Nonholonomic Frames in Finsler Geometry

Ioan Bucătaru

Abstract

We determine a nonholonomic Finsler frame for a class of Generalized Lagrange spaces, for a class of Lagrange spaces with (α, β)-metric and for Finsler spaces with (α, β)-metric. Then, a special Finsler connection induced by such a nonholonomic frame is determined. Finally we study the integrability conditions for Cartan’s structure equations of a Finsler connection.

Mathematics Subject Classification: 53C60 Keywords: Finsler frame, Finsler connection, generalized Lagrange metric, Cartan’s structure equation.

Introduction

In [8,9] P.R. Holland studies a unified formalism that uses a nonholonomic Finsler frame on space-time arising from consideration of a charged particle moving in an external electromagnetic field. In fact, R.S. Ingarden in [10] was first to point out that the Lorentz force law, in this case, could be written as geodesic equations on a Finsler space called Randers space ([16]). In [5,6] a gauge transformation is viewed as a nonholonomic frame on the tangent bundle of a four dimensional base manifold. The geometry that follows from these considerations gives a more unified approach to gravitation and gauge symmetries. In the above mentioned papers, the common Finsler idea used by the physicists R.G. Beil and P.R. Holland is the existence of a nonholonomic frame on the vertical subbundle V T M of the tangent bundle of a base manifold M. This nonholonomic frame relates a semi-Riemannian metric (the Minkowski or the Lorentz metric) with an induced Finsler metric. In [2,3], with P.L.Antonelli we found such a nonholonomic frame for two important classes of Finsler spaces that are dual in the sense of [7]: Randers and Kropina spaces. As Randers and Kropina spaces are members of a bigger class of Finsler spaces, namely the Finsler spaces with (α, β)-metric, it appears a natural question: does a Finsler space with (α, β)-metric have such a nonholonomic frame? As the fundamental tensor of a Finsler space with (α, β)-metric is not so easy to handle with, we didn’t find so far, a direct method to determine a nonholonomic frame for these spaces.

In this paper we find a nonholonomic Finsler frame for a class of Generalized Lagrange spaces introduced and studied by M.Anastasiei and H.Shimada. In [1], the metric tensor of such a Generalized Lagrange space has been called the Beil metric. The Beil metric can be viewed also as a deformation of a Riemannian metric. In this work we consider the most general case of Beil’s metric and we find a nonholonomic frame for it. This frame reduces in a particular case to that considered by R.G.Beil in [5,6]. Then we can use these ideas to find a nonholonomic frame for a class of Lagrange spaces proposed by R.G. Beil, the so-called Lagrange spaces with (α, β)-metric. We prove that the fundamental metric tensor of a Finsler space with (α, β)-metric can be derived from a Riemannian metric using two Beil deformations (1.5). Using these ideas we can find a nonholonomic frame for a Finsler space with (α, β)-metric. As Randers and Kropina spaces are Finsler spaces with (α, β)-metric we may use these techniques to find nonholonomic Finsler frames for these Finsler spaces.

Balkan Journal of Geometry and Its Applications, Vol.7, No.1, 2002, pp. 13-27.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2002.

We prove that every nonholonomic frame induces a special linear connection on the total space of the tangent bundle of the base manifold M. This linear connection has no curvature and the frame is parallel with respect to it. Using the Cartan’s structure equations we show that a special linear connection, called a Finsler connection, has no curvature if and only if it is induced by a nonholonomic Finsler frame. The frame is holonomic if and only if a set of two forms of torsions vanishes.

R.Miron have been studied nonholonomic Finsler frames and the induced Finsler connection in [15] for the so-called strongly non-Riemannian Finsler spaces. M. Matsumoto studied these nonholonomic frames also, in [11], where he called such frames the Miron frames of a strongly non-Riemannian Finsler space. The Miron frame is a natural generalization of the Berwald frame for a two dimensional Finsler space or the Moor frame for a Finsler space of dimension three.

1 Finsler spaces and related Finsler objects

As the Finsler geometry is a part of the geometry of the tangent bundle of a manifold M, we present first some natural geometric objects that live on T M as the vertical distribution, the almost tangent structure. An important tool in the geometry of the tangent bundle is the nonlinear connection. Metric structures on T M are defined and we prove that in some conditions, Lagrange spaces with (α, β)-metric are generalized Lagrange spaces with Beil metric.

We start with a real n-dimensional manifold M of C^{∞}-class. Denote by (T M, π, M) the tangent bundle of the base manifold M and by (T̃ M, π, M), the tangent bundle with the null cross-section removed. For every point p ∈ M, there exist local charts (U, φ = (x^{i})) on p ∈ M and (π^{−1}(U), ϕ = (x^{i}, y^{i})) on u ∈ π^{−1}(p) ⊂ T M such that with respect to these the canonical submersion π has the equations π : (x^{i}, y^{i}) ∈ π^{−1}(U) ↦ (x^{i}) ∈ U. The local charts on T M of the form (π^{−1}(U), ϕ = (x^{i}, y^{i})) are called induced local charts, (y^{i}) are coordinates of vectors y^{i} \frac{∂}{∂x^{i}} |_{p} from T_{p}M, and \frac{∂}{∂x^{i}} |_{p} is the natural basis of T_{p}M.

Denote by π_{∗} the linear map induced by the canonical submersion π : T M → M. As for every u ∈ T M, π_{∗,u} : T_{u}T M → T_{π(u)}M is an epimorphism, then its kernel determines a n-dimensional distribution V : u ∈ T M ↦ V_{u}T M = Kerπ_{∗,u} ⊂ T_{u}T M. We call it the vertical distribution of the tangent bundle. This is the tangent space to the natural foliation induced by the submersion π and consequently we have that the vertical distribution is integrable. If the natural basis of T_{u}T M induced by a local chart (π^{−1}(U), ϕ = (x^{i}, y^{i})) at u is denoted by \{\frac{∂}{∂x^{i}} |_{u}, \frac{∂}{∂y^{i}} |_{u}\}, then \{\frac{∂}{∂y^{i}} |_{u}\} is a basis of V_{u}T M.

For every u ∈ T M we consider the linear map J_{u} : T_{u}T M → T_{u}T M, J_{u} = \frac{∂}{∂y^{i}} |_{u} ⊗ dx^{i}|_{u}^{1}. It is called the almost tangent structure of the tangent bundle (or the vertical endomorphism) and it has the properties: J_{u}^{2} = 0 and KerJ_{u} = ImJ_{u} = V_{u}T M.

We denote by F(T M) the ring of C^{∞}-functions over T M and by X (T M) the F(T M)-module of vector fields over T M. With respect to the Poisson bracket, X (T M) is a real Lie algebra. Then the almost tangent structure J may be taught as an F(T M)-linear map J : X (T M) → X (T M) with the local expression J = \frac{∂}{∂y^{i}} ⊗dx^{i}. 1.1. Definition We call a nonlinear connection on T M a n-dimensional distribution HT M : u ∈ T M ↦ H_{u}T M ⊂ T_{u}T M that is supplementary to the vertical distribution, which means that we have the direct sum:

(1.1)  T_{u}T M = H_{u}T M ⊕ V_{u}T M, ∀u ∈ T M.

As π_{∗,u} : T_{u}T M → T_{π(u)}M is an epimorphism, ∀u ∈ T M, then the restriction of it to H_{u}T M gives us an isomorphism between H_{u}T M and T_{π(u)}M. The inverse map of this isomorphism is denoted by l_{h,u} : T_{π(u)}M → H_{u}T M and it is called the horizontal lift induced by the given nonlinear connection HT M. If we fix an induced local chart (π^{−1}(U), ϕ = (x^{i}, y^{i})) at u ∈ T M, because π_{∗,u} ∘ l_{h,u} = Id_{H_{u}T M} we have that

l_{h,u} (\frac{∂}{∂x^{i}} |_{π(u)}) = \frac{∂}{∂x^{i}} |_{u} − N_{i}^{j}(u) \frac{∂}{∂y^{j}} |_{u} = : \frac{δ}{δx^{i}} |_{u} .

The functions N_{j}^{i} are defined over π^{−1}(U) and are called the local coefficients of the nonlinear connection HT M. For every u ∈ T M and a local chart (π^{−1}(U), ϕ = (x^{i}, y^{i})) at u we have now a basis \{\frac{δ}{δx^{i}} |_{u}, \frac{∂}{∂y^{i}} |_{u}\} of T_{u}T M adapted to the decomposition (1.1). We call it the Berwald basis of the given nonlinear connection. We may remark here that if we change induced local charts from (π^{−1}(U), ϕ = (x^{i}, y^{i})) to (π^{−1}(V), ψ = (x̃^{i}, ỹ^{i})) then the corresponding Berwald base and the local coefficients of the nonlinear connection are related as follows:

\begin{matrix}\frac{δ}{δx^{i}} = \frac{∂x̃^{j}}{∂x^{i}} \frac{δ}{δx̃^{j}} , \frac{∂}{∂y^{i}} = \frac{∂x̃^{j}}{∂x^{i}} \frac{∂}{∂ỹ^{j}} , rank(\frac{∂x̃^{j}}{∂x^{i}}) = n; \\ N_{i}^{k} \frac{∂x̃^{j}}{∂x^{k}} = \frac{∂x̃^{k}}{∂x^{i}} Ñ_{k}^{j} + \frac{∂ỹ^{j}}{∂x^{i}} . \end{matrix}

At every point u ∈ T M we denote by T_{u}^{∗}T M the cotangent space at u to T M, that is the dual space of T_{u}T M over IR. Then {dx^{i}|_{u}, δy^{i}|_{u} = dy^{i}|_{u} + N_{j}^{i}(u)dx^{j}|_{u}} is a basis of T_{u}^{∗}T M, that is called the Berwald cobasis of the nonlinear connection (it is the dual basis of the Berwald basis).

For a nonlinear connection HT M we define the map θ : X (T M) → X (T M) locally given by

(1.2)  θ = \frac{δ}{δx^{i}} ⊗ δy^{i}.

1In this paper the summation convention on upper and lower repeated indices is implied

We have that θ is globally defined and it has the properties: θ^{2} = 0, Kerθ = Imθ = HT M. The maps h_{u} = θ_{u} ∘ J_{u} and v_{u} = J_{u} ∘ θ_{u} are the horizontal and the vertical projectors that correspond to the decomposition (1.1). 1.2. Definition A generalized Lagrange metric (or a GL-metric for short) is a metric g on the vertical subbundle V T M of the tangent space T M. This means that for every u ∈ T M, g_{u} : V_{u}T M × V_{u}T M → IR is bilinear, symmetric, of rank n and of constant signature. A pair GL^{n} = (M, g), with g a GL-metric is called a generalized Lagrange space, or a GL-space for short.

If (π^{−1}(U), ϕ = (x^{i}, y^{i})) is an induced local chart at u = (x, y) ∈ T M, we denote by g_{ij}(u) = g_{u}(\frac{∂}{∂y^{i}} |_{u}, \frac{∂}{∂y^{j}} |_{u}). Then a GL-metric may be given by a collection of functions g_{ij}(x, y) such that we have: 1^{o} rank(g_{ij}) = n, g_{ij}(x, y) = g_{ji}(x, y); 2^{o} the quadratic form g_{ij}(x, y)ξ^{i}ξ^{j} has constant signature on T M; 3^{o} if another local chart (π^{−1}(V), ψ = (x̃^{i}, ỹ^{i})) at u ∈ T M is given and g̃_{kl}(x, y) = g_{u}(\frac{∂}{∂ỹ^{k}} |_{u}, \frac{∂}{∂ỹ^{l}} |_{u}) then g_{ij} and g̃_{kl} are related by

(1.3)  g_{ij} = \frac{∂x̃^{k}}{∂x^{i}} \frac{∂x̃^{l}}{∂x^{j}} g̃_{kl}.

A tensor field of (r, s)-type on T M whose components transform under a change of local coordinates on T M like the components of a tensor field of (r, s)-type on the base manifold is called a Finsler tensor field. From (1.3) we can see that a GL-metric is a Finsler tensor field of (0,2)-type.

If a nonlinear connection is given on a GL-space, then we may extend the metric g to the whole T M by taking:

(1.4) G_{u}(X_{u}, Y_{u}) = g_{u}(J_{u}X_{u}, J_{u}Y_{u}) + g_{u}(J_{u}θ_{u}X_{u}, J_{u}θ_{u}Y_{u}), ∀X_{u}, Y_{u} ∈ T_{u}T M.

With respect to this metric, the vertical and horizontal distributions are orthogonal. In general, a GL-space doesn’t have a canonical nonlinear connection. 1.3. Example Consider a_{ij}(x) the components of a Riemannian metric on the base manifold M, a(x, y) > 0 and b(x, y) ≥ 0 two Finsler scalars and B(x, y) = B_{i}(x, y)dx^{i} a Finsler 1-form. Then:

(1.5)  g_{ij}(x, y) = a(x, y)a_{ij}(x) + b(x, y)B_{i}(x, y)B_{j}(x, y)

is a generalized Lagrange metric ([1]), called the Beil metric. We say also that the metric tensor g_{ij} is a Beil deformation of the Riemannian metric a_{ij}. It has been studied and applied by R.Miron and R.K.Tavakol in General Relativity for a(x, y) = exp(2σ(x, y)) and b = 0. The case a(x, y) = 1 with various choices of b and B_{i} was introduced and studied by R.G.Beil for constructing a new unified field theory in [5]. 1.4. Definition A Finsler metric on T M is a function F : T M → IR with the properties: 1^{o} F is a positive function of C^{∞}-class on T̃ M and only continuous on the null crosssection of the tangent bundle; 2^{o} F is positively homogeneous of degree one on T̃ M with respect to y^{i}; 3^{o} The matrix with the entries:

(1.6)  g_{ij} = \frac{1}{2} \frac{∂^{2}F^{2}}{∂y^{i}∂y^{j}}

has rank n on T̃ M and the quadratic form g_{ij}(x, y)ξ^{i}ξ^{j} has constant signature on T̃ M.

A Finsler space is a pair F^{n} = (M, F) with F a Finsler metric. The tensor field with the components given by (1.6) is called the metric tensor of the Finsler space. We denote by g^{ij} the components of the inverse matrix of g_{ij}, that is g_{ij}g^{jk} = δ_{i}^{k}.

If we do not ask for the homogeneity condition 2^{o}, then F is called a Lagrange metric. The pair (M, F) is called a Lagrange space. The geometry of these spaces was intensively studied by R.Miron in [14].

For a Lagrange space F^{n}, the metric tensor (1.6) determine a GL-metric. The converse of this is not true and the Beil metric (1.5) is an example of GL-metric that is not reducible to a Finsler or Lagrange metric.

It is well known that every Lagrange space induces a canonical nonlinear connection, namely the Cartan nonlinear connection ([14]). This has the local coefficients given by:

\begin{matrix}N_{j}^{i} = \frac{∂G^{i}}{∂y^{j}} , with \\ 4G^{i} = g^{ik} (\frac{∂^{2}F^{2}}{∂y^{k}∂x^{m}} y^{m} − \frac{∂F^{2}}{∂x^{k}} ) . \end{matrix}

Then a Lagrange space F^{n} has a canonical metric G given by formula (1.4).

An important class of Finsler spaces is the class of Finsler spaces with (α, β)- metrics ([12]). The first Finsler spaces with (α, β)-metric were introduced in forties by the physicist G.Randers and them are called the Randers spaces, [16]. Recently, R.G. Beil suggested to consider a more general case, the class of Lagrange spaces with (α, β)-metric. 1.5. Definition A Finsler space F^{n} = (M, F (x, y)) is called with (α, β)-metric if there exists a 2-homogeneous function L of two variables such that the Finsler metric F : T M → IR is given by:

(1.7)  F^{2}(x, y) = L(α(x, y), β(x, y)), where

\begin{matrix}α^{2}(x, y) = a_{ij}(x)y^{i}y^{j}, & a_{ij}(x) is a Riemannian metric on M; \\ β(x, y) = b_{i}(x)y^{i}, & b_{i}(x)dx^{i} is a 1 − form on M. \end{matrix}

If we do not ask for the function L to be homogeneous of order two with respect to (α, β) variables, then we have a Lagrange space with (α, β)-metric. 1.6. Example 1^{o} If L(α, β) = (α + β)^{2}, then the Finsler space with Finsler metric F (x, y) = (a_{ij}(x)y^{i}y^{j})^{\frac{1}{2}} + b_{i}(x)y^{i} is called a Randers space.

2^{o} If L(α, β) = \frac{α^{4}}{β^{2}} , then the Finsler space with Finsler metric

F (x, y) = \frac{a_{ij}(x)y^{i}y^{j}}{|b_{i}(x)y^{i}|} is called a Kropina space.

These classes of Finsler spaces play an important role in Finsler geometry and they are dual in the sense of [7]. 3^{o} If L(α, β) = α^{n}β^{m}, then we have a Lagrange space with (α, β)-metric, where the Lagrange metric is F (x, y) = (a_{ij}(x)y^{i}y^{j})^{\frac{n}{2}} (b_{i}(x)y^{i})^{m}. This Lagrange spaces reduces to a Finsler spaces with (α, β)-metric if and only if n + m = 2.

Throughout this paper we shall rise and lower indices only with the Riemannian metric a_{ij}(x), that is y_{i} = a_{ij}y^{j}, b^{i} = a^{ij}b_{j}, and so on.

For a Lagrange space with (α, β)-metric F^{2}(x, y) = L(α(x, y), β(x, y)) it is usual to denote ([11]):

(1.8)  \begin{matrix}ρ = \frac{1}{2α} \frac{∂L}{∂α}; & ρ_{0} = \frac{1}{2} \frac{∂^{2}L}{∂β^{2}}; \\ ρ_{−1} = \frac{1}{2α} \frac{∂^{2}L}{∂α∂β}; & ρ_{−2} = \frac{1}{2α^{2}} (\frac{∂^{2}L}{∂α^{2}} − \frac{1}{α} \frac{∂L}{∂α}) . \end{matrix}

For a Finsler space with (α, β)-metric, that is L is homogeneous of degree two with respect to α and β we have:

(1.8)′  ρ_{−1}β + ρ_{−2}α^{2} = 0.

With respect to these notations we have that the metric tensor g_{ij} of a Lagrange space with (α, β)-metric is given by ([12]):

(1.9)  g_{ij}(x, y) = ρa_{ij}(x) + ρ_{0}b_{i}(x)b_{j}(x) + ρ_{−1}(b_{i}(x)y_{j} + b_{j}(x)y_{i}) + ρ_{−2}y_{i}y_{j}.

We may remark here that the formula (1.9) was determined in [12] for Finsler spaces with (α, β)-metric but it works more generally for Lagrange spaces with (α, β)-metric. The metric tensor g_{ij} of a Lagrange space with (α, β)-metric can be arranged into the form:

(1.9)′ g_{ij} = ρa_{ij} + \frac{1}{ρ_{−2}} (ρ_{−1}b_{i} + ρ_{−2}y_{i})(ρ_{−1}b_{j} + ρ_{−2}y_{j}) + \frac{1}{ρ_{−2}} (ρ_{0}ρ_{−2} − ρ_{−1}^{2})b_{i}b_{j}.

If the b_{i}b_{j} coefficient vanishes we have: 1.7. Proposition If for a Lagrange space with (α, β)-metric the condition:

(1.10)  ρ_{−1}^{2} = ρ_{0}ρ_{−2}

holds true, then the metric tensor g_{ij} can be written in the equivalent form:

(1.11)  g_{ij}(x, y) = ρ(x, y)a_{ij}(x) + \frac{1}{ρ_{−2}} B_{i}(x, y)B_{j}(x, y), where

B_{i}(x, y) = ρ_{−1}(x, y)b_{i}(x) + ρ_{−2}(x, y)y_{i}.

If we compare (1.11) to (1.5) we have the following result: 1.8. Corollary If for a Lagrange space with (α, β)-metric the condition (1.10) holds true, then its fundamental metric tensor is a Beil metric. 1.9. Remark For the Lagrange space with (α, β)-metric suggested by R.G.Beil, L(α, β) = α^{n}β^{m}, the condition (1.10) is true if and only if m^{2}n^{2} = mn(m − 1)(n − 2). An example of Lagrange space with (α, β)-metric that satisfies the condition (1.10) has the Lagrange metric L(α, β) = \frac{α^{4}}{β}.

2 Nonholonomic Finsler frames for special metrics

The physicists R.G.Beil in [5,6] and P.R. Holland in [8,9] are using nonholonomic Finsler frames to develop unified field theories. In this section, we determine a nonholonomic Finsler frame for a Beil metric (1.5). In the particular case when a(x, y) = 1 and b(x, y) is a constant k we get the frame used by R.G. Beil in [5]. In the previous section, we found conditions in which the fundamental metric of a Lagrange space with (α, β)-metric is a Beil metric. Then we can determine a nonholonomic Finsler frame for a Lagrange space with (α, β)-metric from the nonholonomic Finsler frame of a Beil metric. From (1.9)’ we can see that the fundamental metric tensor of a Finsler space with (α, β)-metric can be derived from a Riemannian metric a_{ij} using the Beil deformation (1.5) in two steps. Using this idea we can determine a nonholonomic frame for a Finsler space with (α, β)-metric as a product of two nonholonomic frames, each of these being determined by a Beil deformation. Let U be an open set of T M and

V_{i} : u ∈ U ↦ V_{i}(u) ∈ V_{u}T M, i ∈ {1, ..., n}

be a vertical frame over U. If V_{i}(u) = V_{i}^{j}(u)\frac{∂}{∂y^{j}} |_{u}, then V_{i}^{j}(u) are the entries of a invertible matrix for all u ∈ U. Denote by Ṽ_{k}^{j}(u) the inverse of this matrix. This means that:

V_{j}^{i}Ṽ_{k}^{j} = δ_{k}^{i}, Ṽ_{j}^{i}V_{k}^{j} = δ_{k}^{i}.

We call V_{j}^{i} a nonholonomic Finsler frame. 2.1. Theorem Consider a GL-space with Beil metric (1.5) and denote by B^{2}(x, y) = a_{ij}(x)B^{i}(x, y)B^{j}(x, y). Then:

(2.1)  V_{j}^{i} = \sqrt{a}δ_{j}^{i} − \frac{1}{B^{2}} (\sqrt{a} ± \sqrt{a + bB^{2}})B^{i}B_{j}

is a nonholonomic Finsler frame. The Beil metric (1.5) and the Riemannian metric a_{ij}(x) are related by:

(2.2)  g_{ij}(x, y) = V_{i}^{k}(x, y)V_{j}^{l}(x, y)a_{kl}(x).

Proof. Consider also:

(2.1)′  Ṽ_{k}^{j} = \frac{1}{\sqrt{a}}δ_{k}^{j} − \frac{1}{B^{2}} (\frac{1}{\sqrt{a}} ± \frac{1}{\sqrt{a + bB^{2}}} ) B^{j}B_{k}.

It is a direct calculation to check that Ṽ_{k}^{j} is the inverse of V_{j}^{i}, that is V_{j}^{i} is a nonholonomic frame. Next we have that V_{i}^{k}V_{j}^{l}a_{kl} = aa_{ij} + bB_{i}B_{j} = g_{ij} so the formula (2.2) holds true. 2.2. Corollary The Beil metric (1.5) is positive definite on T̃ M. Proof. As the Finsler scalars a(x, y) and b(x, y) that define the metric (1.5) are positive and the metric a_{ij} is positive definite from (2.1)’ we can see that Ṽ_{k}^{i} is well defined on T̃ M. Then V_{j}^{i} from (2.1) is a nonholonomic Finsler frame on T̃ M. From (2.2) we have that g_{ij} and a_{ij} have the same signature, so g_{ij} is positive definite on T̃ M.

2.3. Remark If we take a(x, y) = 1 and b(x, y) = k, the nonholonomic Finsler frame (2.1) is the frame used by R.G.Beil in [5], formula (5.1). 2.4. Theorem Let F^{2}(x, y) = L(α(x, y), β(x, y)) be the metric function of a Lagrange space with (α, β)-metric for which the condition ρ_{−1}^{2} = ρ_{0}ρ_{−2} is true. Then:

(2.3)  V_{j}^{i} = \sqrt{ρ}δ_{j}^{i} − \frac{1}{B^{2}} (\sqrt{ρ} ± \sqrt{ρ + \frac{B^{2}}{ρ_{−2}}} ) (ρ_{−1}b^{i} + ρ_{−2}y^{i})(ρ_{−1}b_{j} + ρ_{−2}y_{j})

is a nonholonomic Finsler frame, where B^{2} = ρ_{−1}^{2}b^{2} + ρ_{−2}^{2}α^{2} + 2βρ_{−1}ρ_{−2}, ρ, ρ_{0}, ρ_{−1} and ρ_{−2} are the invariants of the Lagrange space with (α, β)-metric defined in (1.8).

For a Lagrange space with (α, β)-metric L = \frac{α^{4}}{β} we have:

ρ = \frac{2α^{2}}{β} , ρ_{0} = \frac{α^{4}}{β^{3}} , ρ_{−1} = \frac{−2α^{2}}{β^{2}} , ρ_{−2} = \frac{4}{β} .

We have then that the condition (1.10) is true and B^{2} = \frac{4α^{4}b^{2}}{β^{4}} . Consequently a nonholonomic frame for the given Lagrange space with (α, β)-metric is given by:

V_{j}^{i} = α\sqrt{\frac{2}{β}} δ_{j}^{i} − \frac{1}{α^{3}b^{2}} (\sqrt{\frac{2}{β}} ± \sqrt{\frac{2}{β} + \frac{α^{2}b^{2}}{β^{3}}} ) (2βy^{i} − α^{2}b^{i})(2βy_{j} − α^{2}b_{j}).

Consider now a Finsler space with (α, β)-metric. From (1.9)’ we can see that g_{ij} is the result of two Beil deformations:

(2.4)  \begin{matrix}a_{ij} ↦ h_{ij} = ρa_{ij} + \frac{1}{ρ_{−2}} (ρ_{−1}b_{i} + ρ_{−2}y_{i})(ρ_{−1}b_{j} + ρ_{−2}y_{j}) and \\ h_{ij} ↦ g_{ij} = h_{ij} + \frac{1}{ρ_{−2}} (ρ_{0}ρ_{−2} − ρ_{−1}^{2})b_{i}b_{j}. \end{matrix}

The nonholonomic Finsler frame that corresponds to the first deformation (2.4) is, according to the Theorem 2.1, given by:

(2.5)  X_{j}^{i} = \sqrt{ρ}δ_{j}^{i} − \frac{1}{B^{2}} (\sqrt{ρ} ± \sqrt{ρ + \frac{B^{2}}{ρ_{−2}}} ) (ρ_{−1}b^{i} + ρ_{−2}y^{i})(ρ_{−1}b_{j} + ρ_{−2}y_{j}),

where B^{2} = a_{ij}(ρ_{−1}b^{i}+ρ_{−2}y^{i})(ρ_{−1}b^{j} +ρ_{−2}y^{j}) = ρ_{−1}^{2}b^{2}+βρ_{−1}ρ_{−2}. The metric tensors a_{ij} and h_{ij} are related by:

(2.6)  h_{ij} = X_{i}^{k}X_{j}^{l}a_{kl}.

According to the Theorem 2.1, the nonholonomic Finsler frame that corresponds to the second deformation (2.4) is given by:

(2.5)′  Y_{j}^{i} = δ_{j}^{i} − \frac{1}{C^{2}} (1 ± \sqrt{1 + \frac{ρ_{−2}C^{2}}{ρ_{0}ρ_{−2} − ρ_{−1}^{2}}} ) b^{i}b_{j},

where C^{2} = h_{ij}b^{i}b^{j} = ρb^{2} + \frac{1}{ρ_{−2}} (ρ_{−1}b^{2} + ρ_{−2}β)^{2}. The metric tensors h_{ij} and g_{ij} are related by the formula:

(2.6)′  g_{mn} = Y_{m}^{i}Y_{n}^{j}h_{ij}.

From (2.6) and (2.6)’ we have that V_{m}^{k} = X_{i}^{k}Y_{m}^{i}, with X_{i}^{k} given by (2.5) and Y_{m}^{i} given by (2.5)’, is a nonholonomic Finsler frame of the Finsler space with (α, β)-metric.

For a Randers space with the fundamental function L = (α+β)^{2} = F^{2}, the Finsler invariants (1.8) are given by:

\begin{matrix}ρ = \frac{α + β}{α} = \frac{F}{α} , & ρ_{0} = 1, & ρ_{−1} = \frac{1}{α}, & ρ_{−2} = \frac{−β}{α^{3}} , \\ B^{2} = \frac{b^{2}α^{2} − β^{2}}{α^{4}} . \end{matrix}

We have then that the condition (1.10) is not satisfied. If we use the previous idea, then V_{m}^{k} = X_{i}^{k}Y_{m}^{i} is a nonholonomic Finsler frame of a Randers space, where:

X_{j}^{i} = \sqrt{\frac{α + β}{α}} δ_{j}^{i} − \frac{α^{2}}{α^{2}b^{2} − β^{2}} [\sqrt{\frac{α + β}{α}} ± \sqrt{\frac{αβ + 2β^{2} − b^{2}α^{2}}{αβ}} ] (b^{i} − \frac{βy^{i}}{α^{2}} )(b_{j} − \frac{βy_{j}}{α^{2}} ),

\begin{matrix}Y_{j}^{i} = δ_{j}^{i} − \frac{1}{C^{2}} (1 ± \sqrt{1 + \frac{βC^{2}}{α + β}} ) b^{i}b_{j}, and \\ C^{2} = \frac{(α + β)b^{2}}{α} − \frac{α}{β} (b^{2} − \frac{β^{2}}{α^{2}} )^{2} . \end{matrix}

In a similar way we may find a nonholonomic Finsler frame for a Kropina space

with the fundamental function L = \frac{α^{4}}{β^{2}} = F^{2}. In this case, the Finsler invariants are

given by:

\begin{matrix}ρ = \frac{2α^{2}}{β^{2}} , ρ_{0} = 3\frac{α^{4}}{β^{4}} , ρ_{−1} = \frac{−4α^{2}}{β^{3}} , ρ_{−2} = \frac{4}{β^{2}} , \\ B^{2} = 16\frac{α^{2}}{β^{4}} (\frac{α^{2}b^{2}}{β^{2}} − 1) . \end{matrix}

2.5. Remark One may use also the two steps deformations (2.4) to determine the contravariant tensor (g^{ij}) of a Finsler space with (α, β)-metric.

3 Finsler connections induced by a nonholonomic Finsler Frame

Consider now that on the tangent bundle of a manifold M we have a nonlinear connection HT M. Then we consider a special linear connection on T M that preserves by parallelism the horizontal and the vertical distributions and we call it a Finsler connection. We prove that a nonholonomic Finsler frame determine a Finsler connection with no curvature. We study the integrability conditions of the Cartan’s structure equations of a Finsler connection. Using these, we can prove that if a Finsler connection has no curvature then it is induced by a nonholonomic Finsler frame.

3.1. Definition A linear connection D on T M is called a Finsler connection if: 1^{∘} D preserves by parallelism the horizontal distribution HT M; 2^{∘} The almost tangent structure J is absolutely parallel with respect to D.

For a Finsler connection D it is immediate that D preserves also the vertical distribution. With respect to the Berwald basis (\frac{δ}{δx^{i}} , \frac{∂}{∂y^{i}}) of the nonlinear connection a Finsler connection can be expressed as:

(3.1)  \{ \begin{matrix}D_{\frac{δ}{δx^{i}}} \frac{δ}{δx^{j}} & = F_{ji}^{k} \frac{δ}{δx^{k}}; & D_{\frac{δ}{δx^{i}}} \frac{∂}{∂y^{j}} & = F_{ji}^{k} \frac{∂}{∂y^{k}}; \\ D_{\frac{∂}{∂y^{i}}} \frac{δ}{δx^{j}} & = C_{ji}^{k} \frac{δ}{δx^{k}}; & D_{\frac{∂}{∂y^{i}}} \frac{∂}{∂y^{j}} & = C_{ji}^{k} \frac{∂}{∂y^{k}} . \end{matrix}

Observe that under a change of induced coordinates on T M the functions F_{ji}^{k} transform like the coefficients of a linear connection on the base manifold M and C_{ji}^{k} are the components of a Finsler tensor field of (1,2)-type.

If (T_{j_{1}···j_{s}}^{i_{1}···i_{r}}) are the components of a (r, s)-type Finsler tensor field T, then the absolute differential of T with respect to the Finsler connection D is given by:

DT_{j_{1}···j_{s}}^{i_{1}···i_{r}} = dT_{j_{1}···j_{s}}^{i_{1}···i_{r}} +ω_{p}^{i_{1}}T_{j_{1}···j_{s}}^{pi_{2}···i_{r}} +⋯+ω_{p}^{i_{r}}T_{j_{1}···j_{s}}^{i_{1}···i_{r−1}p} −ω_{j_{1}}^{p}T_{pj_{1}···j_{s}}^{i_{1}···i_{r}} −⋯−ω_{j_{s}}^{p}T_{j_{1}···j_{s−1}p}^{i_{1}···i_{r}},

where ω_{j}^{i} = F_{jk}^{i}dx^{k} + C_{jk}^{i}δy^{k} are the connection 1-forms of D.

We can write the previous formula in an equivalent form:

DT_{j_{1}···j_{s}}^{i_{1}···i_{r}} = T_{j_{1}···j_{s}|k}^{i_{1}···i_{r}}dx^{k} + T_{j_{1}···j_{s}}^{i_{1}···i_{r}}|_{k}δy^{k}.

Here T_{j_{1}···j_{s}|k}^{i_{1}···i_{r}} and T_{j_{1}···j_{s}}^{i_{1}···i_{r}}|_{k} stand for horizontal and vertical covariant derivatives of T_{j_{1}···j_{s}}^{i_{1}···i_{r}}, ([14]).

For a Finsler connection D one considers typically:

\begin{matrix}T (X, Y) = D_{X}Y − D_{Y} X − [X, Y], \\ R(X, Y)Z = D_{X}D_{Y} Z − D_{Y} D_{X}Z − D_{[X,Y]}Z \end{matrix}

the torsion and the curvature. It is well known ([4], [14]) that with respect to the Berwald basis \{\frac{δ}{δx^{i}} , \frac{∂}{∂y^{i}} \} there are only five nonzero components of torsion and three components of curvature. The five nonzero components of torsion are:

(3.2) \{ \begin{matrix}hT (\frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ) =: T_{ij}^{k} \frac{δ}{δx^{k}} = (F_{ji}^{k} − F_{ij}^{k})\frac{δ}{δx^{k}}; & (h)h−torsion \\ vT (\frac{δ}{δx^{i}} , \frac{δ}{δx^{j}} ) =: R_{ij}^{k} \frac{∂}{∂y^{k}} = (\frac{δN_{i}^{k}}{δx^{j}} − \frac{δN_{j}^{k}}{δx^{i}} ) \frac{∂}{∂y^{k}}; & (v)h−torsion \\ hT (\frac{∂}{∂y^{i}} , \frac{δ}{δx^{j}} ) = C_{ji}^{k} \frac{δ}{δx^{k}}; & (h)hv−torsion \\ vT (\frac{∂}{∂y^{i}} , \frac{δ}{δx^{j}} ) =: P_{ij}^{k} \frac{∂}{∂y^{k}} = (\frac{∂N_{j}^{k}}{∂y^{i}} − F_{ij}^{k}) \frac{∂}{∂y^{k}}; & (v)hv−torsion \\ vT (\frac{∂}{∂y^{i}} , \frac{∂}{∂y^{j}} ) =: S_{ij}^{k} \frac{∂}{∂y^{k}} = (C_{ji}^{k} − C_{ij}^{k})\frac{∂}{∂y^{k}}; & (v)v−torsion. \end{matrix}

The three components of curvature are given by:

(3.3)  \begin{matrix}R_{j kh}^{i} = \frac{δF_{jk}^{i}}{δx^{h}} − \frac{δF_{jh}^{i}}{δx^{k}} + F_{jk}^{m}F_{mh}^{i} − F_{jh}^{m}F_{mk}^{i} + C_{jm}^{i}R_{kh}^{m}; \\ P_{j kh}^{i} = \frac{∂F_{jk}^{i}}{∂y^{h}} − C_{jk|h}^{i} + C_{jm}^{i}P_{kh}^{m}; \\ S_{j kh}^{i} = \frac{∂C_{jk}^{i}}{∂y^{h}} − \frac{∂C_{jh}^{i}}{∂y^{k}} + C_{jk}^{m}C_{mh}^{i} − C_{jh}^{m}C_{mk}^{i}. \end{matrix}

For a Finsler connection D we have the following Ricci identities:

(3.4)  \{ \begin{matrix}X_{|k|r}^{i} − X_{|r|k}^{i} = X^{m}R_{m kr}^{i} − X_{|m}^{i}T_{kr}^{m} − X^{i}|_{m}R_{kr}^{m}; \\ X_{|k}^{i}|_{r} − X^{i}|_{r|k} = X^{m}P_{m kr}^{i} − X_{|m}^{i}C_{kr}^{m} − X^{i}|_{m}P_{kr}^{m}; \\ X^{i}|_{k}|_{r} − X^{i}|_{r}|_{k} = X^{m}S_{m kr}^{i} − X^{i}|_{m}S_{kr}^{m}. \end{matrix}

Consider now a nonholonomic Finsler frame V_{j} = V_{j}^{i} \frac{∂}{∂y^{i}} on a open set U of T M. That is V_{j}^{i}(u) are the entries of a nonsingular matrices over U. We denote by Ṽ_{k}^{j} the inverse matrix of V_{j}^{i}. 3.2. Theorem There exists a unique Finsler connection D on T M such that the absolute differential of the given nonholonomic frame V_{j} = V_{j}^{i} \frac{∂}{∂y^{i}} with respect to D, is zero. For this Finsler connection D all components of curvature are zero. Proof. The absolute differential of the given nonholonomic frame V_{j} with respect to D is given by DV_{j}^{i} = V_{j|k}^{i}dx^{k} + V_{j}^{i}|_{k}δy^{k} for every fixed j ∈ {1, 2, ..., n}. So, DV_{j}^{i} = 0 if and only if the frame is h− and v−covariant constant with respect to D.

The nonholonomic frame V_{j} = V_{j}^{i} \frac{∂}{∂y^{i}} is h-covariant constant if for all j ∈ \{1, ..., n\}

we have V_{j|k}^{i} = 0. This is equivalent to \frac{δV_{j}^{i}}{δx^{k}} + F_{mk}^{i}V_{j}^{m} = 0. If we solve this for F_{mk}^{i} we have

F_{mk}^{i} = −\frac{δV_{j}^{i}}{δx^{k}} Ṽ_{m}^{j} = V_{j}^{i} \frac{δṼ_{m}^{j}}{δx^{k}} .

Similarly, the nonholonomic frame V_{j} is v-covariant constant if for all j ∈ {1, ..., n}

we have V_{j}^{i}|_{k} = 0. This is equivalent to \frac{∂V_{j}^{i}}{∂y^{k}} + C_{mk}^{i}V_{j}^{m} = 0. If we solve this for C_{mk}^{i} we have

C_{mk}^{i} = −\frac{∂V_{j}^{i}}{∂y^{k}} Ṽ_{m}^{j} = V_{j}^{i} \frac{∂Ṽ_{m}^{j}}{∂y^{k}} .

If we use the Ricci identities (3.4) for V_{j}, we have: R_{m kj}^{i}V_{j}^{m} = 0, P_{m kj}^{i}V_{j}^{m} = 0, and S_{m kj}^{i}V_{j}^{m} = 0, ∀j ∈ {1, ..., n}. As V_{j}^{m} is invertible one obtain: R_{m kj}^{i} = P_{m kj}^{i} = S_{m kj}^{i} = 0.

The Finsler connection we have defined in Theorem 3.1 is called the Crystallographic connection of the nonholonomic frame V_{j}^{i} ([2]).

Next we denote by \{X_{a}\}_{a=\overline{1,2n}} the vector fields of the Berwald basis \{\frac{δ}{δx^{i}} , \frac{∂}{∂y^{i}} \} induced by a nonlinear connection HT M and by \{θ^{a}\}_{a=\overline{1,2n}} the dual basis \{dx^{i}, δy^{i}\}. For a Finsler connection D, the connection 1-forms (ω_{b}^{a}) corresponding to these base are defined as follows:

ω_{b}^{a}(X) = θ^{a}(D_{X}X_{b}), ∀X ∈ χ(T M).

It is a straightforward calculation to check that the connection 1-forms are given by

ω_{b}^{a} = ( \begin{matrix}ω_{j}^{i} & 0 \\ 0 & ω_{j}^{i} \end{matrix} ), where ω_{j}^{i} = F_{jk}^{i}dx^{k} + C_{jk}^{i}δy^{k}. For a vector field W = W^{a}X_{a} ∈

χ(T M) we have that

\begin{matrix}D_{V} W = (V (W^{a}) + W^{b}ω_{b}^{a}(V))X_{a}, that is \\ θ^{a}(D_{V} W) = V (θ^{a}(W)) + θ^{b}(W)ω_{b}^{a}(V). \end{matrix}

3.3. Theorem The Cartan’s first structure equations of a Finsler connection D are given by:

(3.5)  \{ \begin{matrix}& −dx^{h} ∧ ω_{h}^{i} = −Θ^{i}, \\ d(δy^{i}) & −δy^{h} ∧ ω_{h}^{i} = −Θ̃^{i}, \end{matrix}

where the 2-forms of torsions Θ^{a} = (Θ^{i}, Θ̃^{i}) are defined by:

Θ^{a}(X, Y) = θ^{a}(T (X, Y)), and are given by :

(3.6)  \{ \begin{matrix}Θ^{i} = \frac{1}{2}T_{jk}^{i}dx^{j} ∧ dx^{k} + C_{jk}^{i}dx^{j} ∧ δy^{k}, \\ Θ̃^{i} = \frac{1}{2}R_{jk}^{i}dx^{j} ∧ dx^{k} + P_{jk}^{i}dx^{j} ∧ δy^{k} + \frac{1}{2}S_{jk}^{i}δy^{j} ∧ δy^{k}. \end{matrix}

The Cartan’s second structure equations of a Finsler connection D are given by:

(3.7)  dω_{j}^{i} − ω_{j}^{h} ∧ ω_{h}^{i} = −Ω_{j}^{i},

where the curvature 2-forms (Ω_{b}^{a}) = ( \begin{matrix}Ω_{j}^{i} & 0 \\ 0 & Ω_{j}^{i} \end{matrix} ), are defined by:

Ω_{b}^{a}(X, Y) = θ^{a}(R(X, Y)X_{b}), and are given by :

(3.8)  Ω_{j}^{i} = \frac{1}{2}R_{j kh}^{i}dx^{k} ∧ dx^{h} + P_{j kh}^{i}dx^{k} ∧ δy^{h} + \frac{1}{2}S_{j kh}^{i}δy^{k} ∧ δy^{h}.

Proof. We have that

Θ^{a}(X, Y) = θ^{a}(T (X, Y)) = θ^{a}(D_{X}Y) − θ^{a}(D_{Y} X) − θ^{a}([X, Y]) =
= X(θ^{a}(Y)) + θ^{b}(Y)ω_{b}^{a}(X) − Y (θ^{a}(X)) − θ^{b}(X)ω_{b}^{a}(Y) − θ^{a}([X, Y]) =
= dθ^{a}(X, Y) + (ω_{b}^{a} ∧ θ^{b})(X, Y).

If we take θ^{a} to be dx^{i} and δy^{i}, respectively, then we get the Cartan’s first structure equations (3.5).

From Ω_{b}^{a}(X, Y) = θ^{a}(R(X, Y)X_{b}) = dω_{b}^{a}(X, Y) + (ω_{c}^{a} ∧ ω_{b}^{c})(X, Y) we have the Cartan’s second structure equations (3.7). 3.4. Theorem If for a Finsler connection D on T M the curvature 2-forms Ω_{j}^{i} vanish, then there exists a nonholonomic Finsler frame V_{j}^{i} such that the local coefficients of the connection D are given by:

(3.9)  \{ \begin{matrix}F_{jk}^{i} = −\frac{δV_{m}^{i}}{δx^{k}} Ṽ_{j}^{m} = V_{m}^{i} \frac{δṼ_{j}^{m}}{δx^{k}} \\ C_{jk}^{i} = −\frac{∂V_{m}^{i}}{∂y^{k}} Ṽ_{j}^{m} = V_{m}^{i} \frac{∂Ṽ_{j}^{m}}{∂y^{k}} . \end{matrix}

Proof. If the curvature two-forms of D vanish, then the Cartan’s second structure equations are:

dω_{b}^{a} + ω_{c}^{a} ∧ ω_{b}^{c} = 0.

Then there exists a frame V_{b}^{a}(x, y) on the tangent space T M such that

(3.10)  dV_{b}^{a} + ω_{c}^{a}V_{b}^{c} = 0.

As ω_{b}^{a} = ( \begin{matrix}ω_{j}^{i} & 0 \\ 0 & ω_{j}^{i} \end{matrix} ) and if we denote V_{b}^{a} = ( \begin{matrix}V_{j}^{i} & V i ; \overline{j} \\ V \overline{i} ; j & V \overline{i} ; \overline{j} \end{matrix} ) then, the equations (3.10)

are equivalent to:

(3.10)′  \{ \begin{matrix}dV_{j}^{i} + ω_{k}^{i}V_{j}^{k} = 0, \\ dV i ; \overline{j} + ω_{k}^{i}V k ; \overline{j} = 0, \\ dV \overline{i} ; j + ω_{k}^{i}V \overline{k} ; j = 0, \\ dV \overline{i} ; \overline{j} + ω_{k}^{i}V \overline{k} ; \overline{j} = 0. \end{matrix}

As (V_{b}^{a}) are the entries of a non-singular matrix of order 2n, whose blocks are solutions of (3.10)’ we have that at least two of these blocks are invertible. Suppose (V_{j}^{i}) is one of them and Ṽ_{j}^{i} := (V_{j}^{i})^{−1}. Then ω_{j}^{i} = −Ṽ_{k}^{i}dV_{j}^{k} = dṼ_{k}^{i}V_{j}^{k} and consequently we have that the local coefficients of D are given by (3.9).

The Theorems 3.2 and 3.4 say that the only Finsler connections that have zero curvature are induced by nonholonomic Finsler frames.

The frame \{H_{j} = V_{j}^{i} \frac{δ}{δx^{i}} , V_{j} = V_{j}^{i} \frac{∂}{∂y^{i}} \} is said to be holonomic if there exist n

functions ϕ^{j} on the base manifold M such that Ṽ_{i}^{j} = \frac{∂ϕ^{j}}{∂x^{i}}, that is equivalent to say that the one-forms η^{j} = V_{i}^{j}dx^{i} are exact. 3.5. Proposition A frame V_{j}^{i} is holonomic if and only if the torsion two-forms Θ^{i}, defined by (3.6)_{1} of the Crystallographic connection induced by V_{j}^{i}, vanish. Proof. From (3.6)_{1} we have that Θ^{i} = 0 if and only if T_{jk}^{i} = 0 and C_{jk}^{i} = 0, where T_{jk}^{i} = F_{kj}^{i} − F_{jk}^{i}, and F_{jk}^{i} and C_{jk}^{i} are given by (3.9). However C_{jk}^{i} = 0 if and only

if V_{j}^{i} are functions of (x) only. Then T_{jk}^{i} = 0 if and only if \frac{∂_{V_{j}^{i}}}{∂x^{k}} = \frac{∂_{V_{k}^{i}}}{∂x^{j}} and this is equivalent to the fact that Ṽ_{i}^{j} are the gradient of n functions ϕ^{j} on the base manifold M. 3.6. Proposition If for a Finsler connection D on T M the torsion two-forms Θ^{i} and the curvature two-forms Ω_{j}^{i} vanish, then local coordinates may be found on the base manifold M such that with respect to the induced coordinates on T M we have F_{jk}^{i} = C_{jk}^{i} = 0.

Proof. If the curvature two-forms Ω_{j}^{i} of the Finsler connection D vanish then according to the Theorem 3.4 there is a frame V_{j}^{i} such that the local coefficients of the Finsler connection D are given by (3.9). From Proposition 3.5 we have that the frame

V_{i}^{j} is holonomic, that is there exist n functions ϕ^{j} such that Ṽ_{i}^{j} = \frac{∂ϕ^{j}}{∂x^{i}}. Then, ϕ^{j} are coordinate functions on M and with respect to the induced coordinates on T M, the local coefficients of the Finsler connection D, vanish.

Acknowledgement The author would like to thanks Dr. R.G. Beil for many ideas during the elaboration of the paper.

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Faculty of Mathematics ”Al.I.Cuza” University 6600 Iasi, Romania, bucataru@uaic.ro