Finsler Connections in Generalized Lagrange Spaces

Mihai Anastasiei

Abstract The Chern–Rund connection from Finsler geometry is settled in the generalized Lagrange spaces. For the geometry of these spaces, we refer to [5].

Mathematics Subject Classification: 53C60 Key words: Finsler connections, generalized Lagrange spaces, Chern–Rund connection.

Introduction

In a recent paper, [1], we showed that in a Finsler space the connection introduced by S.S. Chern in 1948 is the same with the connection proposed by H. Rund ten years later and bearing his name. Accordingly, we proposed the name of Rund be replaced with that of Chern, but several geometers including S.S. Chern himself, suggested to call it from now on a Chern–Rund connection.

As S.S. Chern and D. Bao showed in [2], the Chern–Rund connection is very convenient in treating of many global problems in Finsler geometry. This fact determined us to come back to the subject.

The efforts made in defining a covariant derivative and accordingly, a parallel displacement in Finsler space led to a concept generically called a Finsler connection. Among the Finsler connections there exist four, which are remarkable by their properties named the Cartan, Berwald, Chern–Rund and Hashiguchi connections, respectively. These are usually put together in a nice commutative diagram (cf. [3, Ch. III]).

The most utilized is the Cartan connection because it is fully metrical i.e. h– and v–metrical, in spite of the fact it has torsion.

But there are some problems involving the Berwald connection which is by no means metrical or the Hashiguchi connection which is only v–metrical.

The Chern–Rund connection being h–metrical and free of torsion is the nearest to the Levi–Civita connection a fact which explains its adequacy for global problems in Finsler geometry.

Balkan Journal of Geometry and Its Applications, Vol.1, No.2, 1996, pp. 1-9 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

The Finsler connections are also suitable for the geometries more general than the Finslerian one as the Lagrange geometry or generalized Lagrange geometry. Our purpose is to review Finsler connections and to settle the Chern– Rund connection in this more general framework.

First, we give in §1 a definition of Finsler connection by local components and introduce its compatibility with a generalized Lagrange metric. Then, in §2, a Finsler connection is defined as a pair (N, ∇), where N is a nonlinear connection on T M and ∇ is a linear connection in the pull–back bundle τ^{−1} : M⟶T M with τ : T M⟶M, the tangent bundle over a manifold M. These definitions are equivalent. The four remarkable connections mentioned above are characterized. A special attention is paid to the possibility of determining N from ∇.

1 Finsler connections. A definition by local components

Let M be a smooth i.e. C^{∞} manifold of finite dimension n and τ : T M → M its tangent bundle. A local chart (U, (x^{i})) on M induces a local chart (τ^{−1}(U), (x^{i}, y^{i})) on T M, where x^{i} ≡ x^{i} ∘ τ and (y^{i}) are provided by

u = y^{i} \frac{∂}{∂x^{i}} |_{p}, p = τ(u).

A change of coordinates (x^{i}, y^{i})⟶(x̃^{i}, ỹ^{i}) on T M has the form

(1.1)  \begin{matrix}x̃^{i} = x̃^{i}(x^{1}, ..., x^{n}), rank (\frac{∂x̃^{i}}{∂x^{j}} ) = n \\ ỹ^{i} = \frac{∂x̃^{i}}{∂x^{j}} (x)y^{j}. \end{matrix}

The indices i, j, k, ... will run from 1 to n and Einstein’s convention on summation is implied.

Let L : T M⟶R be a scalar function on T M. Then L̃(x̃(x), ỹ(y)) = L(x, y), from which, taking partial derivatives and using (1.1), one gets

(1.2)  \frac{∂L}{∂y^{i}} = \frac{∂x̃^{k}}{∂x^{i}} \frac{∂L}{∂ỹ^{k}} ,

(1.3)  \frac{∂L}{∂x^{i}} = \frac{∂x̃^{k}}{∂x^{i}} \frac{∂L}{∂x̃^{k}} + \frac{∂^{2}x̃^{k}}{∂x^{j}∂x^{i}} y^{j} \frac{∂L}{∂ỹ^{k}} ·

According to (1.2), the set of functions (\frac{∂L}{∂y^{i}} (x, y)) may be regarded as the com-

ponents of a covector field on T M. From (1.2), it follows that (\frac{∂^{2}L}{∂y^{i}∂y^{j}} (x, y))

may be also viewed as the components of a (symmetric) tensor field on T M.

Thus on T M there exist geometrical objects whose law of transformation under (1.1) is the same as of the corresponding objects on M. These were called d–objects (d is from distinguished) in [4], Finsler objects in [3] and sometimes M–objects.

The geometry of d–objects is essentially involved in the study of those metrical structures which are more general than Riemannian structures i.e. Finsler structures, Lagrange structures, generalized Lagrange structures (see [4]).

Coming back to (1.3), we see that the behaviour of the operators \frac{∂}{∂x^{i}} is

drastically different from that of \frac{∂}{∂y^{i}} · Let us introduce a correction of \frac{∂}{∂x^{i}} := ∂_{i},

(1.4)  δ_{i}L = ∂_{i}L + N_{i}^{k}(x, y) ˙∂_{k}, ˙∂k̇ := \frac{∂}{∂yk̇} ,

such that, with respect to (1.1):

(1.5)  δ_{i}L = \frac{∂x̃^{k}}{∂x^{i}} δ̃_{k}L,

i.e. (δ_{i}L) to appear as the components of a covector field on T M. Then the functions (N_{i}^{k}(x, y)) have to satisfy

(1.6)  \frac{∂x̃^{j}}{∂x^{i}} Ñ_{j}^{h} = N_{i}^{j} \frac{∂x̃^{h}}{∂x^{j}} + \frac{∂^{2}x̃^{h}}{∂x^{i}∂x^{j}} y^{j}.

Note that (N_{i}^{j}(x, y)) are not the components of a (1, 1)–tensor field on T M but the difference of two sets of this type is so.

As it is well–known, when M is paracompact, there exists on M a linear connection, say of local coefficients (Γ_{jk}^{i}(x)). Then N_{k}^{i}(x, y) = Γ_{jk}^{i}(x)y^{j} verify (1.6). This example assures also the existence of a nonlinear connection within a generally accepted hypothesis on M.

The local vector fields (δ_{i}), i = 1, 2, ..., n, given by (1.4) are linearly independent and in a point u ∈ T M they span an n–dimensional subspace H_{u}T M of T_{u}T M.

Let τ_{∗,u} be the tangent mapping (the Jacobian) of τ. Then V_{u}T M = ker τ_{∗,u} is called the vertical subspace of T_{u}T M. A vertical vector is of the form X^{k}(x, y) ˙∂_{k} such that under (1.1) one has

(1.7)  X̃^{k} = \frac{∂x^{k}}{∂x^{i}} X^{i}.

We immediately have

(1.8)  T_{u}T M = V_{u}T M ⊕ H_{u}T M.

Furthermore, τ_{∗,u} restricted to H_{u}T M gives an isomorphism of it with T_{τ(u)}M such that τ_{∗,u}(δ_{i}) = ∂_{i}|_{τ(u)}.

Conversely, if a supplement of V_{u}T M in T_{u}T M is specified by a basis (δ_{i}), i = 1, 2, ..., n, which is carried by τ_{∗} to (∂_{i}), then letting δ_{i} = ∂_{i} − N_{i}^{k} ˙∂_{i}, the condition δ_{i} = \frac{∂x̃^{k}}{∂x^{i}} δ̃_{k} implies (1.6) for (N_{i}^{k}). One says that (N_{i}^{k}(x, y)) are the coefficients of a nonlinear connection.

A reason for this term is that when (N_{i}^{k}) are linear with respect to (y) i.e. N_{i}^{k}(x, y) = G_{ji}^{k}(x)y^{j}, then (G_{ji}^{k}) are the coefficients of a linear connection on M.

Summarizing the foregoing discussion we may formulate the following two equivalent definitions for a nonlinear connections. Definition 1.1. A nonlinear connection is a set of functions (N_{j}^{i}(x, y)) defined on each domain of local chart on T M such that an overlaps, (1.6) holds good. Definition 1.2. A nonlinear connection is a smooth distribution u⟶H_{u}T M supplementary to the vertical distribution u⟶V_{u}T M i.e. (1.7) holds good for every u ∈ T M.

Let (v^{i}(x, y)) be the components of a d–vector field. Then (\frac{∂v^{i}}{∂y^{j}} (x, y))

are the components of a d–tensor field of type (1, 1). In other words the partial derivatives with respect to (y^{i}) are covariant. However, in some circumstances, these have to be replaced by

(1.9)  v^{i}_{|j} = \frac{∂v^{i}}{∂y^{j}} + C_{kj}^{i}(x, y)v^{k},

where (C_{kj}^{i}(x, y)) are the components of a d–tensor field. One of them is as follows.

First, we introduce Definition 1.3. A d–tensor field of type (0, 2) of components (g_{ij}(x, y)) which is a) symmetric, i.e. g_{ij} = g_{ji}, b) nondegenerate i.e. det(g_{ij}(x, y)) ≠ 0 and c) the quadratic form g_{ij}(x, y)ξ^{i}ξ^{j} (ξ ∈ IR^{n}) has constant signature is called a generalized Lagrange metric (GL–metric for brevity).

Extending (1.9), the covariant derivative of (g_{ij}) is given by

(1.10)  g_{ij|k} = ∂_{j}v^{i} − C_{ik}^{h}g_{hj} − C_{jk}^{h}g_{ih}.

One says that the GL–metric (gij(x, y)) is v–covariant constant if g_{ij|k} = 0. For

the general v^{i}_{|j}, the condition g_{ij|k} = 0 can be fulfilled with

(1.11)  C^{c} _{ij}^{h} = \frac{1}{2}g^{hk}( ˙∂_{i}g_{kj} + ˙∂_{j}g_{ik} − ˙∂_{k}g_{ij}).

The partial derivatives with respect to (x^{i}) are far to be covariant derivatives. A correction of them could be ∂_{j}v^{i} + H_{kj}^{i}(x, y)v^{k}, but (H_{kj}^{i}(x, y)) have a complicated law of transformation A better one is

(1.12)  v^{i}_{|j} = δ_{j}v^{i} + F_{kj}^{i}(x, y)v^{k},

since then (F_{kj}^{i}(x, y)) changes under (1.1) as the local coefficients of a linear connection on M. These derivatives can be extended to any d–tensor field. For instance, the v–covariant derivative of (g_{ij}(x, y)) is given by (1.10) and its h– covariant derivative is

(1.13)  g_{ij|k} = δ_{k}g_{ij} − F_{ik}^{h}g_{hj} − F_{jk}^{h}g_{ih}.

The GL–metric (g_{ij}(x, y)) is said to be h–covariant constant if g_{ij|h} = 0. It is easy to check that the equation g_{ij|h} = 0 is satisfied with

(1.14)  F^{c} _{ij}^{k} = \frac{1}{2}g^{kh}(δ_{i}g_{hj} + δ_{j}g_{ih} − δ_{h}g_{ij}).

The foregoing discussions suggest Definition 1.4. A Finsler connection is a triad F Γ = (N_{j}^{i}(x, y), F_{jk}^{i}(x, y), C_{jk}^{i}(x, y)), where N_{j}^{i}(x, y) are the coefficients of a nonlinear connection, F_{jk}^{i}(x, y) are like the coefficients of a linear connection on M and C_{jk}^{i}(x, y) are the components of a d–tensor field.

We have also got a first example of Finsler connection CΓ = (N_{j}^{i}(x, y),

F^{c} _{jk}^{i}(x, y), C^{c} _{jk}^{i}(x, y)).

Definition 1.5. Let F Γ be a Finsler connection and (g_{ij}(x, y)) a GL–metric. F Γ is said to be h–metrical if g_{ij|h} = 0, v–metrical if g_{ij|h} = 0 and metrical if

the both equations hold.

In the above we have proved Proposition 1.1. The Finsler connection CΓ is metrical.

The following d–tensor fields are called the torsions of F Γ :

(1.15)  \begin{matrix}T_{jk}^{i} = F_{jk}^{i} − F_{kj}^{i}, & R_{jk}^{i} = δ_{k}N_{j}^{i} − δ_{j}N_{k}^{i}, C_{jk}^{i}, \\ P_{jk}^{i} = ˙∂_{k}N_{j}^{i} − F_{kj}^{i}, & S_{jk}^{i} = C_{jk}^{i} − C_{kj}^{i}. \end{matrix}

Remark 1.1. R_{jk}^{i} is the integrability tensor of the horizontal distribution. It measures also the curvature of the nonlinear connection N.

The d–tensor fields

(1.16)  D_{j}^{i} = F_{kj}^{i}y^{k} − N_{j}^{i}, d_{j}^{i} = δ_{j}^{i} + C_{kj}^{i}y^{k},

where (δ_{j}^{i}) is Kronecker’ symbol, are called h–deflection and v–deflection of F Γ, respectively.

From (1.6) we infer that G_{jk}^{i} = ˙∂_{j}N_{k}^{i} transform under (1.1) as F_{jk}^{i}. Thus BΓ = (N_{j}^{i}, G_{jk}^{i}, 0) is a Finsler connection. It will be called the Berwald connection. This connection is neither v–metrical nor h–metrical and is free of torsions if and only if N is integrable (R_{jk}^{i} = 0) and symmetric ( ˙∂_{j}N_{k}^{i} = ˙∂_{k}N_{j}^{i}).

The connection CΓ will be called the Cartan connection. It is h–metrical, h–symmetric (F^{c} _{jk}^{i}(x, y) =F^{c} _{kj}^{i}(x, y)), v–metrical and v–symmetric. The Finsler

connection HΓ = (N_{j}^{i}, G_{jk}^{i}(x, y), C^{c} _{kj}^{i}(x, y)) will be called the Hashiguchi connection. This is v–metrical, no h–metrical and has torsion. The Finsler connection CRΓ = (N_{j}^{i}, F^{c} _{jk}^{i}(x, y), 0) will be called the Chern–Rund connection. This is h–metrical but not v–metrical.

Summarizing, for a fixed nonlinear connection N and a GL–metric (g_{ij}(x, y)) we have four typical Finsler connections: BΓ, CΓ, HΓ and CRΓ.

Let us replace T M by T_{0}M = T M \ 0.

A GL–metric (g_{ij}(x, y)) on T_{0}M reduces to a Finsler metric if there exists a fundamental Finsler function F : T_{0}M⟶R_{+} such that g_{ij}(x, y) =

\frac{1}{2} ˙∂_{i} ˙∂_{j}F^{2}(x, y)). Taking as N the Cartan nonlinear connection of coefficients

N^{c} _{j}^{i} = \frac{1}{2} ˙∂_{j}γ_{oo}^{i}, γ_{oo}^{i} = γ_{jk}^{i}y^{j}y^{k}, γ_{jk}^{i} = \frac{1}{2}g^{ih}(∂_{j}g_{hk} + ∂_{k}g_{jh} − ∂_{h}g_{jk}), the afore men-

tioned Finsler connections reduce to the four remarkable connections in Finsler geometry (3, [Ch. III]).

The form of D_{j}^{i} in (1.6) shows that one may associate to any F Γ a new Finsler connection (F_{kj}^{i}y^{k}−D_{j}^{i}, F_{kj}^{i}, C_{kj}^{i}) whose h–deflection is just D_{j}^{i}, when this is prescribed. In particular, for D_{j}^{i} = 0 a Finsler connection without h–deflection is obtained. In Finsler geometry BΓ, CΓ, HΓ and CRΓ are h–deflection free. So we have an explanation why the nonlinear connection was noted quite late in Finsler geometry.

2 Another definition of Finsler connections

Let be τ^{−1}T M = {(u, v) ∈ T M × T M, τ(u) = τ(v)} fibered over T M by π(u, v) = u. The local fiber in (u, v) is T_{τ(u)}M. A section in (τ^{−1}T M, π, T M) is locally of the form X̄ = X̄^{i}(x, y)¯∂_{i} with (¯∂_{i}) the natural basis in T_{τ(u)}M. It follows that under (1.1) we have

(2.1)  ˜  X̄^{i} = \frac{∂x̃^{i}}{∂x^{k}} X̄^{k}.

X̄ will be called a τ–vector field on T M. It can be identified with the d–vector field (X̄^{i}(x, y)). More general, the tensorial algebra of the pull–back bundle τ^{−1}T M can be thought of as algebra of d–tensor fields on T M. There exists a remarkable τ–vector field C C : u⟶(u, u), which locally is y^{i} ¯∂_{i} and so it can be identified to the Liouville vector field C C = y^{i} ˙∂_{i}. Theorem 2.1. There exists a one–to–one correspondence between the set of Finsler connections F Γ and the set of pairs (N, ∇) with N a nonlinear connection on T M and ∇ a linear connection in the pull–back bundle τ^{−1}T M. Proof. If F Γ is specified by (N_{j}^{i}, F_{jk}^{i}, C_{jk}^{i}), we take N = (N_{j}^{i}) and define ∇ by

(2.2)  ∇_{δ_{k}} ¯∂_{i} = F_{jk}^{i} ¯∂_{i}, ∇ ˙∂_{k} ¯∂_{j} = C_{jk}^{i} ¯∂_{i}.

In the natural basis ∇ takes the form

(2.3)  ∇_{∂_{k}} ¯∂_{j} = Γ_{jk}^{i} ¯∂_{i}, ∇ ˙∂_{k} ¯∂_{i} = C_{jk}^{i} ¯∂_{i}.

(2.4)  Γ_{jk}^{i} = F_{jk}^{i} + N_{k}^{h}C_{jh}^{i}.

Conversely, given N = (N_{j}^{i}) and ∇ specified by (2.3) it results that (N_{j}^{i}, F_{jk}^{i}, C_{jk}^{i}) with F_{jk}^{i} given by (2.4) is a Finsler connection.

A GL–metric (g_{ij}(x, y)) defines a metrical structure g in the bundle τ^{−1}T M:

(2.5)  g = g_{ij}(x, y)dx^{i} ⊗ dx^{j}.

Conversely, any metrical structure in the bundle τ^{−1}T M defines by (2.5) a GL–metric.

One easily checks Theorem 2.2. In the correspondence F Γ ⟷ (N, ∇) we have

a) F Γ is h–metrical if and only if ∇_{hX}g = 0,

b) F Γ is v–metrical if and only if ∇_{vX}g = 0,

c) F Γ is metrical if and only if ∇_{X}g = 0,

for every X ∈ X (T M).

Let ρ : T T M⟶τ^{−1}T M be the morphism of vector bundles given by ρ(X_{u}) = (u, τ_{∗,u}X_{u}), X_{u} = T_{u}T M, u ∈ T M. It follows that ker ρ_{u} = V_{u}T M i.e. ρ( ˙∂_{i}) = 0 and ρ(δ_{i}) = ¯∂_{i}. Alternatively, we may define a morphism σ : T T M⟶τ^{−1}T M on basis by σ(δ_{i}) = 0, σ( ˙∂_{i}) = ¯∂_{i}. We say that

(2.6)  \begin{matrix}TT_{ρ}(X, Y) = ∇_{X}ρ(Y) − ∇_{Y} ρ(X) − ρ[X, Y], \\ TT_{σ}(X, Y) = ∇_{X}σ(Y) − ∇_{Y} σ(X) − σ[X, Y], X, Y ∈ X (T M), \end{matrix}

are torsions of ∇.

The following characterizations of the Finsler connections BΓ, HΓ, CRΓ and CΓ follow. Theorem 2.3. In the correspondence F Γ ⟷ (N, ∇) we have

a) BΓ ⟷ (N, ∇) with TT_{σ}(hX, vY) = 0, TT_{ρ}(hX, vY) = 0;

b) HΓ ⟷ (N, ∇) with TT_{σ}(hX, vY) = 0, TT_{σ}(vX, vY) = 0, ∇_{vX}g = 0;

c) CRΓ ⟷ (N, ∇) with TT_{ρ}(hX, vY) = 0, TT_{ρ}(hX, hY) = 0, ∇_{hX}g = 0;

d) CΓ ⟷ (N, ∇) with TT_{ρ}(hX, vY) = 0, TT_{σ}(vX, vY) = 0, ∇_{X}g = 0. Proof. The local expressions of TT_{ρ} and TT_{σ} in conjunction with Theorem 2.2 give the desired results.

Now the following question appears. Which conditions sould satisfy ∇ in order to determine N such that the pair (N, ∇) to correspond to a Finsler connection. An answer is as follows. Definition 2.1. A linear connection ∇ in the pull–back bundle τ^{−1}T M is said to be regular if the subspace {X_{u} | ∇_{X_{u}}C C = 0, X ∈ X (T M)} of T_{u}T M is supplementary to V_{u}T M for every u ∈ T M.

By the definition, every regular connection ∇ induces a nonlinear connection N on T M. The pair (N, ∇), as we have seen before, corresponds to a Finsler connection F Γ. This F Γ has to be of a particular form. Indeed, one has Theorem 2.4. There exists a bijection between the set of regular connections in τ^{−1}T M and the set of Finsler connections F Γ = (N_{j}^{i}, F_{jk}^{i}, C_{jk}^{i}) satisfying D_{j}^{i} = 0 and det(d_{h}^{i}) ≠ 0. Proof. Let ∇ be specified by (2.3). Using N = (N_{j}^{i}) provided by the regularity of ∇, we define F_{jk}^{i} as in (2.4). Then 0 = ∇_{δ_{k}}C C = (y^{j}F_{jk}^{i} − N_{k}^{i})¯∂_{k} implies D_{k}^{i} = 0. Contracting (2.4) by y^{j} we get N_{k}^{h}(d_{h}^{i}) = y^{j}Γ_{jk}^{i} and as (N_{k}^{h}) is specified this equation has to have an unique solution. Hence with necessity det(d_{h}^{i}) ≠ 0.

Conversely, let (N, ∇) be in correspondence with F Γ. The condition D_{j}^{i} =0 assures that the subspace {X_{u}|∇_{X_{u}}C C = 0, X ∈ X (T M), u ∈ T M} is contained in the horizontal subspace H_{u}T M of N. The condition det(d_{k}^{i})≠0 implies that this subspace is supplementary to V_{u}T M. Thus ∇ is regular and the nonlinear connection derived from it coincides with N.

Let us assume that (g_{ij}) reduces to a Finsler metric on T_{0}M. Then CΓ is characterized by the following Matsumoto’s axioms:

(∗)  T_{jk}^{i} = 0, g_{ij|k} = 0, S_{jk}^{i} = 0, g_{ij|k} = 0, D_{j}^{i} = 0.

It results d_{j}^{i} = δ_{j}^{i}.

Combining these with Theorems 2.4 and 2.3, one obtains Theorem 2.5. Let F^{n} = (M, F) be a Finsler space. There exists a unique regular connection ∇ in π^{−1}T_{0}M satisfying the conditions:

TT_{ρ}(hX, hY) = 0, TT_{σ}(vX, vY) = 0, ∇_{X}g = 0, X, Y ∈ X (T_{0}M)

where h and v are projectors of N induced by ∇.

We note that ∇ is determined by F only.

According to 4 the Chern–Rund connection in a Finsler space is characterized by the following axioms:

T_{jk}^{i} = 0, g_{ij|k} = 0, C_{jk}^{i} = 0, D_{j}^{i} = 0.

We have again d_{j}^{i} = δ_{j}^{i}. By Theorems 2.3 and 2.4 we have Theorem 2.6. Let F^{n} = (M, F) be a Finsler space. There exists a unique regular connection ∇ in π^{−1}T_{0}M satisfying the conditions:

TT_{ρ}(hX, hY) = 0, TT_{ρ}(hX, vY) = 0, ∇_{hX}g = 0, X, Y ∈ X (T_{0}M)

where h and v are projectors of N induced by ∇.

The systems of axioms for HΓ and BΓ discussed for minimality in 4 give similar results in view of Theorems 2.3 and 2.4.

The Finsler connections may be viewed also as special liner connections on T M or in the Finsler bundle π^{−1}LM, where LM is the principal bundle of linear frames on M. We refer to 4 and 3, respectively. Aknowledgement. We are indebted to Prof.Dr. Radu Miron who suggested us several improvements of the first version of this paper.

References

[1] M. Anastasiei, On Chern’s connection in Finsler geometry, to appear.

[2] D. Bao, S.S. Chern, On a notable connection in Finsler geometry, Houston J. Math. 19(1), 1993, 137–182.

[3] M. Matsumoto, Foundations of Finsler geometry and special Finsler spaces, Kaiseisha Press, Japan, 1986.

[4] R. Miron, M. Anastasiei, The geometry of Lagrange spaces: Theory and Applications. Kluwer Academic Publishers, FTPH 59, 1994.

[5] R. Miron, T. Aikou, M. Hashiguchi, On minimality of axiomatic systems of remarkable Finsler connections, Rep. Fac. Sci. Kagoshima Univ. (Math. Phys. & Chem.) 26, 41–51, 1993.

Faculty of Mathematics, ”Al.I.Cuza” University of Iaşi R-6600 Iaşi, Romania