}_{h}(x, y), V_{j}^{i}_{h}(x, y) we have (1). In this case the base manifold T M has dimension 2n. Its coordinates can be denoted by u^{A}, A = 1, …, 2n; u^{i} = x^{i}, u^{n+i} = y^{i}. E(§, †) has the equation a_{ij}(x, y)ξ^{i}ξ^{j} = 1, and the equation of Ê(x + dx) is

a_{ij}(x + dx, y + dy)ξ^{i}(x + dx, y + dy)ξ^{j}(x + dx, y + dy) = 1.

Here

a_{ij}(x + dx, y + dy) = a_{ij}(x) + M_{ij}^{rs}_{h}(x, y)a_{rs}(x, y)dx^{h} + M_{ij}^{rs}_{n+k}(x, y)a_{rs}dy^{h}.

Contrasting with (9), here the last index of M runs from 1 to 2n the other indices from 1 to n. Considerations and calculations similar to those done above yield

\begin{aligned}M_{ij}^{rs}_{h} & = & F_{j}^{s}_{h}δ_{i}^{r} + F_{i}^{r}_{h}δ_{j}^{s} \\ M_{ij}^{rs}_{n+k} & = & V_{j}^{s}_{k}δ_{i}^{r} + V_{i}^{r}_{k}δ_{j}^{s}, \end{aligned}

and furthermore

\begin{aligned}T_{ij}^{rs}_{kh} & = & ^{F} R_{i}^{r}_{kh}δ_{j}^{s} +^{F} R_{j}^{s}_{kh}δ_{i}^{r} \\ T_{ij}^{rs}_{n+k n+h} & = & ^{V} R_{i}^{r}_{kh}δ_{j}^{s} +^{V} R_{j}^{s}_{kh}δ_{i}^{r}, \end{aligned}

where^{F} R and^{V} R are formed from F_{j}^{s}_{i} and V_{j}^{s}_{i} resp. like common curvature tensors. Finally

\begin{matrix}T_{ij}^{rs}_{n+h k} = \frac{∂M_{ij}^{rs}_{n+h}}{∂x^{k}} − \frac{∂M_{ij}^{rs}_{k}}{∂y^{h}} + (V_{j}^{s}_{k}F_{s}^{c}_{h} − F_{j}^{s}_{k}V_{s}^{c}_{h})δ_{i}^{b}+ \\ +V_{j}^{c}_{k}F_{i}^{b}_{h} − F_{j}^{c}_{k}V_{i}^{b}_{h} + V_{i}^{b}_{k}F_{j}^{c}_{h} − F_{i}^{b}_{k}V_{j}^{c}_{h} + (V_{i}^{r}_{k}F_{r}^{b}_{h} − F_{i}^{r}_{k}V_{r}^{b}_{h})δ_{j}^{c}. \end{matrix}

One can use other connections, e.g. a pre-Finsler connection F Γ(F_{j}^{i}_{k}, N^{i}_{j}, V_{j}^{i}_{h}) and h- and v-covariant derivatives

\begin{matrix}ξ^{i}_{|k} = \frac{∂ξ^{i}}{∂x^{k}} − \frac{∂ξ^{i}}{∂y^{r}} N^{r}_{k} + F_{j}^{i}_{k}ξ^{j} \\ ξ^{i}|_{k} = \frac{∂ξ^{i}}{∂y^{k}} + V_{j}^{i}_{k}ξ^{j}. \end{matrix}

In this case (1) becomes

d_{Γ}ξ^{i} = (F_{j}^{i}_{k} − V_{j}^{i}_{r}N^{r}_{k})ξ^{j}dx^{k} + V_{j}^{i}_{k}ξ^{j}dy^{k},

or

d_{Γ}ξ^{i} = [(F_{j}^{i}_{k} − V_{j}^{i}_{r}F_{s}^{r}_{k}y^{s})dx^{k} + V_{j}^{i}_{k}dy^{k}] ξ^{j}

if F Γ is without deflection. These lead to other formulae for M_{ij}^{rs}_{A} and T_{ij}^{rs}_{AB}. If F_{j}^{i}_{k} and V_{j}^{i}_{k} are symmetric, F Γ is without deflection and metrizable, then F Γ is the Cartan connection.

Finally we mention still another affine connection introduced by M. Matsumoto [7], [8] (see also [2], [3]) which is an ordinary affine connection derived from a Finsler connection F Γ(F_{j}^{i}_{k}, N^{i}_{j}, V_{j}^{i}_{k}). Starting with an F Γ and a nonvanishing vector field Y (x) which depends on the point x only

(10)  \underline{Γ}_{j}^{i}_{k}(x) := F_{j}^{i}_{k}(x, Y (x)) + V_{j}^{i}_{r}(x, Y (x)) (\frac{∂Y^{r}}{∂x^{k}} + Y^{s}(x)F_{s}^{r}_{k}(x, Y (x)))

turn out to be connection coefficients of an ordinary affine connection. Using the vector field Y (x) one can associate to any Finsler vector field ξ^{i}(x, y) an ordinary vector field \underline{ξ}^{i}(x) := ξ^{i}(x, Y (x)). Then there exists a nice relation among the covariant derivative \underline{ξ}^{i}_{;k} constructed with \underline{Γ}, and the h- and v-covariant derivatives with respect to F Γ, namely

\underline{ξ}^{i}_{;k} = [ξ^{i}_{|k} + ξ^{i}|_{k} (\frac{∂Y^{r}}{∂x^{k}} + Y^{s}F_{s}^{r}_{k})]_{|_{y=Y (x)}} .

Given a \underline{Γ} and a Y (x), there are many F Γ which satisfy (10). Then we can use our method to search for metrizable ones among these F Γ, e.g. for such, where F Γ satisfies (10) with the given \underline{Γ} and Y (x) and g_{ij|k} = g_{ij}|_{k} = 0 with respect to this F Γ.

Acknowledgements. This research was supported by OTKA: T - 17261.

Erratum. Received August 4, 1994; Revised August 15, 1995. Editorial Board decided to reprint this paper since the first printing in BJGA, 1,1 (1996), pp.83-90, contains several missprints due to e-mail and computer transmission.

References

[1] S. Golab, Uber die Metrisierbarkeit der affin-zusammenhangenden Raume, Tensor N.S., 9 (1959), 1-7.

[2] R. S. Ingarden, M. Matsumoto and L. Tamássy, Vector-field relative Finsler space in two physical examples, to appear.

[3] R. S. Ingarden and M. Matsumoto, On the 1953 Barthel connection of a Finsler space and its physical aspects, Publ. Math. Debrecen, vol. 43, 87-90, 1993.

[4] A. Jakubowicz, Uber die Metrisierbarkeit der affin-zusammenhangenden Raume I., II., III. Tensor N. S., vol 14, 132–137, 1963; vol 17, 28-43, 1966; vol 18, 259-270, [5] O. Kowalski, On regular curvature structures, Math. Z., vol 125, 129-138, 1972.

[6] D. Krupka and A. E. Sattarov The inverse problem of the calculus of variations for Finsler structures, Math. Slovaca, vol 35, 217-222, 1985.

[7] M. Matsumoto, Contributions of prescribed supporting element and the Cartan connection, Tensor N.S., vol 49, 9-17, 1990.

[8] M. Matsumoto, Theory of Y-extremal and minimal hypersurfaces in a Finsler space, J. Math. Kyoto Univ., vol 26, 647-665, 1986.

[9] Z. I. Szabó, Positive definite Berwald spaces, Tensor N.S., vol 35, 25-39, 1981.

[10] L. Tamássy and L. Kozma, Connections in k-vector bundles, Math. Pannonica, vol 6, 105-114, 1995.

[11] L. Tamássy, Induced areal spaces and reduction of the connection in an areal space to connection in a vector bundle, An. Sţ. Univ. Ovidius Constantza, vol 3, to appear, 1996.

[12] T. Y. Thomas, The differential invariants of generalized spaces, Cambridge, 1934.

Institute of Mathematics Kossuth Lajos University Pf.12, Debrecen H-4010 Hungary