Theory of Conformally Berwald Finsler Spaces and Its Applications to (α, β)-Metrics

Shun-ichi Hojo, M.Matsumoto and K.Okubo

Dedicated to Prof.Dr. Constantin UDRIŞTE on the occasion of his sixtieth birthday

Abstract

The theory of conformal changes of Finsler metrics has been studied by M. Hashiguchi [2] in 1976 and some of the Japanese school have directed their efforts to find conformally invariant curvature tensors similar to the Weyl conformal curvature tensor of a Riemannian space and to establish the condition for a Finsler space to be conformally flat. Finally, about five years ago, S.Kikuchi [6] succeeded in finding a conformally invariant Finsler connection and giving the conformally flat condition.

We have, however, a strange and objectionable in Kikuchi’s theory. His conformally invariant connection can be only defined on an essential assumption. Whether this assumption holds or not in a Finsler space under consideration poses newly a difficult problem. Since we have not a conformally invariant connection in the Riemannian case, the assumption is, of course, not satisfied by any Riemannian space.

About ten years ago, Y.Ichijyo and M.Hashiguchi [5] defined a conformally invariant Finsler connection in a Finsler space with (α, β)–metric, where α = (a_{ij}(x)y^{i}y^{j})^{1/2} is a Riemannian metric and β = b_{i}(x)y^{i} is a one-form in y^{i}, on the assumption b^{2} = a^{ij}b_{i}b_{j} ≠ 0. They gave the condition for a Randers space with the metric α + β to be conformally flat based on their connection. M.Matsumoto [11] showed that their theory can be applied to a Kropina space with the metric α^{2}/β.

The main purpose of the present paper is to consider Kikuchi’s conformally invariant Finsler connections of Finsler spaces with (α, β)–metric. Since our main interest is Kikuchi’s assumption, it is sufficient to stop our studies halfway to Finsler spaces conformal to locally Minkowski spaces. Thus we shall propose a new notion of conformally Berwald Finsler space.

Mathematics Subject Classification: 53C05, 53C60 Key words: Berwald connection, (α, β)-metric, Kropina space

Balkan Journal of Geometry and Its Applications, Vol.5, No.1, 2000, pp. 107-118 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1 Conformally Berwald connections

In this section, we give a conformally Berwald connection which is induced from a scalar field S with a regularity condition.

Let us consider a Finsler space F^{n} = F^{n}(M^{n}, L) with the Berwald connection BΓ = (G^{i}_{j}, G_{j}^{i}_{k}, 0) and a conformal change L → L̄ = e^{c(x)}L. The quantities of the conformally changed space F̄^{n} will be denoted by putting a bar.

We have first the conformally invariant tensors B_{ij} and B^{ij};

B_{ij} = (\frac{2}{L^{2}} ) (g_{ij} − 2l_{i}l_{j}), B^{ij} = (\frac{L^{2}}{2} ) (g^{ij} − 2l^{i}l^{j}).

The matrix (B^{ij}) is the inverse of (B_{ij}) [2]. In the following we denote by subscripts of B^{ij} the partial differentiations of B^{ij} by y^{h} : B^{ij}_{h...k} = ˙∂ … ˙∂_{k}B^{ij}.

Putting F = \frac{L^{2}}{2} and 2G^{i} = g^{ij}(( ˙∂_{j}∂_{r}F)y^{r} − ∂_{j}F), we have G^{i}_{j} = ˙∂_{j}G^{i} and

G_{j}^{i}_{k} = ˙∂_{k}G^{i}_{j}. If we put c_{i} = ∂_{i}c, on account of the paper [2] we get Ḡ^{h} = G^{h} −B^{hr}c_{r} and

(1.1)  Ḡ^{h}_{i} = G^{h}_{i} − B^{hr}_{i}c_{r}, Ḡ_{i}^{h}_{j} = G_{i}^{h}_{j} − B^{hr}_{ij}c_{r}.

Then we obtain the relations between the hv-curvature tensors

(1.2)  Ḡ_{i}^{h}_{jk} = G_{i}^{h}_{jk} − B^{hr}_{ijk}c_{r}.

Assume that we have a conformally invariant scalar field S(x, y) which is (r)p– homogeneous in (y^{i}). Denoting by (; ) the h-covariant differentiation in BΓ, (1.1) yields

S̄_{;i} = ∂_{i}S̄ − ( ˙∂_{r}S)G^{r}_{i} = ∂_{i}S − ( ˙∂_{r}S)(G^{r}_{i} − B^{rs}_{i}c_{s}),

and hence

(1.3)  S̄_{;i} − S_{;i} = W^{r}_{i}c_{r}, W^{j}_{i} := ( ˙∂_{r}S)B^{rj}_{i}.

Along the lines of S.Kikuchi [6] and F.Ikeda [4] we shall suppose that

(1.4)  det(W^{j}_{i}) ≠ 0, and let (V^{i}_{j}) be the inverse matrix of (W^{j}_{i}).

V^{i}_{j}(x, y) are (−r)p–homogeneous in y^{i} and (1.3) can be written in the form

(1.5)  V̄_{j} − V_{j} = c_{j}, V_{j} := S_{;r}V^{r}_{j}.

V_{j}(x, y) are (0)p–homogeneous in y^{i}. Since c_{j} are functions of position, we must have

(1.6)  ˙∂_{i}(V̄_{j} − V_{j}) = 0.

Substituting from (1.5) in (1.1), we get the invariant quantities

(1.7)  ^{c}G^{h}_{i} = G^{h}_{i} + B^{hr}V_{r}, ^{c}G_{i}^{h}_{j} = G_{i}^{h}_{j} + B^{hr}V_{r}.

Consequently we obtain the conformally invariant Finsler connection^{c}BΓ = (^{c}G^{h}_{i},^{c}G_{i}^{h}_{j}, 0). This is called the conformal Berwald connection with respect to S. On the other hand, (1.2) yields a conformally invariant tensor

(1.8)  ^{c}G_{i}^{h}_{jk} = G_{i}^{h}_{jk} + B^{hr}_{ijk}V_{r}.

It is remarked that the hv-curvature tensor ( ˙∂_{k}(^{c}G_{i}^{h}_{j}) of^{c}BΓ is conformally invariant, but it is different from ^{c}G_{i}^{h}_{jk}:

(1.9)  ˙∂_{k}(^{c}G_{i}^{h}_{j}) = ^{c}G_{i}^{h}_{jk} + B^{hr}_{ij} ˙∂_{k}V_{r}.

According to the Berwald expression (Theorem 3.4) of a Finsler connection given by T.Aikou and M.Hashiguchi [1], the set (L_{k}, D^{i}_{k}, T_{j}^{i}_{k}, P^{i}_{jk}, C_{j}^{i}_{k}) of the essential tensor fields of ^{c}BΓ are

(1.10)  L_{k} = −l_{r}B^{rs}_{k}V_{s}, D^{i}_{k} = 0, T_{j}^{i}_{k} = 0, P^{i}_{jk} = B^{ir}_{j} ˙∂_{k}V_{r}, C_{j}^{i}_{k} = 0.

The conditions (1) and (2) mentioned in their theorem are satisfied because V_{i}(x, y) are (0)p−homogeneous in y^{i}.

A Finsler space F^{n} is called a Berwald space if G_{j}^{i}_{k} are functions of position alone, or G_{i}^{h}_{jk} = 0. In the Cartan connection CΓ = (G^{i}_{j}, F_{j}^{i}_{k}, C_{j}^{i}_{k}), F^{n} is a Berwald space if and only if F_{j}^{i}_{k} are functions of position alone, or C_{j}^{i}_{k/l} = 0 in terms of the hcovariant differentiation in CΓ. Definition. A Finsler space F^{n} = (M^{n}, L) is called conformally Berwald, if there exists a conformal change L → L̄ = e^{c(x)}L such that the changed space F̄^{n} = (M^{n}, L̄) is a Berwald space.

We deal with a conformally invariant scalar S which satisfies S_{;i} = 0 for a Berwald space. Such an invariant S is called of parallel type [4], (Theorem 2.1). The supposition det(W^{j}_{i}) ≠ 0 with respect to S of parallel type is called the Kikuchi condition. Then we get

(1.11)  c_{j} = V̄_{j} − V_{j}, V_{j} = S_{;r}V^{r}_{j},

on the Kikuchi condition.

Now we consider a Finsler space F^{n} having S satisfying the Kikuchi condition and suppose that F^{n} is conformal to a Berwald space F̄^{n}. Then S̄_{;i} = 0 and V̄_{j} = 0, and hence (1.11) is reduced to

(1.12)  c_{j} = −V_{j},

which implies that V_{j} = V_{j}(x) is a gradient vector:

(1.13)  (a) ˙∂_{j}V_{i} = 0, (b) V_{i;j} − V_{j;i} = 0.

Next, since F̄^{n} is a Berwald space, we have Ḡ_{i}^{h}_{jk} = 0 and hence (1.8) implies cḠ_{i}^{h}_{jk} = 0. Consequently we have

(1.14)  ^{c}G_{i}^{h}_{jk} = 0.

Therefore (1.13a), (1.14) and (1.9) lead to the fact that the hv–curvature tensor of cBΓ vanishes.

Conversely, we consider a Finsler space F^{n} having S satisfying the Kikuchi condition such that V_{j} with respect to S satisfies (1.13) and ^{c}BΓ has the vanishing hv–curvature tensor. (1.9) with (1.13a) show ^{c}G_{i}^{h}_{jk} = 0. (1.13) gives the function c(x) satisfying (1.12) and hence we have the conformal change L → L̄ = e^{c(x)}L. Then cG_{i}^{h}_{jk} = 0 and (1.5) with (1.12) give V̄_{j} = 0. Then (1.8) leads to Ḡ_{i}^{h}_{jk} = 0 and thus the changed space F̄^{n} is a Berwald space.

We denote by ^{c}∇ the h–covariant differentiation in ^{c}BΓ, the v–covariant one in cBΓ is ˙∂. Then (1.13) are written in terms of ^{c}BΓ as follows:

(1.13′)  (a) ˙∂_{j}V_{i} = 0, (b) ^{c}∇_{j}V_{i} −^{c}∇_{i}V_{j} = 0.

Therefore we have Theorem 1. Let F^{n} be a Finsler space having an S satisfying the Kikuchi condition. F^{n} is a conformally Berwald space, if and only if its conformal Berwald connection with respect to S has the vanishing hv–curvature tensor and satisfies (1.13′).

2 Kikuchi’s assumption for (α, β)–metrics

To do justice Kikuchi’s assumption (1.4), that is the condition det(W^{i}_{j}) ≠ 0, we shall be concerned with Finsler space with (α, β)–metrics.

Let us consider a Finsler space F^{n} = (M^{n}, L(α, β)) with (α, β)–metric where α^{2} = a_{ij}(x)y^{i}y^{j} is a Riemannian fundamental form and β = b_{i}(x)y^{i} is a 1–form in y^{i}. We put α_{i...j} = ˙∂_{i} … ˙∂_{j}α and have

(2.1)  \begin{matrix}αα_{i} = Y_{i}, Y_{i} := a_{ir}y^{r}, \\ αα_{ij} = a_{ij} − Y_{i}Y_{j}/α^{2} := k_{ij}, \end{matrix}

where k_{ij} is the angular metric tensor of the Riemannian space R^{n} = (M^{n}, α) associated with F^{n}. Next we have

(2.2)  αα_{ijk} = −(k_{ij}Y_{k} + (ijk))/α^{2},

where +(ijk) denotes cyclic permutations with respect to indices and their sum.

Further we put F = L^{2}/2 and the derivatives of F with respect to (y^{i}, α, β) are denoted by the subscripts (i, 1, 2). Then F_{i} = F_{1}α_{i} + F_{2}v_{i} and

(2.3)  F_{ij} = F_{1}α_{ij} + F_{11}α_{i}α_{j} + F_{12}(α_{i}b_{j} + α_{j}b_{i}) + F_{22}b_{i}b_{j}.

Since F_{ij} is the fundamental tensor g_{ij} of F^{n}, we have from (2.1) and F_{1} = F_{11}α + F_{12}β,

(2.3)′  g_{ij} = (F_{1}/α)a_{ij} + F_{22}b_{i}b_{j} + (F_{12}/α)(b_{i}Y_{j} + b_{j}Y_{i}) − (βF_{12}/α^{3})Y_{i}Y_{j}.

We have shown [8], [12].

(2.4)  \begin{matrix}det(g_{ij}) = (F_{1}/α)^{n−2}T det(a_{ij}), \\ T := DB + 2F F_{1}/α^{3}, \\ D := F_{11}F_{22} − F_{12}^{2}, \\ B := b^{2} − (β/α)^{2}. \end{matrix}

We suppose F_{1} ≠ 0, of course. As a consequence F^{n} has the irregular metric (det(g_{ij}) = 0), if and only if T = 0 [7]. In the following we are concerned only with F^{n} having regular metric.

Then the inverse matrix (g^{ij}) of (g_{ij}) may be put

(2.5)  g^{ij} = (α/F_{1})a^{ij} − s_{0}B^{i}B^{j} − s_{−1}(B^{i}y^{j} + B^{j}y^{i}) − s_{−2}y^{i}y^{j},

where B^{i} = a^{ir}b_{r}. The condition g^{ik}g_{ij} = δ^{k}_{j} for g^{ij} leads to

(2.6)  \begin{matrix}T_{s_{0}} = αD/F_{1}, T_{s_{−1}} = 2F F_{12}/α^{2}F_{1}, \\ T_{s_{−2}} = −(F_{12}/α^{2}F_{1})(BF_{2} + 2F β/α^{2}). \end{matrix}

From (2.3) and F_{ijk} = 2C_{ijk} we have

2C_{ijk} = C_{1} + C_{2} + C_{3} + C_{4} + C_{5},
C_{1} = F_{1}α_{ijk} + F_{11}(α_{ij}α_{k} + (ijk)) + F_{12}(α_{ij}b_{k} + (ijk)),
C_{2} = (F_{111}α_{k} + F_{112}b_{k})α_{i}α_{j}, C_{3} = (F_{112}α_{k} + F_{122}b_{k})b_{i}α_{j},
C_{4} = (F_{112}α_{k} + F_{122}b_{k})α_{i}b_{j}, C_{5} = (F_{112}α_{k} + F_{222}b_{k})b_{i}b_{j}.

From (2.1) and (2.2), and putting

(2.7)  p_{i} := b_{i} − (β/α^{2})Y_{i},

we have C_{1} = (F_{12}/α)(k_{ij}p_{k} + (ijk)). Next, from F_{ab1}α + F_{ab2}β = 0, for a, b = 1, 2, we have F_{122} = −(β/α)F_{222}, F_{112} = (β/α)^{2}F_{222} and F_{111} = −(β/α)^{3}F_{222}. Then

C_{2} = (β^{2}/α^{4})F_{222}Y_{i}Y_{j}p_{k}, C_{3} = −(β/α^{2})F_{222}b_{i}Y_{j}p_{k},
C_{2} + C_{3} = −(β/α^{2})F_{222}p_{i}Y_{j}p_{k},
C_{4} = −(β/α^{2})F_{222}Y_{i}b_{j}p_{k}, C_{5} = F_{222}b_{i}b_{j}p_{k},
C_{4} + C_{5} = F_{222}p_{i}b_{j}p_{k},
C_{2} + C_{3} + C_{4} + C_{5} = F_{222}p_{i}p_{j}p_{k}.

Consequently we have [12], [11]

(2.8)  C_{ijk} = (F_{12}/2α)(k_{ij}p_{k} + (ijk)) + (F_{222}/2)p_{i}p_{j}p_{k}.

Now as a conformally invariant scalar S in the section 1, we shall study two cases, that is, A^{2} = g^{ij}(LC_{i})(LC_{j}) in the case of a Berwald F^{n} in this section and β/α in the next section.

In the following of this section we study the condition (1.4) i.e. Kikuchi’s assumption for (α, β)–metric.

Let us find A^{2} = g^{ij}(LC_{i})(LC_{j}) for F^{n} with (α, β)–metric. From (2.5) and (2.8) we have

C_{i} = C_{ijk}g^{jk} = C_{ijk}(αa^{jk}/F_{1} − s_{0}B^{j}B^{k}).

Also we have

p_{k}a^{jk} = B^{j} − (β/α^{2})y^{j}, k_{ij}p_{k}a^{jk} = p_{i}, p_{j}p_{k}a^{jk} = B,
C_{ijk}a^{jk} = ((n + 1)F_{12}/2α + F_{222}β/2)p_{i},
k_{ij}B_{j} = p_{i}, p_{k}B^{k} = B, k_{jk}B^{j}B^{k} = B,
C_{ijk}B^{j}B^{k} = (3F_{12}B/2α + F_{222}B^{2}/2)p_{i}.

Thus we get

(2.9)  \begin{matrix}C_{i} = Ep_{i}, \\ E = (F_{12}/α)((n + 1)α/2F_{1} − s_{0}B) + (F_{222}B/2)(α/F_{1} − s_{0}B). \end{matrix}

Consequently we have

(2.10)  A^{2} = 4F^{2}E^{2}B/α^{2}T.

For the later use we are concerned with two examples where we put t = β/α. Ex. 1. Randers metric L = α + β,

\begin{matrix}T = (1 + t)^{3}, & βE = (n + 1)t/2(1 + t), \\ A^{2} = (n + 1)^{2}B/4(1 + t), & B = b^{2} − t^{2}. \end{matrix}

Ex.2 Kropina metric L = α^{2}/β,

T = 2b^{2}/t^{6},
βE = −n − 2 + t^{2}/b^{2},
A^{2} = (B/2b^{2})(t^{2}/b^{2} − n − 2)^{2}.

It is remarked that ˙∂_{i}t = p_{i}/α, ∂_{i}A^{2} = (∂A^{2}/∂t)p_{i}/α for the above two examples.

We now approach to our problem by another way from the homogeneity of F (α, β). If we put

F (α, β) = α^{2}f(t), f(t) = F (1, t), t = β/α,

then we have the following:

(F_{1}, F_{2}) = α(ϕ(t), f′(t)), ϕ(t) = 2f − tf′,
(F_{11}, F_{12}, F_{22}) = (ϕ − tϕ′, ϕ′, f′′),
(F_{111}, F_{112}, F_{122}, F_{222}) = (f′′′/α)(−t^{3}, t^{2}, −t, 1).
D = δ(t) := 2ff′′ − (f′)^{2}, B = B(b, t) := b^{2} − t^{2},
T = δ(t)B(b, t) + 2f(t)ϕ(t) = T (b, t),
T s_{0} = δ(t)/ϕ(t), E = Ψ(b, t)/α,
Ψ(b, t) = [(n + 1)fϕ′ + B{(n − 1)δϕ/2ϕ′ + ff′′′}]/T,
A^{2} = 4(fΨ)^{2}B/T := Π(b, t).

Consequently we have

˙∂_{i}A^{2} = Π_{t}p_{i}/α.

Putting W^{j}_{i} = ( ˙∂_{r}A^{2})B^{rj}_{i}, W_{0}^{j}_{i} = ( ˙∂_{r}t)B^{rj}_{i}, we obtain

W^{j}_{i} = Π_{t}W_{0}^{j}_{i}.

As an example of this process we show a case of Kropina metric L = α^{2}/β,

f(t) = 1/2t^{2}, ϕ(t) = 2/t^{2}, δ(t) = 2/t^{6}, T = 2b^{2}/t^{6},
Ψ = (t^{2} − (n + 2)b^{2})/b^{2}t, A^{2} = (B/2b^{2})(t^{2}/b^{2} − (n + 2))^{2}.

Thus we have Theorem 2. If a non-Riemannian Finsler space F^{n} with L(α, β) has the non-zero Π_{t}, then S = A^{2} satisfies the Kikuchi’s condition and Theorem 1 can be applicable to F^{n}.

3 Another assumption for (α, β)–metrics

As conformally changed L̄(ᾱ, β̄) = e^{c(x)}L(α, β) = L(e^{c(x)}α, e^{c(x)}β) by (1)p–homogeneity of L, β/α is conformally invariant [5]. Let us take S = β/α and put W_{ij} = g_{ir}W^{r}_{j} = g_{ir}( ˙∂_{s}S)B^{sr}_{j}. Then we have ˙∂_{s}S = p_{s}/α and

\begin{aligned}W_{ij} & = & g_{ir}(p_{s}/α)(y_{j}g^{rs} − δ^{s}_{j}y^{r} − δ^{r}_{j}y^{s} − L^{2}C^{rs}_{j}) = \\ & = & (p_{i}y_{j} − p_{j}y_{i} − 2F p^{r}C_{rij})/α. \end{aligned}

We put P^{r} = a^{ri}p_{i}, and have from (2.5), (2.7) and (2.6)

\begin{aligned}p^{r} & = & g^{ri}p_{i} = (α/F_{1} − s_{0}B)p^{r} − (s_{0}β/α^{2} + s_{−1})By^{r} \\ & = & (2F/α^{2}T)P^{r} − (B/α^{2}F T)(αβD + 2F F_{12})y^{r}. \end{aligned}

Since [7] shows αβD + 2F F_{12} = F_{1}F_{2}, we find

(3.1)  p^{r} = (2F P^{r} − BF_{2}y^{r})/α^{2}T.

Thus (2.8) together with P^{i}p_{i} = B and P^{i}k_{ij} = p_{j} leads to

(3.2)  p^{r}C_{rij} = (F/α^{3}T)(F_{12}Bk_{ij} + (2F_{12} + αF_{222}B)p_{i}p_{j}).

From (2.3′) we have

(3.3)  y_{i} = F_{2}p_{i} + (2F/α^{2})Y_{i}.

Consequently we obtain

(3.4)  \begin{matrix}W_{ij} = Qa_{ij} + Q_{0}p_{i}p_{j} + Q_{−1}(p_{i}Y_{j} − p_{j}Y_{i}) + Q_{−2}Y_{i}Y_{j}, \\ Q = −2F^{2}F_{12}B/α^{4}T, \\ Q_{0} = −(2F^{2}/α^{4}T)(2F_{12} + αBF_{222}), \\ Q_{−1} = 2F/α^{3}, \\ Q_{−2} = −Q/α^{2}. \end{matrix}

Q = 0, if and only if F_{12} = 0, that is, F is of the form c_{1}α^{2} + c_{2}β^{2} with constant c’s. Thus, suppose that F^{n} is not Riemannian, then Q ≠ 0.

Next, we put V^{jk} = V^{j}_{r}g^{rk}. Then

(3.5)  W_{ij}V^{jk} = δ_{i}^{k}.

Let us put

(3.6)  V^{jk} = a^{jk}/Q + R_{0}P^{j}P^{k} + R_{−1}(P^{j}y^{k} − P^{k}y^{j}) + R_{−2}y^{j}y^{k}.

Then (3.5) yields as coefficients of the following terms,

p_{i}P^{k} : (Q + Q_{0}B)R_{0} − α^{2}Q_{−1}R_{−1} = −Q_{0}/Q,
Y_{i}P^{k} : −Q_{−1}BR_{0} = Q_{−1}/Q,
p_{i}y^{k} : (Q + Q_{0}B)R_{−1} + α^{2}Q_{−1}R_{−2} = −Q_{−1}/Q,
Y_{i}y^{k} : −Q_{−1}BR_{−1} = 1/α^{2}.

Therefore we obtain

(3.7)  \begin{matrix}R_{0} = −1/BQ, R_{−1} = −1/α^{2}BQ_{−1}, \\ R_{−2} = −1/α^{2}Q + (Q + Q_{0}B)/α^{4}(Q_{−1})^{2}B. \end{matrix}

In an interesting paper [3] concerned with Finsler spaces equipped with a linear connection, M.Kashiguchi and Y.Ichijyo showed that if b_{i} of a Finsler space F^{n} with (α, β)–metric is parallel with respect to the Levi-Civita connection γ = (γ_{j}^{i}_{k}(x)) of the associated Riemannian space, then F_{j}^{i}_{k} of the Cartan connection CΓ = (F_{j}^{i}_{k}, G^{i}_{j}, C_{j}^{i}_{k}) coincide with γ_{j}^{i}_{k}(x) and hence F^{n} is a Berwald space. This is also shown directly from the equation which gives the difference B_{j}^{i}_{k} = G_{j}^{i}_{k} −γ_{j}^{i}_{k} [9].

L_{α}B_{j}^{k}_{i}y^{j}y_{k} = αL_{β}(b_{j;i} − B_{j}^{k}_{i}b_{k})y^{j}.

If b_{j;i} = 0, then the uniqueness of the theorem leads to B_{j}^{k}_{i} = 0 immediately. The converse is not true; b_{i;j} = 0 is not necessary for F^{n} to be a Berwald space. For instance, as has been shown in [9], a Randers space with L = α+β is a Berwald space, if and only if b_{i;j} = 0, while a Kropina space with L = α^{2}/β is a Berwald space, if and only if there exists a vector field f_{i}(x) satisfying b_{i;j} = (f^{r}b_{r})a_{ij} + b_{i}f_{j} − b_{j}f_{i}. Definition. Let a Finsler space F^{n} with L(α, β)–metric (α, β) be a Berwald space. If b_{i} is necessarily parallel in the associated Riemannian space, then F^{n} is called a parallel Berwald space and L(α, β) is of parallel type.

We give here some example of parallel Berwald spaces. Ex. 1 [13]

L = (α^{s} + … + c_{k}α^{s−k}β^{k} + … + β^{s})^{r},

where rs = 1 and const. c’s, is of parallel type. Ex. 2 [9], [10].

L = c_{1}α + c_{2}β + β^{2}/α, c_{2} ≠ 0,
L = c_{1}α + c_{2}β + α^{2}/β, c_{1} ≠ 0,

where const. c’s, are of parallel type.

We consider a Finsler space F^{n} with L(α, β) of parallel type and conformal to a Berwald space F̄^{n}. Then S = β/α is conformally invariant and S̄_{;i} = 0 in the Levi-Civita connection γ̄ of R̄^{n}. Therefore we have Theorem 3. Let F^{n} = (M^{n}, L(α, β)) be a Finsler space with (α, β)–metric of parallel type. F^{n} is a conformally Berwald space, if and only if the conformal Berwald connection with respect to β/α has the vanishing hv–curvature tensor and satisfies (1.12′).

4 Conformally Berwald Randers spaces and Kropina spaces

The last section is devoted to the conditions for Finsler spaces of Randers type and Kropina type to be conformally Berwald. We shall use the symbols

r_{ij} = (b_{i;j} + b_{j;i})/2, s_{ij} = (b_{i;j} − b_{j;i})/2, s_{j} = b^{i}s_{ij},

where the covariant differentiation (; ) is the one with respect to the associated Riemannian space with the metric α. By a conformal change L → L̄ = e^{c(x)}L various quantities are changed as follows:

ā_{ij} = e^{2c}a_{ij}, b̄_{i} = e^{c}b_{i}.

Putting c_{i} = ∂_{i}c and c^{i} = a^{ir}c_{r}, the Christoffel symbols γ_{j}^{i}_{k} constructed from a_{ij} are changed to

γ̄_{j}^{i}_{k} = γ_{j}^{i}_{k} + δ^{i}_{j}c_{k} + δ^{i}_{k}c_{j} − c^{i}a_{jk},

and hence we obtain

b̄_{i;j} = e^{c}(b_{i;j} − c_{i}b_{j} + b^{r}c_{r}a_{ij}).

First we are concerned with a Randers space with the metric L = α + β. It is a Berwald space, if and only if b_{i;j} = 0 [9]. Consequently the space is conformally Berwald, if and only if there exists a gradient c_{i}(x) satisfying

(4.1)  b_{i;j} − c_{i}b_{j} + b^{r}c_{r}a_{ij} = 0.

From (4.1) we get

b^{j}b_{i;j} = b^{2}c_{i} − b^{r}c_{r}b_{i}, a^{ij}b_{i;j} = −(n − 1)b^{r}c_{r}.

Consequently we have

(4.2)  c_{i} = (b^{r}b_{i;r} − a^{rs}b_{r;s}b_{i}/(n − 1))/b^{2}.

Since c_{i} is a gradient vector, we have

(4.3)  c_{i;j} − c_{j;i} = 0.

(4.1) can be written as

r_{ij} = (c_{i}b_{j} + c_{j}b_{i})/2 − b^{r}c_{r}a_{ij}, s_{ij} = (c_{i}b_{j} − c_{j}b_{i})/2.

These give respectively

a^{rs}r_{rs} = −(n − 1)b^{r}c_{r}, s_{j} = (b^{r}c_{r}b_{j} − b^{2}c_{j})/2.

Hence we have

(4.4)  r_{ij} = (r^{s}_{s}/(n − 1))(a_{ij} − b_{i}b_{j}/b^{2}) − (b_{i}s_{j} + b_{j}s_{i})/b^{2}

(4.5)  s_{ij} = (b_{i}s_{j} − b_{j}s_{i})/b^{2}.

Now (4.2) can be written as

c_{i} = (b^{r}r_{ir} − s_{i} − a^{rs}r_{rs}b_{i}/(n − 1))/b^{2},

and (4.4) gives b^{r}r_{ir} = −s_{i}. Therefore we have

(4.6)  c_{i} = −(2s_{i} + r^{s}_{s}b_{i}/(n − 1))/b^{2}.

Therefore we have Theorem 4. A Randers space is conformally Berwald, if and only if (4.4) and (4.5) hold and c_{i} given by (4.6) is gradient, that is, satisfies (4.3).

Let F^{n} = (M^{n}, L = α^{2}/β) be a Kropina space and F̄^{n} = (M^{n}, L̄) a conformally changed space with L̄ = e^{c(x)}L. The latter is a Berwald space [9], if and only if there exists f_{i} satisfying

b̄_{i;j} = (b̄^{r}f_{r})(ā_{ij} + b̄_{i}f_{j} − b̄_{j}f_{i}.

From b̄_{i;j} = e^{c}(b_{i;j} − c_{i}b_{j} + b^{r}c_{r}a_{ij}) and b̄^{i} = e^{−c}b^{i} the above is written as

(4.7)  b_{i;j} − c_{i}b_{j} + b^{r}c_{r}a_{ij} = b^{r}f_{r}a_{ij} + b_{i}f_{j} − b_{j}f_{i},

which is equivalent to

(4.8)  r_{ij} − (b_{i}c_{j} + b_{j}c_{i})/2 + b^{r}c_{r}a_{ij} = b^{r}f_{r}a_{ij}.

(4.9)  s_{ij} + (b_{i}c_{j} + b_{j}c_{i})/2 = b_{i}f_{j} − b_{j}f_{i}.

Multiplying b^{i} to (4.9) yields

(4.10)  s_{j} = b^{2}(f_{j} − c_{j}/2) − b_{j}(f_{i} − c_{i}/2)b^{i}.

Consequently, eliminating f_{i} from (4.9) we obtain

(4.11)  s_{ij} = (b_{i}s_{j} − b_{j}s_{i})/b^{2}.

Next we deal with (4.8). Put

(4.12)  u = b^{r}(c_{r} − f_{r}), b^{i}r_{ij} = br_{j} and b^{j}r_{j} = br.

(4.8), transvected with b^{i} yields

(4.13)  br_{j} − (b^{i}c_{i}b_{j} + b^{2}c_{j})/2 + ub_{j} = 0.

Multiplying b^{j} and from b^{2} ≠ 0 for the Kropina space, we obtain r + u = b^{i}c_{i}. Then (4.13) gives

(4.14)  c_{j} = (2br_{j} + (u − r)b_{j})/b^{2}.

As a consequence (4.8) may be written in the form

(4.15)  r_{ij} = (b_{i}r_{j} + b_{j}r_{i})/b + (u − r)b_{i}b_{j}/b^{2} − ua_{ij}.

(4.14) gives b^{j}c_{j} = u + r, and hence (4.12) yields b^{r}f_{r} = r. Consequently (4.10) yields

(4.16)  f_{i} = s_{i}/b^{2} + r_{i}/b.

Conversely, we consider a Kropina space F^{n} such that (4.15) and (4.11) are satisfied and c_{j} of (4.14) is gradient (c_{j} = ∂_{j}c(x)). We make the conformally changed F̄^{n} from F^{n} by the conformal change L → L̄ = e^{c(x)}L. Then (4.15), (4.11) and (4.14) lead to

\begin{matrix}b_{i;j} − c_{i}b_{j} + b^{r}c_{r}a_{ij} = r_{ij} + s_{ij} − c_{i}b_{j} + b^{r}c_{r}a_{ij} = \\ = ra_{ij} + b_{i}(r_{j}/b + s_{j}/b^{2}) − b_{j}(r_{i}/b + s_{i}/b^{2}). \end{matrix}

Thus, (4.16) immediately leads to (4.7). Theorem 5. A Kropina space is conformally Berwald, if and only if (4.15) and (4.11) hold and c_{j} of (4.14) is gradient.

References

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Shun-ichi Hojo  Makoto Matsumoto
Dept. of Applied Math.  15 Zenbu-cho, Shomogamo
Konan University  Sakyo-ku, 606-0815
Okamoto, Higashinada  Kyoto, Japan

Kobe, 658, Japan

Katsumi Okubo Faculty of Education, Shiga University 2-5-1 Hiratsu, Otsu, 520, Japan