Beil Metrics Associated to a Finsler Space

M. Anastasiei and H. Shimada

Mathematics Subject Classification: 53C60 Key words: Finsler metric, Beil metric, metrical connections, concurrent Finsler vector field.

1 Introduction

Let F^{n} = (M, F) be a Finsler space with M a smooth i.e. C^{∞} manifold and F : T M → R, (x, y) → F (x, y). Assume that F^{n} is endowed with a Finsler 1–form β_{i}(x, y) and set β = β_{i}(x, y)y^{i}. Here i, j, k, ... will run from 1 to n = dim M and the Einstein convention on summation is implied. Then ^{∗}F = L(F, β) in some conditions on L is so that ^{∗}F^{n} = (M,^{∗} F) is a new Finsler space. It is said that ^{∗}F^{n} is obtained from F^{n} by a β–change [?],[?].

Typical for ^{∗}F^{n} are the Randers and Kropina spaces which are obtained from a Riemannian space by particular β–changes.

Let g_{ij}(x, y) be the Finsler metric tensor of F^{n}. If one wishes the construction of a new Finsler metric ^{∗}g_{ij} which depends on g_{ij}(x, y), then because of the linear structure of the set of Finsler tensor fields of a given type, the most general choice is

(1.1)  ^{∗}g_{ij}(x, y) = ρ(x, y)g_{ij}(x, y) + σ(x, y)B_{ij}(x, y),

for ρ and σ two Finsler scalars and B_{ij}(x, y) a symmetric Finsler tensor field of type (0, 2). We may say that ^{∗}g_{ij} is obtained from g_{ij} by a B–change.

It is clear that^{∗}g_{ij} is no longer a Finsler metric except if some strong conditions on ρ, σ and B_{ij} are imposed. Metrics similar to (1.1) appear in [?] and [?] from physical considerations. See also [?].

In order to relax such conditions we do not ask ^{∗}g_{ij} be a Finsler metric but a generalized Lagrange metric in Miron’ sense, shortly a GL–metric. For the theory of the GL–metrics we refer to [?], ch.X.

As such (^{∗}g_{ij}) has to satisfy

a) det(^{∗}g_{ij}) ≠ 0 and

b) the quadratic from ^{∗}g_{ij}(x, y)ξ^{i}ξ^{j}, (ξ^{i}) ∈ IR^{n}, to be of constant signature.

Balkan Journal of Geometry and Its Applications, Vol.3, No.2, 1998, pp. 1-16 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

Even this minimal requirements are not easy to be fullfiled except for some particular σ, ρ and B_{ij}.

By our best knowledge the following two particular forms of the GL–metric (1.1) were studied:

(1.2)  ^{∗}g_{ij}(x, y) = e^{2α(x,y)}g_{ij}(x, y).

This class of GL–metrics contains the Miron–Tavakol metrics used by them in General Relativity and the Antonelli metrics which were introduced by P.L. Antonelli for some studies in Biology and Ecology. For details see [?], ch.XI, and reference therein;

(1.3)  ^{∗}g_{ij}(x, y) = g_{ij}(x, y) + σ(x, y)y_{i}y_{j}, y_{i} = g_{ij}(x, y)y^{j}.

Particular forms of the GL–metric (1.3) were used by R. Miron in Relativistic Geometrical Optics. See also [?], ch.XII.

Some particular forms of the GL–metric

(1.4)  ^{∗}g_{ij}(x, y) = g_{ij}(x, y) + σ(x, y)B_{i}(x, y)B_{j}(x, y),

with B_{i}(x, y) = g_{ij}(x, y)B^{j}(x, y), for B^{j}(x, y) a given Finsler vector field, were introduced by R.G. Beil in order to develop his interesting unified field theory ([?]). These were called Beil metrics. As such we refer to ^{∗}g_{ij} in (1.4) as to the Beil metric, too. The following comment of R.G.Beil is illuminating on (1.4). ”Since in my unified theory the quantity k which correspond to your σ is related to the gravitational constant, this means that a possible physical interpretation of your theory with a y-dependent σ is that gravitation itself is velocity dependent. This possibility is mentioned , for example, in Section 40.8 of the famous book ”Gravitation” by Misner, Thorne and Wheeler”. See [?].

The particular form of (1.4) obtained for σ = 1 and B_{i} = \frac{∂f}{∂x^{i}} , f : M → IR was

considered by C. Udrişte in [?]. He proved that if f is proper i.e. f^{−1}(K) is a compact set whenever K is compact, then the Finsler manifold (M,^{∗} g_{ij}(x, y)) is complete. A Riemannian version of (1.1), that is, was used by T. Aubin in order to prove that any compact Riemannian manifold of dimension greater then 2 admits a metric whose scalar curvature is a negative constant. See [?] and for other connected results.

The geometry of the GL–metrics (1.4) was not investigated in a systematic way. It is our purpose to fill this gap. After some preliminaries in Section 2, we show in Section 3 that (^{∗}g_{ij}) from (1.4) is a GL–metric and we point out cases when it reduces to a Lagrange or to a Finsler metric. In Section 4 we discuss possibilities for introducing metrical connections for the GL–space (M,^{∗} g_{ij}). In Section 5 we digress on parallel and resp. concurrent Finsler vector fields showing that the usual definitions for these notions are also justified from the viewpoint of the almost Hermitian model of a GL–space. For such a model see [?], ch.X. Section 6 is devoted to the analysis of the GL–metric (1.4) for B^{i} a concurent Finsler vector field. For σ a constant we rediscover a modification of a Finsler function studied by M. Matsumoto and K. Eguchi in [?]. The case when σ is a solution of the so–called Tavakol–Van der Berg equation is investigated, too. In Section 7 we treat a Beil metric associated to a Finsler space with (α, β)–metric. It is a future task to find properties of the GL–metric (1.4) when F^{n} is a particular Finsler space or its dimension is low (2 or 3).

2 Preliminaries

Let M be a smooth i.e. C^{∞} manifold, paracompact and of dimension n, T M its tangent manifold and τ : T M → M its tangent bundle. If x = (x^{i}), i, j, k, ... = 1, ..., n are local coordinates on M, then the induced coordinates on T M will be (x, y) =

(x^{i} : x^{i} ∘ τ, y^{i}) with (y^{i}) provided by u_{x} = y^{i} \frac{∂}{∂x^{i}} |_{x} , u ∈ T_{x}M, x ∈ M. The change of

coordinates (x, y) → (x̃, ỹ) on T M are as follows.

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

The geometrical objects on T M whose local components change by (2.1) as on M i.e. ignoring their dependence on y, will be called Finsler objects as in [?] or d–objects as in [?].

We set ∂_{i} := \frac{∂}{∂x^{i}} , ˙∂_{i} := \frac{∂}{∂y^{i}} and notice that the vertical subspace of T_{u}T M i.e.

V_{u}T M = Ker (Dτ)_{u}, u ∈ T M, where Dτ means the differential of τ, is spanned by ( ˙∂_{i}). The d–objects can be expressed using ( ˙∂_{i}).

A function F : T M → IR which is positive, smooth on T M \0 and only continuous in the rest, positively homogeneous of degree 1 with respect to y i.e. F (x, λy) = λF (x, y), λ > 0 and with the quadratic form g_{ij}(x, y)ξ^{i}ξ^{j}, (ξ^{i}) ∈ IR^{n} nondegenerate and of constant signature, where

(2.2)  g_{ij}(x, y) = \frac{1}{2} ˙∂_{i} ˙∂_{j}F^{2},

is called a fundamental Finsler function. The pair F^{n} = (M, F) is called a Finsler space.

The function g_{ij}(x, y) are the components of a Finsler tensor field called the Finsler metric of F^{n}.

A supplement H_{u}T M of V_{u}T M i.e. the decomposition in a direct sum T_{u}T M = H_{u}T M ⊕ V_{u}T M holds, will be called the horizontal space and the distribution u → H_{u}T M will be called a horizontal distribution. A basis of it of the form δ_{i} = ∂_{i} − N_{i}^{k}(x, y) ˙∂_{k}, provides the functions (N_{i}^{k}(x, y)) called the local coefficients. These functions have a special rule of change by (2.1) and in turn they completely determine the horizontal distribution called also a nonlinear connection. Then (δ_{i}, ˙∂_{i}) is a basis adapted to the previous decomposition of T_{u}T M. The Finsler objects may be also expressed by using (δ_{i}). We notice that (δ_{i}) are Finsler vector fields. For more details we refer to [?],[?].

3 The Beil metric

Let F^{n} = (M, F) be a Finsler space and g_{ij}(x, y) its Finsler metric. Assume that F^{n} is endowed with a Finsler vector field B = B^{i}(x, y) ˙∂_{i} and let B_{i}(x, y)dx^{i} the Finsler 1-form with B_{i} = g_{ik}B^{k}. The lowering and rising of indices will be done with (g_{ij}) and (g^{jk}), where g^{jk}g_{ki} = δ_{i}^{j}, respectively. Let σ : T M → IR, (x, y) → σ(x, y) a Finsler scalar. We set

(3.1)  ^{∗}g_{ij}(x, y) = g_{ij}(x, y) + σ(x, y)B_{i}(x, y)B_{j}(x, y).

The functions (^{∗}g_{ij}) from (3.1) define for σ > 0 a positive definite GL–metric called the Beil metric.

It is clear that (^{∗}g_{ij}) are the components of a symmetric d–tensor field. We look for the inverse of the matrix (^{∗}g_{ij}) in the form ^{∗}g^{jk} = ^{∗}g^{jk} −^{∗}σB^{j}B^{k} with ^{∗}σ to be determined. From ^{∗}g_{ij}^{∗}g^{jk} = δ_{i}^{k} it follows that ^{∗}σ = \frac{σ}{1 + σB^{2}} , with B^{2} = B_{i}B^{i} =

g_{ij}B^{i}B^{j} (the length of B with respect to g_{ij}). Thus we have

(3.2)  ^{∗}g^{jk} = g^{jk} − \frac{σ}{1 + σB^{2}} B^{j}B^{k}.

Consequently, we have det(g_{ij}) ≠ 0.

The quadratic from Φ(ξ) = ^{∗}g_{ij}ξ^{i}ξ^{j} = g_{ij}ξ^{i}ξ^{j} + σ(B_{k}ξ^{k})^{2} is clear positive definite in our hypothesis. q.e.d.

We notice that (3.2) holds in the weaker condition σ ≠ −\frac{1}{B^{2}} and if g_{ij}ξ^{i}ξ^{j} is only

of constant signature, the signature of Φ(ξ) will be constant for some σ and (B^{k}) at least locally.

Remark 3.1. The GL–metric (3.1) appears in papers by R.G. Beil ([?]) for F^{n} a pseudo–Riemannian space or a Minkowski space. It was called Beil’s metric.

We notice that for B^{i} = y^{i} in (3.1) one obtains a general version of the Synge metric which was used by R. Miron for a geometrical theory of Relativistic Optics (cf. [?], ch.XI).

In the following we shall assume B^{i} ≠ y^{i} and use the ideas and techniques from [?], ch.XI.

One says that ^{∗}g_{ij} is reducible to a Lagrange metric, shortly an L–metric if there

exists a Lagrangian L : T M → IR such that^{∗}g_{ij} = \frac{1}{2} ˙∂_{i} ˙∂_{j}L. A necessary and sufficient condition for ^{∗}g_{ij} be reducible to an L–metric is the symmetry in all indices of the

Cartan tensor field ^{∗}C_{ijk} = \frac{1}{2} ˙∂_{k}^{∗}g_{ij} i.e.

(3.3)  ˙∂_{k}^{∗}g_{ij} = ˙∂_{i}^{∗}g_{kj}.

Using (3.1) this condition becomes

(3.4)  \begin{matrix}σ̇_{k}B_{i}B_{j} − σ̇_{i}B_{k}B_{j} + σ( ˙∂_{k}B_{i} · B_{j} − ˙∂_{i}B_{k} · B_{j})+ \\ +σ(B_{i} · ˙∂_{k}B_{j} − B_{k} · ˙∂_{i}B_{j}) = 0, σ̇_{k} := ˙∂_{k}σ. \end{matrix}

Multiplying it by B^{j} one gets

(3.5)  \begin{matrix}B^{2}(σ̇_{k}B_{i} − σ̇_{i}B_{k}) + σB^{2}( ˙∂_{k}B_{i} − ˙∂_{i}B_{k})+ \\ +σ(B_{i} · ˙∂_{k}B_{j} · B^{j} − B_{k} ˙∂_{i}B_{j} · B^{j}) = 0. \end{matrix}

If (3.4) is an identity, then (3.5) should be an identity for any σ and B_{i}. But for B_{i} = B_{i}(x) and σ = F^{2}, (3.5) reduces to y_{k}B_{i} − y_{i}B_{k} = 0 which is not an identity for any B_{i}. Thus in general ^{∗}g_{ij}(x, y) is not reducible to an L–metric.

We have a case when ^{∗}g_{ij}(x, y) is an L–metric as follows.

Proposition 3.1. Assume B_{i} = B_{i}(x). If σ(x, y) = f(B_{i}(x)y^{i}) for a smooth function f : IR → IR, then ^{∗}g_{ij} is an L–metric.

Indeed, it is easy to check that in these hypothesis (3.4) identically holds. Notice

that we do not know which is L such that ^{∗}g_{ij} = \frac{1}{2} ˙∂_{i} ˙∂_{j}L.

It is said that ^{∗}g_{ij}(x, y) is weakly regular if its absolute energy

(3.6)  E(x, y) := ^{∗}g_{ij}(x, y)y^{i}y^{j} = F^{2}(x, y) + σ(x, y)(B_{i}y^{i})^{2}

is a regular Lagrangian i.e. the matrix with the entries

(3.7)  a_{kh}(x, y) = \frac{1}{2} ˙∂_{k} ˙∂_{h}E,

is of rank n.

A direct calculation yields

(3.8)  a_{kh} = g_{kh} + \frac{1}{2}σ̇_{kh}β^{2} + β(σ̇_{k}β̇_{h} + σ̇_{h}β̇_{k}) + σβ̇_{k}β̇_{h} + σββ̇_{kh},

(3.8)′  β := B_{i}(x, y)y^{i}, β̇_{k} := ˙∂_{k}β, β̇_{kh} := ˙∂_{k} ˙∂_{h}β, σ̇_{kh} := ˙∂_{k} ˙∂_{h}σ, σ̇_{k} := ˙∂_{k}σ

It is hopeless to decide if a_{kh} is invertible or not. However we have some interesting particular cases.

Proposition 3.2

a) If B is orthogonal to the Liouville vector field C = y^{i} ˙∂_{i}, then ^{∗}g_{ij} is weakly regular and a_{kh}(x, y) = g_{kh}(x, y).

b) If B_{i} = B_{i}(x) and σ(x, y) = f(β) for some smooth function f : R → R, then ^{∗}g_{ij} is weakly regular if and only if 1+φ(β)B^{2} ≠ 0, where 2φ(β) = β^{2}f′′ +4βf′ +2f,

f′ = \frac{df}{dβ} , f′′ = \frac{d^{2}f}{dβ^{2}} and we have

((3.9)  a_{kh}(x, y) = g_{kh}(x, y) + φ(x, y)B_{k}(x)B_{h}(x).

Proof. a) The condition B orthogonal to C is equivalent to β = 0. Thus E(x, y) = F^{2}(x, y) and so a_{kh} = g_{kh}.

b) By a direct calculation one finds ((3.9). Hence (a_{kh}) has the same form as ^{∗}g_{kh} with σ replaced by φ. The conclusion follows.

We keep the hypothesis B_{i} = B_{i}(x) and σ = f(β), β ≠ 0. From ((3.9) we see that we have again a_{kh} = g_{kh} when φ = 0. The differential equation β^{2}f′′ + 4βf′ + 2f = 0 takes the form (β^{2}f′ + 2βf)′ = 0 and so its general solution is f(β) = \frac{a}{β} + \frac{b}{β^{2}} ,

a, b ∈ IR. The metric ^{∗}g_{ij} becomes

((3.10)  ^{∗}g_{ij} = g_{ij} + (\frac{a}{B_{i}(x)y^{i}} + \frac{b}{(B_{s}(x)y^{s})^{2}} ) B_{i}(x)B_{j}(x).

Notice that although ^{∗}g_{ij} is an L–metric, we do not yet know the Lagrangian L.

The absolute energy of ^{∗}g_{ij} is now E = F^{2} +a(F_{i}(x)y^{i})+b and the Lagrange space L^{n} = (M, E) is called an almost Finslerian–Lagrange space (see Section 6, ch.IX of [?]).

We may put ((3.9) into the form

((3.9)′  a_{kh}(x, y) = ^{∗}g_{kh} + (\frac{1}{2}β^{2}f′′ + 2βf′) B_{k}B_{h}.

Thus we see that a_{kh} = ^{∗}g_{kh} if and only if f is a solution of the differential equation

\frac{1}{2}f′′β^{2} + 2βf′ = 0 i.e. f(β) = c − \frac{d}{β^{3}} , c, d ∈ IR.

We know that ^{∗}g_{kh} is an L–metric (in previous hypothesis). The condition a_{kh} = ^{∗}g_{kh} gives L in the form L(x, y) = E(x, y) + A_{i}(x)y^{i} + ψ(x), where A_{i} is a covector and ψ a scalar. Inserting here E we get

((3.10)′ L(x, y) = F^{2}(x, y) + c(B_{i}(x, y)y^{i})^{2} − \frac{d}{B_{i}(x)y^{i}} + A_{i}(x)y^{i} + ψ(x), c, d ∈ IR.

Therefore we found a case when ^{∗}g_{ij} is an L–metric with L of explicit form ((3.10)′.

Remark 3.2 In the hypothesis of a) in Proposition 3.2, ^{∗}g_{ij} is not necessarily an L– metric. If σ(x, y) and B_{i}(x, y) are positively homogeneous of degree 0, then ^{∗}g_{ij}(x, y) is so and (M,^{∗} g_{ij}) is a generalized Finsler space in Izumi’ sense (see [?]).

Remark 3.3. The condition B orthogonal to C is equivalent with the condition B is tangent to the indicatrix bundle I(M) ⊂ T M.

Caution. The conditions β = 0 and B_{i} = B_{i}(x) are incompatible since they lead to B = 0.

Remark 3.4. If in ((3.10) we take d = 0, A_{i} = 0, ψ = 0, c > 0, then ^{∗}F^{2} := L(x, y) is positively homogeneous of degree 2 and so ^{∗}F^{n} = (M,^{∗} F) becomes a Finsler space. Notice that ^{∗}F is getting from F by a β–change and in this case ^{∗}g_{ij} reduces to a Finsler metric.

Remark 3.5. An interesting Beil metric can be associated to a Finsler space F^{n} with an (α, β)–metric. Here α^{2} = a_{ij}(x)y^{i}y^{j} and β = b_{i}(x)y^{i}, where a_{ij} is a Riemannian metric an b_{i} a covector field on M. One may consider

(3.12)  ^{∗}g_{ij}(x, y) = a_{ij}(x) + σ(x, y)b_{i}(x)b_{j}(x),

where σ is a Finsler scalar such that 1 + σb^{2} ≠ 0 for b^{2} = a^{ij}b_{i}b_{j}. This GL–metric is not reducible to an L–metric or a Finsler metric. The previous discussion applies, too.

4 Metrical connections for GL=(M,^{∗} g_{ij}(x, y))

In Finsler geometry as well as in their generalizations, the nonlinear connections play an important role. For instance these connections allow us to work with d– or Finsler objects and so to keep and check easily the geometrical meaning of calculation in local coordinates.

A nonlinear connection always exists if M is paracompact. But the nonlinear connections derived from or associated in a way to a GL–metric are much more useful. There are no possibilities to find nonlinear connections for any GL–metric. But there are some classes of GL–metrics for which such possibilities exist. One is that of weakly regular GL–metrics and as it is well known there exist nonlinear connections canonically derived from a Lagrangian, a Finslerian or a Riemannian metric. See [?] for details.

We recall here the Cartan nonlinear connection for F^{n}. Set

(4.1)  γ_{jk}^{i}(x, y) = \frac{1}{2}g^{ih}(∂_{j}g_{hk} + ∂_{k}g_{hj} − ∂_{h}g_{jk}), γ_{00}^{i} := γ_{jk}^{i}y^{j}y^{k}.

Then N^{∘} _{j}^{i} = \frac{1}{2} ˙∂_{j}γ_{00}^{i} are the local coefficients of the Cartan nonlinear connection.

For any Finsler connection F Γ(N) we denote by |k and |_{k} its h– and v–covariant derivatives. Then F Γ(N) is called h–metrical if g_{ij|k} = 0 and v–metrical if g_{ij}|_{k} = 0.

We consider

(4.2)  \begin{matrix}F_{jk}^{i} = \frac{1}{2}g^{ih}(δ_{j}g_{hk} + δ_{k}g_{jh} − δ_{h}g_{jk}), \\ C_{jk}^{i} = \frac{1}{2}g^{ih}( ˙∂_{j}g_{hk} + ˙∂_{k}g_{jh} − ˙∂_{h}g_{jk}), \end{matrix}

where δ_{j} = ∂_{j}− N^{∘} _{j}^{k} ˙∂_{k}. For F^{n} we have four remarkable Finsler connections based on

(N^{∘} _{j}^{i}).

We mention here only the Cartan connection CΓ(N^{∘}) = (N^{∘} _{j}^{i}, F_{jk}^{i}, C_{jk}^{i}). This is v– and h–metrical and two torsions of it vanishes.

Let us come back to the GL–metric (3.1). We cannot derive a nonlinear connection from it. But since it is constructed with g_{ij}(x, y), we may take into consideration the

Cartan nonlinear connection (N^{∘} _{j}^{i}) and then all possible nonlinear connections have

the form N_{j}^{i} =N^{∘} _{j}^{i} − A_{j}^{i} with A_{j}^{i}(x, y) an arbitrary Finsler tensor field of type (1, 1). Now we replace in the right side of (4.2) the metric g_{ij} by ^{∗}g_{ij} and the operator δ_{j}

by ^{s}δ_{j} = ∂_{j}− N^{∘} _{j}^{k} ˙∂_{k} + A_{j}^{k} ˙∂_{k} and denote the results in the left side by ^{s}F_{jk}^{i} and ^{s}C_{jk}^{i}, respectively. Thus we get a Finsler connection ^{s}CΓ(N) = (N_{j}^{i},^{s} F_{jk}^{i},^{s} C_{jk}^{i}) which we call standard metrical connection of GL.

This connection is metrical i.e. ^{∗}g_{ij|^{s}k} = 0, ^{∗}g_{ij} |^{s}_{k} = 0 and its h(hh)– torsion and

v(vv)–torsion vanish. It is clear that it depends on A_{j}^{i} but if A_{j}^{i} is given apriori it is the unique Finsler connection with the above properties. For A_{j}^{i} = 0 we set ^{∗}F := ^{s}F and ^{∗}C := ^{s}C. Thus we have

(4.3)  \begin{matrix}^{s}F_{jk}^{i} =^{∗} F_{jk}^{i} + \frac{1}{2}^{∗}g^{ih}(A_{j}^{s} ˙∂_{s}^{∗}g_{hk} + A_{k}^{s} ˙∂_{s}^{∗}g_{hj} − A_{h}^{s} ˙∂_{s}^{∗}g_{jk}) \\ ^{s}C_{jk}^{i} = ^{∗}C_{jk}^{i}. \end{matrix}

The first equation in (4.3) takes also the form

sF_{jik} =^{∗} F_{jik} +^{∗} C_{kis}A_{j}^{s} +^{∗} C_{jis}A_{k}^{s} −^{∗} g^{ih}A_{h}^{l}^{∗}C_{jkl}.

Remark 4.1. If (^{∗}g_{ij}) reduces to an L–metric or to a Finsler metric, (4.3) becomes

(4.3)′  \begin{matrix}^{s}F_{jk}^{i} = ^{∗}F_{jk}^{i} + C_{ks}^{i}A_{j}^{s} \\ ^{s}C_{jk}^{i} = ^{∗}C_{jk}^{i}. \end{matrix}

We notice the following possible choices of A_{j}^{i} : λ(x, y)δ_{j}^{i}, y^{i}y_{j}, B^{i}y_{j}, y^{i}B_{j}, B^{i}B_{j}. By (3.1) we find

(4.4)  \begin{aligned}^{∗}F_{jk}^{i} & = B_{s}^{i}F_{jk}^{s} + \frac{σ}{2} g^{∗}^{ih}[δ_{j}(B_{h}B_{k}) + δ_{k}(B_{h}B_{j}) − δ_{h}(B_{j}B_{k})]+ \\ & +\frac{1}{2}^{∗}g^{ih}(σ_{j}B_{h}B_{k} + σ_{k}B_{h}B_{j} − σ_{h}B_{j}B_{k}), \\ ^{∗}C_{jk}^{i} & = B_{s}^{i}C_{jk}^{s} + \frac{σ}{2} g^{∗}^{ih}[ ˙∂_{j}(B_{h}B_{k}) + ˙∂_{j}(B_{h}B_{j}) − ˙∂_{j}(B_{j}B_{k})]+ \\ & +\frac{1}{2}^{∗}g^{ih}(σ̇_{j}B_{h}B_{k} + σ̇_{k}B_{h}B_{j} − σ̇_{h}B_{j}B_{k}), with \end{aligned}

(4.4)′  B_{s}^{i} = ˙∂_{s}^{i} −^{∗}σB^{i}B_{s}, σ_{k} := δ_{k}σ, σ̇_{k} := ˙∂_{k}σ, ^{∗}σ = σ/(1 + σB^{2}).

Now, ^{s}F_{jk}^{i} and ^{s}C_{jk}^{i} are easily deduced from (4.3).

Remark 4.2. The matrix B_{s}^{i} is invertible. Its inverse is (B^{−1})_{k}^{s} = δ_{k}^{s} + σB^{s}B_{k}. As such from (4.4) we can find F and C as depending on ^{∗}F and ^{∗}C.

In order to evaluate the torsions and curvatures of ^{∗}CΓ(^{c}N) it is more convenient to put (4.4) into the form

(4.5)  \begin{matrix}^{∗}F_{jk}^{i} = F_{jk}^{i} + Λ_{jk}^{i}, \\ ^{∗}C_{jk}^{i} = C_{jk}^{i}+ Λ^{∘} _{jk}^{i}, for \end{matrix}

(4.5)′  \begin{aligned}Λ_{jk}^{i} & = \frac{1}{2}^{∗}g^{ih}[δ_{k}(σB_{j}B_{h}) + δ_{j}(σB_{h}B_{k}) − δ_{h}(σB_{j}B_{k})]+ \\ & −^{∗}σB^{i}B^{h}F_{jhk} \\ Λ^{∘} _{jk}^{i} & = \frac{1}{2}^{∗}g^{ih}[ ˙∂_{k}(σB_{j}B_{h}) + ˙∂_{j}(σB_{h}B_{k}) − ˙∂_{h}(σB_{j}B_{k})]+ \\ & −^{∗}σB^{i}B^{h}C_{ijk}. \end{aligned}

The torsions of ^{∗}CΓ(^{c}N) are as follows.

(4.6)

∗T_{jk}^{i} = 0, ^{∗}R_{jk}^{i} = R_{jk}^{i}, ^{∗}S_{jk}^{i} = 0
∗P_{jk}^{i} = P_{jk}^{i} − Λ_{kj}^{i} and ^{∗}C_{jk}^{i} from (4.5).

As for the curvatures we have

\begin{matrix}(4.7) & ^{∗}S_{j}^{i}_{kh} = S_{j}^{i}_{kh}+ Λ^{∘}_{j}^{i}_{kh} + (C_{jk}^{s} Λ^{∘} _{sh}^{i}+ Λ^{o} _{jk}^{s}C_{sh}^{i} − (k/h)) \\ (4.7)′ & Λ^{∘}_{j}^{i}_{kh} = ˙∂_{h} Λ^{∘} _{jk}^{i}+ Λ^{∘} _{jk}^{s} Λ^{o} _{sh}^{i} − (k/h), \end{matrix}

where −(k/h) means the substraction of the preceeding terms with k replaced by h.

(4.8)

∗P_{j}^{i}_{kh} = P_{j}^{i}_{kh} + ˙∂_{h}Λ_{jk}^{i}− Λ^{∘} _{jh|k}^{i} − C_{jh∥k}^{i}− Λ^{∘} _{jh∥k}^{i}+

+ ˙∂_{k}C_{jh}^{i} + ˙∂_{k} Λ^{o}_{jh}^{i} −C_{js}^{i}Λ_{hk}^{s}+ Λ^{o} _{js}^{i}P_{hk}^{s}− Λ^{o} _{js}^{i}Λ_{kh}^{s},

where ∥k denotes a covariant derivative constructed with Λ_{jk}^{i}.

(4.9)  ^{∗}R_{j}^{i}_{kh} = R_{j}^{i}_{kh} + Λ_{j}^{i}_{kh} + (F_{jk}^{s}Λ_{sh}^{i} + Λ_{jk}^{s}F_{sh}^{i} − (k/h))+ Λ^{∘} _{js}^{s}R_{kh}^{s},

where

(4.9)′  Λ_{j}^{i}_{kh} = δ_{h}Λ_{jk}^{i} + Λ_{jk}^{s}Λ_{sh}^{i} − (k/h).

5 Parallel and concurrent Finsler vector fields

Let B^{i}(x, y) be a Finsler vector field and F Γ(N) be a Finsler connection. Then it is said that (B^{i}) is parallel if

(5.1)  B_{|k}^{i} = 0, B^{i}|_{k} = 0

and (B^{i}) is concurrent if

(5.2)  B_{|k}^{i} = −δ_{k}^{i}, B^{i}|_{k} = 0.

It is our purpose to confirm the correctness of these definitions from the viewpoint of the almost Kählerian model of a Finsler space (see [?], ch.VII for details on this model). A different confirmation of these definitions is given in [?] using the principal Finsler bundle model due to M. Matsumoto. The giving of N is equivalent to the decomposition

(5.3)  T_{u}T M = H_{u}T M ⊕ V_{u}T M, u ∈ T M (Whitney’ sum).

Accordingly we have two projectors h and v and an almost product structure P such that if we put X = hX + vX for a vector field X on T M, then

(5.5)  P (hX) = hX, P (vX) = −vX.

The set of Finsler connections is in a one–to–one correspondence with the set of linear connections on T M which verify

(5.6)  D_{X}P = 0, D_{X}F = 0 for any vector field X on T M.

By the very definition, a vector field B on T M is parallel with respect to D if

(5.7)  D_{X}B = 0,

and is concurrent if

(5.8)  D_{X}B = −X, for any vector field X on T M.

Let (δ_{i}, ˙∂_{i}) be the usual adapted basis for the decomposition (5.3). The above mentioned one–to–one correspondence is established by

(5.9)  \begin{matrix}D_{δ_{k}}δ_{j} = L_{jk}^{i}δ_{i}, & D ˙∂_{k}δ_{j} = V_{jk}^{i}δ_{i}, \\ D_{δ_{k}} ˙∂_{j} = L_{jk}^{i} ˙∂_{i}, & D ˙∂_{k} ˙∂_{j} = V_{jk}^{i} ˙∂_{i}, \end{matrix}

for D ↔ F Γ(N) = (N_{j}^{i}, L_{jk}^{i}, V_{jk}^{i}).

It is obvious that (5.7) is equivalent to

(5.7)′  D_{δ_{k}}B = 0, D ˙∂_{k}B = 0,

and (5.8) is equivalent to

(5.8)′  D_{δ_{k}}B = −δ_{k}, D ˙∂_{k}B = − ˙∂_{k}.

Let now be B = B^{i}(x, y)δ_{i} + B̂^{i}(x, y) ˙∂_{i}. Then (5.7)′ is equivalent by virtue of (5.9) with

(5.7)′′  B_{|k}^{i} = 0, B^{i}|_{k} = 0, B̂_{|k}^{i} = 0, B̂^{i}|_{k} = 0.

One may associate to B^{i}(x, y) at least the following three vector fields on T M : B^{i}δ_{i}, B^{i} ˙∂_{i}, B^{i}δ_{i} + B^{i} ˙∂_{i} and it is obvious by (5.7)′′ that B^{i}(x, y) is parallel in the sense of (5.1) if and only if at least one from these vector fields on T M is parallel with respect to D. Thus (5.1) is in agreement with the usual definition of parallelism.

Let us make a similar analysis for concurrent Finsler vector fields. By (5.8), B is concurrent on T M if and only if

(5.10)  B_{|k}^{i} = −δ_{k}^{i}, B|_{k}^{i} = 0, B̂_{|k}^{i} = 0, B̃^{i}|_{k} = −δ_{k}^{i}.

Now we assume that D or F Γ(N) is of Cartan type, that is,

(5.11)  y_{|k}^{i} = 0, y^{i}|_{k} = δ_{k}^{i}.

The tensors y_{|k}^{i} and y^{i}|_{k} are called h–deflection and v–deflection tensors, respectively. The equations (5.11) hold for all four remarkable connections in Finsler spaces.

If moreover we assume that B̂^{i} is positively homogeneous of degree 1, a natural assumption in Finslerian setting, writing B̂^{i}|_{k} = −δ_{k}^{i} in the form ˙∂_{k}B̂^{i} +V_{jk}^{i}B̂^{j} = −δ_{k}^{i} and contracting it by y^{k} it results using (5.11) that y^{k} ˙∂_{k}B̂^{i} = −y^{i}. Thus by the Euler theorem, B̂^{i} = −y^{i} and then B̂_{|k}^{i} = 0 reduces to y_{|k}^{i} = 0 i.e. the first equation in (5.11). Concluding, if we associate to the Finsler vector field B^{i}(x, y) the vector field B = B^{i}(x, y)δ_{i}−y^{i} ˙∂_{i} on T M, we find that (B^{i}(x, y)) is concurrent in the sense of (5.2) if and only if B is concurrent by the new definition of concurrence on any manifold. In other words, the condition (5.2) is in agreement with the notion of concurrence for vector fields.

6 The metric ^{∗}g_{ij} with B^{i}(x, y) a concurrent Finsler vector field

In this section we are dealing with the GL–metric ^{∗}g_{ij} given by (3.1) for B^{i}(x, y) a concurrent Finsler vector field with respect to the Cartan connection CΓ of F^{n} i.e.

(6.1)  B_{|j}^{i} = −δ_{j}^{i}, B^{i}|_{j} = 0.

First we notice some results on concurrent Finsler vector fields due to M. Matsumoto and K. Eguchi [?].

If B^{i}(x, y) is concurrent we have with respect to CΓ :

(6.2) B_{i|j} = −g_{ij}, B_{i}|_{j} = 0,

(6.3) B^{h}R_{hijk} = 0, B^{h}P_{hijk} + C_{ijk} = 0, B^{h}S_{hijk} = 0,

(6.4) B^{i}C_{ijk} = C_{jk}^{s}B_{s} = 0,

(6.5) B^{i} = B^{i}(x) and B_{i} = B_{i}(x) i.e. B^{i} and B_{i} are functions on position only,

(6.6) ∂_{i}B_{j} = ∂_{j}B_{i} = F_{ij}^{s}B_{s} − g_{ij}, ∂_{k}B^{i} = −δ_{k}^{i} − F_{sk}^{i}B^{k}.

In these circumstancies a direct calculation yields

(6.7)  \begin{matrix}Λ_{jk}^{i} = \frac{^{∗}σ}{2σ} B^{i}(σ_{k}B_{j} + σ_{j}B_{k} + σ(B^{s}σ_{s})B_{j}B_{k} − 2σg_{jk}) − \frac{1}{2}σ^{i}B_{j}B_{k} \\ Λ^{∘} _{jk}^{i} = \frac{^{∗}σ}{2σ} B^{i}(σ̇_{k}B_{j} + σ̇_{j}B_{k} + σ(B^{s}σ̇_{s})B_{j}B_{k} − \frac{1}{2}σ̇^{i}B_{j}B_{k}, where \end{matrix}

(6.7)′  σ_{k} := δ_{k}σ, σ̇_{k} := ˙∂_{k}σ, σ^{i} = g^{i_{k}}σ_{k}, σ̇^{i} = g^{i_{k}} σ̇_{k}.

Looking at (6.7) we see that the simplest case is given by

(6.8)  σ_{k} = 0, σ̇_{k} = 0.

From (6.8) it results that σ is a constant c. And ^{∗}F^{2} := ^{∗}g_{ij}y^{i}y^{j} takes the form

(6.9)  ^{∗}F^{2} = F^{2} + cβ^{2}, β = B_{i}(x)y^{i}.

Thus, for c > 0, ^{∗}F is a new Finsler function which is obtained from F by a particular β–change.

The case c = 1 is studied in [?].

Further on we have

(6.10)  ^{∗}F_{jk}^{i} = F_{jk}^{i} −^{∗}σB^{i}g_{jk}, ^{∗}C_{jk}^{i} = C_{jk}^{i}.

Remark 6.1. The Cartan nonlinear connection of ^{∗}F^{n} = (M,^{∗} F) is given by N_{j}^{i} =N^{c} _{j}^{i}− σ^{∗} B^{i}y_{j} i.e. the difference tensor is A_{j}^{i} =σ^{∗} B^{i}y_{j}. Inserting it in (4.3)′ we find

sF_{jk}^{i} = ^{∗}F_{jk}^{i}. Therefore, in the geometry of ^{∗}F^{n} we may equally use N^{c} _{j}^{i} or N_{j}^{i}.

By (6.10) we immediately get

(6.11)  ^{∗}S_{ijkh} = S_{ijkh}.

Again by (6.10) but after a long calculation one finds

(6.12)  ^{∗}R_{ijkh} = R_{ijkh} +^{∗}σ(g_{ik}g_{jh} − g_{ih}g_{jk}).

This suggests us to take into consideration the case when F^{n} is h–isotropic i.e. there exists a constant K such that R_{ijkh} = K(g_{ik}g_{jh} − g_{ih}g_{jk}). A contraction of this last equation by B^{i} gives for K ≠ 0, B_{k}g_{jh}−B_{h}g_{jk} = 0 in virtue of (6.3). A new contraction by B^{k} yields B^{2}g_{jh} = B_{j}B_{h} which contradicts the condition rank (g_{ij}) = n > 1. Thus we have

Theorem 6.1. If F^{n} is h–isotropic, then it does not admit any concurrent Finsler vector field.

The proof of the following two theorems are the same as for c = 1 (see Theorems 14 and 15 in [?]).

Theorem 6.2. If F^{n} admits a concurrent Finsler vector field, then there is no a Finsler vector field which to be concurrent with respect to ^{∗}F given by (6.9). Theorem 6.3. If F^{n} admits a concurrent Finsler vector field and is R3–like, then ∗F^{n} = (M,^{∗} F) with ^{∗}F from (6.5) is also R3–like.

Now we consider a more complicated case

(6.13)  σ_{k} = 0, σ̇_{k} ≠ 0.

Remark 6.2. The equation σ_{k} := \frac{∂σ}{∂x^{h}} − N^{c} _{k}^{s} \frac{∂σ}{∂y^{s}} = 0 is known as Tavakol–Van der

Berg equation. A solution of it is for instance σ = aF^{2} for a ∈ IR. For more details see [?].

Now (6.10) is replaced by

(6.14)  \begin{matrix}^{∗}F_{jk}^{i} = F_{jk}^{i} −^{∗}σB^{i}g_{jk} \\ ^{∗}C_{jk}^{i} = C_{jk}^{i} + \frac{^{∗}σ}{2σ} B^{i}(σ̇_{k}B_{j} + σ̇_{j}B_{k} + σ(B^{s}σ̇_{s})B_{j}B_{k}) − \frac{1}{2}σ̇^{i}B_{j}B_{k}. \end{matrix}

The Remark 6.1 is still valid for this case. Precisely, if we ask for the vanishing of

the h–deflection of ^{∗}F Γ(N^{c}), then ^{∗}N_{j}^{i} =N^{c} _{j}^{i}− σ^{∗} B^{i}y_{j} and so ^{s}F Γ(N^{∗}) coincides with

∗F Γ(N^{c}).

Now we notice

\begin{matrix}(6.15) & ^{∗}C_{j} = C_{j} + \frac{^{∗}σB^{2}}{2σ} σ̇_{j}, C_{j} := C_{ji}^{i}, \\ (6.16) & ^{∗}C_{jik} = C_{jik} + \frac{1}{2}(σ̇_{k}B_{i}B_{j} + σ̇_{j}B_{i}B_{k} − σ̇_{i}B_{j}B_{k}). \end{matrix}

A long calculation yields

(6.17)  \begin{matrix}^{∗}R_{jskh} = R_{jskh} +^{∗}σ(g_{jk}g_{sh} − g_{jh}g_{sk})+ \\ +\frac{^{∗}σ}{σ} B_{s}(∂_{k}σ · g_{jh} − ∂_{h}σ · g_{jk}) + \frac{1}{2}B_{j}B_{s}R_{kh}^{q}σ̇_{q}. \end{matrix}

Let us assume that F^{n} is a locally Minkowski space. Then R_{j}^{i}_{kh} = 0 and C_{jk|h}^{i} = 0. In

a local chart in which g_{ij} do not depend on x we have N^{c} _{j}^{i} = 0 and so ∂_{k}σ =N^{c} _{j}^{p}σ̇_{p} = 0 i.e. σ does not depend on x.

The equation (6.17) reduces to

(6.18)  ^{∗}R_{jskh} = ^{∗}σ(g_{jk}g_{sh} − g_{jh}g_{sk}).

It takes also the form

(6.18)′  \begin{matrix}^{∗}R_{jskh} = ^{∗}σ(^{∗}g_{jk}^{∗}g_{sh} −^{∗}g_{jh}^{∗}g_{sk}) + σ^{∗}σ(B_{j}B_{hsk} + B_{s}B_{kjh}) for \\ B_{hsk} := B_{h}g_{sk} − B_{k}g_{sh}. \end{matrix}

We notice that B_{hsk} is never vanishing since otherwise a contraction by B^{h} gives a contradiction with rank (g_{ij}) = n > 1.

7 A Beil metric for a Finsler space with (α, β)–metric

Here we consider again the Beil metric described in Remark 3.5. Let F^{n} be a Finsler space with an (α, β)–metric. A natural Beil metric is then

(7.1)  ^{∗}g_{ij}(x, y) = a_{ij}(x) + σ(x, y)b_{i}(x)b_{j}(x).

Let γ_{jk}^{i} be the Christoffel symbols for a_{ij}(x). Then N^{c} _{j}^{i} = γ_{jk}^{i}y^{k} =: γ_{j0}^{i} and the triple Γ = (γ_{j0}^{i}, γ_{jk}^{i}, 0) may be thought of as a Finsler connection.

We have

Theorem 7.1. If b_{i}(x) is parallel and σ is covariant constant with respect to Γ, then Γ is like Chern–Rund connection for (^{∗}g_{ij}). Proof. Let _{;k} denote the h–covariant derivative with respect to Γ. Notice that v– covariant derivative is just the derivative with respect to y. Our hypothesis read

(7.2)  b_{i;k} = 0, δ_{k}σ = 0, δ_{k} = ∂_{k} − γ_{k0}^{s} ˙∂_{s}.

Then we easily get

(7.3)  \begin{matrix}^{∗}g_{i;jk} = (δ_{k}σ)b_{i}b_{j} = 0 \\ ^{∗}g_{ij,k} = ( ˙∂_{k}σ)b_{i}b_{j} = 2^{∗}C_{ikj}. \end{matrix}

Thus Γ is h–metrical and no metrical for ^{∗}g_{ij}. Hence it is similar to the Chern–Rund connection from Finsler geometry. q.e.d.

The Chern–Rund connection is a remarkable one in Finsler geometry ([?]). Notice that its h–deflection vanishes.

From now on we assume b_{i;k} = 0 and δ_{k}σ = 0.

A direct calculation yields

(7.4)  \begin{matrix}^{∗}F_{jk}^{i} = γ_{jk}^{i}, \\ ^{∗}C_{jk}^{i} = \frac{^{∗}σ}{2σ} b^{i}(σ̇_{k}b_{j} + σ̇_{j}b_{k} + σ(b^{h}σ̇_{h})b_{j}b_{k}) − \frac{1}{2}σ̇^{i}b_{j}b_{k}. \end{matrix}

The first equation in (7.4) is important in many respects. For instance using it we

find the h–curvature of ^{∗}F Γ(N^{c}) in the form

(7.5)  ^{∗}R_{h}^{i}_{jh} = γ_{h}^{i}_{jh}+ Λ^{∘}_{h} ^{i}_{s}R_{jk}^{s},

where γ_{h}^{i}_{jh} is the curvature tensor of a_{ij}(x) and R_{jk}^{i} = γ_{0}^{i}_{jk}. Here, as before, the index 0 indicates the contraction by y. Consequently, (7.5) takes the form

(7.6)  ^{∗}R_{h}^{i}_{jk} = (δ_{s}^{i}δ_{h}^{r}+ Λ^{∘}^{i}_{hs}y^{r})γ_{r}^{s}_{jk}.

From Ricci identities we find γ_{i}^{s}_{jk}b_{s} = 0 and from (7.5) we deduce

(7.7)  ^{∗}R_{hijk} = γ_{hijk} + \frac{1}{2}b_{h}b_{i}γ_{0}^{s}_{jk}σ̇_{s}.

As for Ricci curvatures one finds

(7.8)  ^{∗}R_{ij} = r_{ij},

where r_{ij} is the Ricci curvature for (a_{ij}(x)). From here it results

(7.9)  ^{∗}R = r,

where ^{∗}R and r are the scalar curvatures for (^{∗}g_{ij}) and (a_{ij}(x)), respectively.

So, the h–Einstein tensor field of ^{∗}g_{ij} i.e. ^{∗}E_{ij} =^{∗} R_{ij} − \frac{1}{2}^{∗}R^{∗}g_{ij} is related to the Einstein tensor E_{ij} of a_{ij}(x) by

(7.10)  ^{∗}E_{ij} = E_{ij} + \frac{σr}{2} b_{i}b_{j}.

Consequently, the h–Einstein equation for GL i.e. ^{∗}E_{ij} = κ^{∗}τ_{ij} with κ ∈ IR reduces to

(7.11)  r_{ij} − \frac{r}{2}a_{ij} = κτ_{ij},

where

(7.12)  τ_{ij} = ^{∗}τ_{ij} − \frac{σr}{2κ}b_{i}b_{j}.

The equation (7.11) is the Einstein equation for (M, a_{ij}(x)) but with the energy– momentum tensor influenced by a field described by b_{i}. In the the unified theory of R.G. Beil the term b_{i}b_{j} in (7.12) is a ”matter term” which could be the energy density of the self-field of a charged object.

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University ”Al.I.Cuza” Iaşi Faculty of mathematics 6600, Iaşi, Romania

Hokkaido Tokai University Minami-ku, Minami-sawa 5-1, Sapporo 005, Japan